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60% of the freshman ate pizza [URL='http://www.mathcelebrity.com/community/x-apple-data-detectors://3']at lunch today[/URL]. If 180 freshman ate pizza, how many freshman are enrolled at our school? 60% of x = 180 We write this as 0.6x = 180 Divide each side by 0.6 to isolate x. We get x = 300 freshman

(4x - 20)/8 = 9y for x Cross multiply: 4x - 20 = 8 * 9y 4x - 20 = 72y Add 20 to each side to isolate x: 4x - 20 + 20 = 72y + 20 Cancel the 20 on the left side, we get: 4x = 72y + 20 Divide each side by 4: 4x/4 = (72y + 20)/4 Cancel the 4 on the left side: x = [B](72y + 20)/4[/B]

-2 <= x +4 < 9 Subtract 4 from each piece: -2 - 4 <= x < 5 Simplify: [B]-6 <= x < 5 [/B] To find the interval notation, we set up our notation: [LIST] [*]The left side has a solid bracket, since we have an equal sign: [*]The right side has an open parentheses, since we have no equal sign [*][B][-6, 5)[/B] [/LIST]

-3x<= -9 or 5+x<6 Take each piece: -3x<= -9 Divide each side by -3: x>=3 Now take 5 + x < 6 5 + x < 6 Subtract 5 from each side: x < 1 Joining together the two inequalities, we have: x<1 or x>=3 Use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3C1orx%3E%3D3&pl=Show+Interval+Notation']interval notion calculator[/URL] to find the interval notation of this compound inequality

-5n - 5n - 5 = 5 Solve for [I]n[/I] in the equation -5n - 5n - 5 = 5 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (-5 - 5)n = -10n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] -10n - 5 = + 5 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -5 and 5. To do that, we add 5 to both sides -10n - 5 + 5 = 5 + 5 [SIZE=5][B]Step 4: Cancel 5 on the left side:[/B][/SIZE] -10n = 10 [SIZE=5][B]Step 5: Divide each side of the equation by -10[/B][/SIZE] -10n/-10 = 10/-10 n = [B]- 1 [URL='https://www.mathcelebrity.com/1unk.php?num=-5n-5n-5%3D5&pl=Solve']Source[/URL][/B]

-g + 3/4a = y for a Add g to each side: -g + g + 3/4a = y + g Cancel the g terms on the left side: 3/4a = y + g Cross multiply: 3a = 4(y + g) Divide each side by 3 to isolate a: 3a/3 = 4(y + g)/3 a = [B]4(y + g)/3[/B]

-g+F/A=h^3 for A Add g to each side: -g + g+F/A=h^3 + g Cancel the g's on the left side: F/A = h^3 + g Cross multiply: F = A(h^3 + g) Divide each side by (h^3 + g) F/(h^3 + g) = A(h^3 + g)/(h^3 + g) Cancel (h^3 + g) on the right side: A = [B]F/(h^3 + g)[/B]

-n = -50 Multiply each side by -1: -1*-n = -1 * -50 n = [B]50[/B]

-n = n Add n to each side: -n + n = n + n Cancel the n's on the left side: 0 = 2n Only number that solves this is [B]n = 0[/B]

-n/8 = 80 Cross multiply: -n = 80 * 8 -n = 640 Multiply each side by -1: -1 * -n = -1 * 640 n = [B]-640[/B]

1 - n = n - 1 Solve for [I]n[/I] in the equation 1 - n = n - 1 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables -n and n. To do that, we subtract n from both sides -n + 1 - n = n - 1 - n [SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE] -2n + 1 = -1 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 1 and -1. To do that, we subtract 1 from both sides -2n + 1 - 1 = -1 - 1 [SIZE=5][B]Step 4: Cancel 1 on the left side:[/B][/SIZE] -2n = -2 [SIZE=5][B]Step 5: Divide each side of the equation by -2[/B][/SIZE] -2n/-2 = -2/-2 n = [B]1 [URL='https://www.mathcelebrity.com/1unk.php?num=1-n%3Dn-1&pl=Solve']Source[/URL][/B]

1 integer is 7 times another. If the product of the 2 integers is 448, then find the integers. Let the first integer be x and the second integer be y. We have the following two equations: [LIST=1] [*]x = 7y [*]xy = 448 [/LIST] Substitute (1) into (2), we have: (7y)y = 448 7y^2 = 448 Divide each side by 7 y^2 = 64 y = -8, 8 We use 8, since 8*7 = 56, and 56*8 =448. So the answer is [B](x, y) = (8, 56)[/B]

1/2a-10b=c solve for a Multiply each side of the equation by 2: 2/2a - 2(10)b = 2c Simplify: a - 20b = 2c Add 20b to each side: a - 20b + 20b = 2c + 20b Cancel the 20b on the left side: [B]a = 2c + 20b [/B] You can also factor out a 2 on the left side for another version of this answer: [B]a = 2(c + 10b)[/B]

1/2n + 1&1/2n = -10 [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%261%2F2&frac2=3%2F8&pl=Simplify']1&1/2 = 3/2[/URL] so we have: n/2 + 3n/2 = -10 4n/2 = -10 2n = -10 Divide each side by 2: 2n/2 = -10/2 n = [B]-5[/B]

1/3ab^2=6 for a Multiply each side by 3: ab^2 = 18 Divide each side by b^2 a = 18/b^2

1/a + 1/b = 1/2 for a Subtract 1/b from each side to solve this literal equation: 1/a + 1/b - 1/b = 1/2 - 1/b Cancel the 1/b on the left side, we get: 1/a = 1/2 - 1/b Rewrite the right side, using 2b as a common denominator: 1/a = (b - 2)/2b Cross multiply: a(b - 2) = 2b Divide each side by (b - 2) a = [B]2b/(b - 2)[/B]

1/n + 3/5 = 1 Subtract 3/5 from each side where 1 = 5/5 1/n + 3/5 - 3/5 = 5/5 - 3/5 1/n = 2/5 Cross multiply: 5 * 1 = 2 * n 2n = 5 Divide each side by 2: n = [B]5/2 or 2.5[/B]

1/n^2 = 3/192 Cross multiply: 192 * 1 = 3 * n^2 3n^2 = 192 Divide each side by 3: 3n^2/3 = 192/3 Cancel the 3's on the left side: n^2 = 64 Take the square root of both sides: n = [B]8 or -8[/B]

100n = 100 Solve for [I]n[/I] in the equation 100n = 100 [SIZE=5][B]Step 1: Divide each side of the equation by 100[/B][/SIZE] 100n/100 = 100/100 n = [B]1[/B]

10ac-x/11=3 for a Add x/11 to each side of the equation to isolate a: 10ac - x/11 + x/11 = 3 + x/11 Cancelling the x/11 on the left side, we get: 10ac = 3 + x/11 Divide each side by 10c to isolate a: 10ac/10c = 3 + x/11 Cancelling the 10c on the left side, we get: a = [B]3/10c + x/110c[/B]

10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6 Solve for [I]n[/I] in the equation 10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (10 - 9 + 8 - 7 + 6)n = 8n [SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE] 10 - 9 + 8 - 7 + 6 = 8 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 8n = + 8 [SIZE=5][B]Step 4: Divide each side of the equation by 8[/B][/SIZE] 8n/8 = 8/8 n = [B]1[/B]

10n = 0.5 Solve for [I]n[/I] in the equation 10n = .5 [SIZE=5][B]Step 1: Divide each side of the equation by 10[/B][/SIZE] 10n/10 = .5/10 n = [B]0.05 [URL='https://www.mathcelebrity.com/1unk.php?num=10n%3D.5&pl=Solve']Source[/URL][/B]

Let G be green eyes and S be Stripes, and SG be Stripes and Green Eyes. [U]Set up an equation[/U] Total Cats = Green Eyes + Stripes - Green Eyes and Stripes 15 = G + 10 - 7 15 = G + 3 [U]Subtract 3 from each side:[/U] [B]G = 12[/B]

15y + 13/c = m for y Subtract 13/c from each side to isolate the y term: 15y + 13/c - 13/c = m - 13/c Cancel the 13/c on the left side and we get 15y = m - 13/c Now, divide each side by 15 to isolate y: 15y/15 = (m - 13/c)/15 Cancel the 15 on the left side, and we get: y = [B](m - 13/c)/15[/B]

175/n = 25 25n = 175 Divide each side by 25 [B]n = 7 classes[/B]

175 students separated into n classes is 25 [U]Divide 175 by n[/U] 175/n [U]The word [I]is[/I] means equal to, so set this expression equal to 25[/U] 175/n = 25 [U]Cross multiply[/U] 25n = 175 [U]Divide each side by 25[/U] [B]n = 7[/B]

Let x be the first even integer. That means the next consecutive even integer must be x + 2. Set up our equation: x + (x + 2) = 118 Group x terms 2x + 2 = 118 Subtract 2 from each side 2x = 116 Divide each side by 2 x = 58 Which means the next consecutive even integer is 58 + 2 = 60 So our two consecutive even integers are [B]58, 60[/B] Check our work: 58 + 60 = 118

2 consecutive odd integers such that their product is 15 more than 3 times their sum. Let the first integer be n. The next odd, consecutive integer is n + 2. We are given the product is 15 more than 3 times their sum: n(n + 2) = 3(n + n + 2) + 15 Simplify each side: n^2 + 2n = 6n + 6 + 15 n^2 + 2n = 6n + 21 Subtract 6n from each side: n^2 - 4n - 21 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-4n-21%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get: n = (-3, 7) If we use -3, then the next consecutive odd integer is -3 + 2 = -1. So we have [B](-3, -1)[/B] If we use 7, then the next consecutive odd integer is 7 + 2 = 9. So we have [B](7, 9)[/B]

2 fair sided die are rolled. How many ways can the dice be rolled to sum exactly 6? [URL='https://www.mathcelebrity.com/2dice.php?gl=1&pl=6&opdice=1&rolist=+2%2C3%2C9%2C10&dby=2%2C3%2C5&ndby=4%2C5&montect=+100']Using our 2 dice calculator[/URL], we get the following options: [LIST] [*]2,4 [*]3,3 [*]4,2 [*]1,5 [*]5,1 [/LIST] The probability of rolling a sum of 6 is [B]5/36[/B]

2 numbers that add up makes 5 but multiplied makes -36 Let the first number be x and the second number be y. We're given two equations: [LIST=1] [*]x + y = 5 [*]xy = -36 [/LIST] Rearrange equation (1) by subtracting y from each side: [LIST=1] [*]x = 5 - y [*]xy = -36 [/LIST] Substitute equation (1) for x into equation (2): (5 - y)y = -36 5y - y^2 = -36 Add 36 to each side: -y^2 + 5y + 36 = 0 We have a quadratic equation. To solve this, we [URL='https://www.mathcelebrity.com/quadratic.php?num=-y%5E2%2B5y%2B36%3D0&pl=Solve+Quadratic+Equation&hintnum=0']type it in our search engine and solve[/URL] to get: y = ([B]-4, 9[/B]) We check our work for each equation: [LIST=1] [*]-4 + 9 = -5 [*]-4(9) = -36 [/LIST] They both check out

2 numbers that are equal have a sum of 60 Let's choose 2 arbitrary variables for the 2 numbers x, y Were given 2 equations: [LIST=1] [*]x = y <-- Because we have the phrase [I]that are equal[/I] [*]x + y = 60 [/LIST] Because x = y in equation (1), we can substitute equation (1) into equation (2) for x: y + y = 60 Add like terms to get: 2y = 60 Divide each side by 2: 2y/2 = 60/2 Cancel the 2's and we get: y = [B]30 [/B] Since x = y, x = y = 30 x = [B]30[/B]

2, 4, 6, 8....1000. What term is the number 1000? Formula for nth term is 2n If 2n = 1000, then dividing each side by 2, we see that: 2n/2 = 1000/2 n = [B]500[/B]

20% of a number is x. What is 100% of the number? Assume x>0. Let the number be n. We're given: 0.2n = x <-- Since 20% = 0.2 To find n, we multiply each side of the equation by 5: 5(0.2)n = 5x n = [B]5x[/B]

26 students 15 like vanilla 16 like chocolate. 3 do not like either flavour. How many like both vanilla and chocolate Define our people: [LIST=1] [*]We have Vanilla Only [*]Chocolate Only [*]Both Vanilla and Chocolate [*]Neither Vanilla Nor Chocolate [*]Add up 1-4 to get our total [/LIST] Total = Vanilla Only + Chocolate Only - Vanilla and Chocolate + Neither 26 = 15 + 16 - V&C + 3 26 = 34 - V&C Subtract 34 from each side -V&C = -8 Multiply each side by -1 [B]V&C = 8[/B]

2d = (a - b)/(b - c) for d Divide each side by 2 to isolate d: 2d/2 = (a - b)/2(b - c) Cancel the 2's on the left side, we get: d = [B](a - b)/2(b - c)[/B]

2m - n/3 = 5m for n Subtract 2m from each side of the equation: 2m-n/3 - 2m = 5m - 2m -n/3 = 3m Multiply each side of the equation by -3 to isolate n: -3 * -n/3 = -3 * 3m n = [B]-9m[/B]

2n + 1 = n + 10 Solve for [I]n[/I] in the equation 2n + 1 = n + 10 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 2n and n. To do that, we subtract n from both sides 2n + 1 - n = n + 10 - n [SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE] n + 1 = 10 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 1 and 10. To do that, we subtract 1 from both sides n + 1 - 1 = 10 - 1 [SIZE=5][B]Step 4: Cancel 1 on the left side:[/B][/SIZE] n = [B]9[/B]

2n + 10 = 3n + 5 Solve for [I]n[/I] in the equation 2n + 10 = 3n + 5 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 2n and 3n. To do that, we subtract 3n from both sides 2n + 10 - 3n = 3n + 5 - 3n [SIZE=5][B]Step 2: Cancel 3n on the right side:[/B][/SIZE] -n + 10 = 5 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 10 and 5. To do that, we subtract 10 from both sides -n + 10 - 10 = 5 - 10 [SIZE=5][B]Step 4: Cancel 10 on the left side:[/B][/SIZE] -n = -5 [SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE] -1n/-1 = -5/-1 n = [B]5[/B]

2n + 8 - n = 20 Solve for [I]n[/I] in the equation 2n + 8 - n = 20 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (2 - 1)n = n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] n + 8 = + 20 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 8 and 20. To do that, we subtract 8 from both sides n + 8 - 8 = 20 - 8 [SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE] n = [B]12[/B]

2n + 8 = 24 Solve for [I]n[/I] in the equation 2n + 8 = 24 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 8 and 24. To do that, we subtract 8 from both sides 2n + 8 - 8 = 24 - 8 [SIZE=5][B]Step 2: Cancel 8 on the left side:[/B][/SIZE] 2n = 16 [SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE] 2n/2 = 16/2 n = [B]8[/B]

2n - 1&1/2n = 59 1&1/2n = 3/2n or 1.5n So we have: 2n - 1.5n = 59 Solve for [I]n[/I] in the equation 2n - 1.5n = 59 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (2 - 1.5)n = 0.5n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 0.5n = + 59 [SIZE=5][B]Step 3: Divide each side of the equation by 0.5[/B][/SIZE] 0.5n/0.5 = 59/0.5 n = [B]118[/B]

2n - 7 = 0 Solve for [I]n[/I] in the equation 2n - 7 = 0 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants -7 and 0. To do that, we add 7 to both sides 2n - 7 + 7 = 0 + 7 [SIZE=5][B]Step 2: Cancel 7 on the left side:[/B][/SIZE] 2n = 7 [SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE] 2n/2 = 7/2 n = [B]3.5[/B]

2n = 4n Solve for [I]n[/I] in the equation 2n = 4n [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 2n and 4n. To do that, we subtract 4n from both sides 2n - 4n = 4n - 4n [SIZE=5][B]Step 2: Cancel 4n on the right side:[/B][/SIZE] -2n = 0 [SIZE=5][B]Step 3: Divide each side of the equation by -2[/B][/SIZE] -2n/-2 = 0/-2 n = [B]0[/B]

2x - b/y = 4c for y Subtract 2x from each side: 2x - 2x - b/y = 4c - 2x Cancel the 2x's on the left side and we get: -b/y = 4c - 2x Cross multiply: -b = y(4c - 2x) Divide each side by (4c - 2x): -b/(4c - 2x) = y(4c - 2x)/(4c - 2x) Cancel the (4c - 2x) on the right side and we get: [B]y = -b/(4c - 2x) [/B]

2x/5 - 9y = 6 for x Add 9y to each side to isolate the x term: 2x/5 - 9y + 9y = 9y + 6 Cancel the 9y's on the left side: 2x/5 = 9y + 6 Multiply each side by 5: 2x * 5/5 = 5(9y + 6) Cancel the 5's on the left side and we get: 2x = 5(9y + 6) Divide each side by 2 to isolate x: 2x/2 = 5/2(9y + 6) Cancel the 2's on the left side and we get our final literal equation of: x = [B]5/2(9y + 6)[/B]

2x/5 - 9y = 6 for x Add 9y to each side of the equation: 2x/5 - 9y + 9y = 6 + 9y Cancel the 9y's on the left side to get: 2x/5 = 6 + 9y Multiply each side of the equation by 5: 5(2x/5) = 5(6 + 9y) Cancel the 5's on the left side to get 2x = 5(6 + 9y) Divide each side of the equation by 2: 2x/2 = 5/2(6 + 9y) Cancel the 2's on the left side to get: x = [B]5/2(6 + 9y)[/B]

2x^2+4x < 3x+6 Subtract 3x from both sides: 2x^2 + x < 6 Subtract 6 from both sides 2x^2 + x - 6 < 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=2x%5E2%2Bx-6&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get: x < 1.5 and x < -2 When we take the intersection of these, it's [B]x < 1.5[/B]

2^n = 4^(n - 3) 2^n = (2^2)^(n - 3) (2^2)^(n - 3) = 2^2(n - 3) 2^n= 2^2(n - 3) Comparing exponents, we see that: n = 2(n - 3) n = 2n - 6 Subtract n from each side: n - n = 2n - n - 6 0 = n - 6 n = [B]6[/B]

36 PAGES AND IT 3/8CM THICK, WHAT IS THE THICKNESS OF 1 PAGE? Set up a proportion in pages to cm: 36 pages /3/8cm = 1 page/x cm Cross multiply: 36x = 3/8 Divide each side by 36 x = 3/(8 * 36) x = 1/(8*12) x = [B]1/96 cm[/B]

3k^3 = rt for t This is a literal equation. Let's divide each side of the equation by r, to isolate t: 3k^3/r = rt/r Cancel the r's on the right side, and we get: t = [B]3k^3/r[/B]

3n/5 = 1 Cross multiply: 3n = 5 * 1 3n = 5 Divide each side by 3: 3n/3 = 5/3 n = 5/3

4 rectangular strips of wood, each 30 cm long and 3 cm wide, are arranged to form the outer section of a picture frame. Determine the area inside the wooden frame. Area inside forms a square, with a length of 30 - 3 - 3 = 24. We subtract 3 twice, because we account for 2 rectangular strips with a width of 3. Area of a square is side * side. So we have 24 * 24 = [B]576cm^2[/B]

4d/a - 9 = g for a Add 9 to each side: 4d/a - 9 + 9 = g + 9 Cancel the 9's on the left side and we get: 4d/a = g + 9 Cross multiply: 4d = a(g + 9) Divide each side of the equation by (g + 9) to isolate a: 4d/(g + 9) = a(g + 9)/(g + 9) Cancel the (g + 9) on the right side, and we get: a = [B]4d/(g + 9)[/B]

4n - 8 = n + 1 Solve for [I]n[/I] in the equation 4n - 8 = n + 1 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 4n and n. To do that, we subtract n from both sides 4n - 8 - n = n + 1 - n [SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE] 3n - 8 = 1 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -8 and 1. To do that, we add 8 to both sides 3n - 8 + 8 = 1 + 8 [SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE] 3n = 9 [SIZE=5][B]Step 5: Divide each side of the equation by 3[/B][/SIZE] 3n/3 = 9/3 n = [B]3[/B]

Subtract 5 from each side: -8|-2n| = -80 Divide each side by -8 |-2n| = 10 Since this is an absolute value equation, we need to setup two equations: -2n = 10 -2n = -10 Solving for the first one by dividing each side by -2, we get: n = -5 Solving for the second one by dividing each side by -2, we get: n = 5

5 books and 5 bags cost $175. What is the cost of 2 books and 2 bags Let the cost of each book be b and the cost of each bag be c. We're given 5b + 5c = 175 We can factor this as: 5(b + c) = 175 Divide each side of the equation by 5, we get: (b + c) = 35 The problem asks for 2b + 2c Factor out 2: 2(b + c) we know from above that (b + c) = 35, so we substitute: 2(35) [B]70[/B]

5/9v+w=z,for v Subtract w from each side: 5/9v = z - w Multiply each side by 9/5 [B]v = 9(z - w)/5 [MEDIA=youtube]zZo_HveA3AY[/MEDIA][/B]

508 people are there, the daily price is $1.25 for kids and $2.00 for adults. The receipts totaled $885.50. How many kids and how many adults were there? Assumptions: [LIST] [*]Let the number of adults be a [*]Let the number of kids be k [/LIST] Given with assumptions: [LIST=1] [*]a + k = 508 [*]2a + 1.25k = 885.50 (since cost = price * quantity) [/LIST] Rearrange equation (1) by subtracting c from each side to isolate a: [LIST=1] [*]a = 508 - k [*]2a + 1.25k = 885.50 [/LIST] Substitute equation (1) into equation (2): 2(508 - k) + 1.25k = 885.50 Multiply through: 1016 - 2k + 1.25k = 885.50 1016 - 0.75k = 885.50 To solve for k, we [URL='https://www.mathcelebrity.com/1unk.php?num=1016-0.75k%3D885.50&pl=Solve']type this equation into our search engine[/URL] and we get: k = [B]174[/B] Now, to solve for a, we substitute k = 174 into equation 1 above: a = 508 - 174 a = [B]334[/B]

5n - 5 = 85 Solve for [I]n[/I] in the equation 5n - 5 = 85 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants -5 and 85. To do that, we add 5 to both sides 5n - 5 + 5 = 85 + 5 [SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE] 5n = 90 [SIZE=5][B]Step 3: Divide each side of the equation by 5[/B][/SIZE] 5n/5 = 90/5 n = [B]18[/B]

5^(n - 1) = 15,625 We know 5^6 = 15,625, so we have: n - 1 = 6 Add 1 to each side: n - 1 + 1 = 6 + 1 Cancel the 1's on the left side: n = [B]7[/B]

6 sided die probability to roll a odd number or a number less than 6 First, we'll find the set of rolling an odd number. [URL='https://www.mathcelebrity.com/1dice.php?gl=1&opdice=1&pl=Odds&rolist=+2%2C3%2C4&dby=+2%2C3%2C5&montect=+100']From this dice calculator[/URL], we get: Odd = {1, 3, 5} Next, we'll find the set of rolling less than a 6. [URL='https://www.mathcelebrity.com/1dice.php?gl=4&pl=6&opdice=1&rolist=+2%2C3%2C4&dby=+2%2C3%2C5&montect=+100']From this dice calculator[/URL], we get: Less than a 6 = {1, 2, 3, 4, 5} The question asks for [B]or[/B]. Which means a Union: {1, 3, 5} U {1, 2, 3, 4, 5} = {1, 2, 3, 4, 5} This probability is [B]5/6[/B]

6 times the sum of a number and 5 is 16 A number represents an arbitrary variable, let's call it x x The sum of x and 5 x + 5 6 times the sum of x and 5 6(x + 5) Is means equal to, so set 6(x + 5) equal to 16 [B]6(x + 5) = 16 <-- This is our algebraic expression Solve for x[/B] Multiply through: 6x + 30 = 16 Subtract 30 from each side: 6x - 30 + 30 = 16 - 30 6x = -14 Divide each side by 6 6x/6 = -14/6 Simplify this fraction by dividing top and bottom by 2: x = [B]-7/3 [MEDIA=youtube]oEx5dsYK7DY[/MEDIA][/B]

Set up an equation where h is Helena's age. h + 24 = 63 [URL='http://www.mathcelebrity.com/1unk.php?num=h%2B24%3D63&pl=Solve']Subtract 24 from each side[/URL] h = 39

7n + 4 + n - 5 = 63 Solve for [I]n[/I] in the equation 7n + 4 + n - 5 = 63 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (7 + 1)n = 8n [SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE] 4 - 5 = -1 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 8n - 1 = + 63 [SIZE=5][B]Step 4: Group constants:[/B][/SIZE] We need to group our constants -1 and 63. To do that, we add 1 to both sides 8n - 1 + 1 = 63 + 1 [SIZE=5][B]Step 5: Cancel 1 on the left side:[/B][/SIZE] 8n = 64 [SIZE=5][B]Step 6: Divide each side of the equation by 8[/B][/SIZE] 8n/8 = 64/8 n = [B]8[/B]

9 rulers cost the same as 11 erasers. One eraser cost 0.09 cents. What is the cost of 1 ruler Let the cost of a ruler be r. We're given: 9r = 11(0.09) 9r = 0.99 Divide each side by 9 and we get: r = [B]0.11[/B]

963 animals on a farm, 159 sheep and 406 cows and pigs. How many are pigs? Set up equation to represent the total animals on the farm Total Animals = Cows + Pigs + Sheep Now plug in what is given 963 = 406 + Pigs + 159 Simplify: Pigs + 565 = 963 Subtract 565 from each side [B]Pigs = 398[/B]

993 cold drinks bottles are to be placed in crates. Each crate can hold 9 bottles. How many crates would be needed and how many bottles will remain? Let c equal the number of crates 9 bottles per crate * c = 993 9c = 993 Solve for [I]c[/I] in the equation 9c = 993 [SIZE=5][B]Step 1: Divide each side of the equation by 9[/B][/SIZE] 9c /9 = 993/9 c = 110.33333333333 Since we can't have fractional crates, we round up 1 to the next full crate c = [B]111[/B]

a +?b +?c =?180 for b We have a literal equation. Subtract (a + c) from each side of the equation to isolate b: a + b + c - (a + c) = 180 - (a + c) The (a + c) cancels on the left side, so we have: [B]b = 180 - (a + c)[/B] or, distributing the negative sign: [B]b = 180 - a - c[/B]

A 12 feet ladder leans against the side of a house. The bottom of the ladder is 9 feet from the side of the house. How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth. We have a right triangle, where 12 is the hypotenuse. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=9&hypinput=12&pl=Solve+Missing+Side']Using our right triangle calculator[/URL], we get: side = [B]7.9[/B]

a 12 sided die is rolled find the probability of rolling a number greater than 7 We assume this is a fair die, not loaded. This means each side 1-12 has an equal probability of 1/12 of being rolled. The problem asks, P(Roll > 7) Greater than 7 means our sample space is {8, 9, 10, 11, 12} If each of these 5 faces have an equal probability of being rolled, then we have: P(Roll > 7) = P(Roll = 8) + P(Roll = 9) + P(Roll = 10) + P(Roll = 11) + P(Roll = 12) P(Roll > 7) = 1/12 + 1/12 + 1/12 + 1/12 + 1/12 P(Roll > 7) =[B] 5/12[/B]

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,,8,9,10,11,12}. Find the probability of rolling a number less than 6. We want a {1, 2, 3, 4, 5} P(X < 6) =[B] 5/12[/B]

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,8,9,10,11,12}. Find the probability of rolling a number less than 6. We have 12 outcomes. Less than 6 means 1, 2, 3, 4, 5. Our probability P(x < 6) is: P(x < 6) = [B]5/12[/B]

A 13 ft. ladder is leaning against a building 12 ft. up from the ground. How far is the base of the ladder from the building? This is a classic 5-12-13 pythagorean triple, where the hypotenuse is 13, and the 2 sides are 5 and 12. The building and the ground form a right triangle. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=12&hypinput=13&pl=Solve+Missing+Side']You can see the proof here[/URL]...

A 13ft ladder leans against the side of a house. The bottom of the ladder is 10ft from the side of the house. How high is the top of the ladder from the ground? If necessary, round your answer to the nearest tenth. We have a right triangle. Hypotenuse = 13, one leg = 10. We use our [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=10&hypinput=13&pl=Solve+Missing+Side']Pythagorean theorem Calculator to solve for the other leg[/URL]: s = [B]8.3066[/B]

A 15 feet piece of string is cut into two pieces so that the longer piece is 3 feet longer than twice the shorter piece. If the shorter piece is x feet long, find the lengths of both pieces. If the shorter piece is x, the longer piece is 20 - x We also are given 15 - x = 2x + 3 Add x to each side: 3x + 3 = 15 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B3%3D15&pl=Solve']equation calculator[/URL], we get a shorter piece of: [B]x = 4[/B] The longer piece is: 15 - x 15 - 4 [B]11[/B]

A 20 feet piece of string is cut into two pieces so that the longer piece is 5 feet longer than twice the shorter piece. If the shorter piece is x feet long, find the lengths of both pieces. If the shorter piece is x, the longer piece is 20 - x We also are given 20 - x = 2x + 5 Add x to each side: 3x + 5 = 20 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B5%3D20&pl=Solve']equation calculator[/URL], we get a shorter piece of: [B]x = 5 [/B] The longer piece is: 20 - x 20 - 5 [B]15[/B]

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? [U]Set up the relationship of still water speed and downstream speed[/U] Speed down stream = Speed in still water + speed of the current Speed down stream = x+2 Therefore: Speed upstream =x - 2 Since distance = rate * time, we rearrange to get time = Distance/rate: 15/(x+ 2) + 15 /(x- 2) = 3 Multiply each side by 1/3 and we get: 5/(x + 2) + 5/(x - 2) = 1 Using a common denominator of (x + 2)(x - 2), we get: 5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2) (5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2) 10x = (x+2)(x-2) We multiply through on the right side to get: 10x = x^2 - 4 Subtract 10x from each side: x^2 - 10x - 4 = 0 This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get: Speed of the boat in still water =X=5 +- sq. Root of 29 kmph We only want the positive solution: x = 5 + sqrt(29) x = 10.38 [U]Calculate time for upstream journey:[/U] Time for upstream journey = 15/(10.38 - 2) Time for upstream journey = 15/(8.38) Time for upstream journey = [B]1.79[/B] [U]Calculate time for downstream journey:[/U] Time for downstream journey = 15/(10.38 + 2) Time for downstream journey = 15/(12.38) Time for downstream journey = [B]1.21[/B]

A 50-foot pole and a 70-foot pole are 30 feet apart. If you were to run a line between the tops of the two poles, what is the minimum length of cord you would need? The difference between the 70 foot and 50 foot pole is: 70 - 50 = 20 foot height difference. So we have a right triangle, with a height of 20, base of 30. We want to know the hypotenuse. Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=20&side2input=30&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator to solve for hypotenuse[/URL], we get: hypotenuse = [B]36.06 feet[/B]

A 6-sided die is rolled once. What is the probability of rolling a number less than 4? Using our [URL='https://www.mathcelebrity.com/1dice.php?gl=4&pl=4&opdice=1&rolist=+2%2C3%2C4&dby=+2%2C3%2C5&montect=+100']one dice calculator[/URL], we get: P(N < 4) = [B]1/2[/B]

A 6000 seat theater has tickets for sale at $24 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $188,800? Let x be the number of $24 tickets, and y be the number of $40 tickets. We have: [LIST=1] [*]24x + 40y = 188,800 [*]x + y = 6,000 [*]Rearrange (2) to solve for x: x = 6000 - y [*]Plug in (3) to (1): [/LIST] 24(6000 - y) + 40y = 188800 144,000 - 24y + 40y = 188,800 16y + 144,000 = 188,800 Subtract 144,000 from each side: 16y = 44,800 Divide each side by 16 y = 2,800 ($40 tickets) Plug this into (2) x + 2,800 = 6000 Subtract 2,800 from each side: x = 3,200 ($24 tickets)

A 74 inch rake is Leaning against a wall. The top of the rake hits the wall 70 inches above the ground. How far is the bottom of the rake from the base of the wall? We have a right triangle. Hypotenuse is the rake length fo 74 inches. One of the legs is 70. We [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=70&hypinput=74&pl=Solve+Missing+Side']use our right triangle calculator to solve for the other leg[/URL]: [B]24 inches[/B]

a 9-foot rope is cut into two pieces one piece is x feet express the length of the other piece in terms of x Piece 1 + Piece 2 = 9 Piece 1 = x x + Piece 2 = 9 Subtracting x from each side, we get: x - x + Piece 2 = 9 - x Cancel the x's on the left side, we get: Piece 2 = [B]9 - x [/B] Check our work: x + 9 - x ? 9 9 = 9

A 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the other. How long should the pieces be? The key phrase in this problem is [B]two pieces[/B]. Declare Variables: [LIST] [*]Let the short piece length be s [*]Let the long piece length be l [/LIST] We're given the following [LIST=1] [*]s = l - 10 [*]s + l = 98 (Because the two pieces add up to 98) [/LIST] Substitute equation (1) into equation (2) for s: l - 10+ l = 98 Group like terms: 2l - 10 = 98 Solve for [I]l[/I] in the equation 2l - 10 = 98 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants -10 and 98. To do that, we add 10 to both sides 2l - 10 + 10 = 98 + 10 [SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE] 2l = 108 [SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE] 2l/2 = 108/2 l = [B]54[/B] To solve for s, we substitute l = 54 into equation (1): s = 54 - 10 s = [B]44[/B] Check our work: The shorter piece is 10 inches shorter than the longer piece since 54 - 44 = 10 Second check: Do both pieces add up to 98 54 + 44 ? 98 98 = 98

a = v^2/r for r Start by cross multiplying to get r out of the denominator: ar = v^2 Divide each side of the equation by a to isolate r: ar/a = v^2/a Cancel the a's on the left side, and we get: r = [B]v^2/a[/B]

A barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks in the barn and a total of 313 legs. How many ducks are there in the barn? [LIST] [*]Let the number of ducks be d. Duck legs = 2 * d = 2d [*]Number of cows = 2d. Cow legs = 4 * 2d = 8d [*]1 dog Tripod has 3 legs [/LIST] Total legs: 2d + 8d + 3 = 313 Solve for [I]d[/I] in the equation 2d + 8d + 3 = 313 [SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE] (2 + 8)d = 10d [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 10d + 3 = + 313 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 3 and 313. To do that, we subtract 3 from both sides 10d + 3 - 3 = 313 - 3 [SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE] 10d = 310 [SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE] 10d/10 = 310/10 d = [B]31[/B] [URL='https://www.mathcelebrity.com/1unk.php?num=2d%2B8d%2B3%3D313&pl=Solve']Source[/URL]

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many more tickets would they need to sell to earn at least $441? Let the number of tickets above 42 be t. A few things to note on this question: [LIST] [*]The phrase [I]at least[/I] means greater than or equal to, so we have an inequality. [*]Earnings = Price * Quantity [/LIST] We're given: Earnings = 4.50 * 42 + 4.5t >= 441 Earnings = 189 + 4.5t >= 441 We want to solve this inequality for t: Solve for [I]t[/I] in the inequality 189 + 4.5t ? 441 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 189 and 441. To do that, we subtract 189 from both sides 4.5t + 189 - 189 ? 441 - 189 [SIZE=5][B]Step 2: Cancel 189 on the left side:[/B][/SIZE] 4.5t ? 252 [SIZE=5][B]Step 3: Divide each side of the inequality by 4.5[/B][/SIZE] 4.5t/4.5 ? 252.4.5 [B]t ? 56[/B]

A bird was sitting 12 meters from the base of an oak tree and flew 15 meters to reach the top of the tree. How tall is the tree? So we have a [U]right triangle[/U]. Hypotenuse is 15. Base is 12. We want the length of the leg. The formula for a right triangle relation of sides is a^2 + b^2 = c^2 where c is the hypotenuse and a, b are the sides Rearranging this equation to isolate a, we get a^2 = c^2 - b^2 Taking the square root of both sides, we get a = sqrt(c^2 - b^2) a = sqrt(15^2 - 12^2) a = sqrt(225 - 144) a = sqrt(81) a = [B]9 meters[/B]

a boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equation to describe this relationship We know the distance (d) equation in terms of rate (r) and time (t) as: d = rt We're given d = 336km and t = 12 hours, so we have: [B]336 km = 12t [/B] <-- this is our equation Divide each side by 12 to solve for t: 12t/12 = 336/12 t = [B]28 km / hour[/B]

A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books they make sell for $40 each. [B][U]Set up Cost Function C(b) where b is the number of books:[/U][/B] C(b) = Fixed Cost + Variable Cost x Number of Units C(b) = 180,000 + 25(b) [B]Set up Revenue Function R(b):[/B] R(b) = 40b Set them equal to each other 180,000 + 25b = 40b Subtract 25b from each side: 15b = 180,000 Divide each side by 15 [B]b = 12,000 for break even[/B]

A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each? Let the boy's age be b and his brother's age be c. We're given two equations: [LIST=1] [*]b = c + 10 [*]b + 4 = 2(c + 4) [/LIST] Substitute equation (1) into equation (2): (c + 10) + 4 = 2(c + 4) Simplify by multiplying the right side through and grouping like terms: c + 14 = 2c + 8 [URL='https://www.mathcelebrity.com/1unk.php?num=c%2B14%3D2c%2B8&pl=Solve']Type this equation into our search engine[/URL] and we get: c = [B]6[/B] Now plug c = 6 into equation (1): b = 6 + 10 b = [B]16[/B]

A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will the car be worth $7300 or less? (Use the calculator provided if necessary.) Write the smallest possible whole number answer. Set up the depreciation equation D(t) where t is the number of years in the life of the car: D(t) = 24,000 * (1 - 0.3)^t D(t) = 24000 * (0.7)^t The problem asks for D(t)<=7300 24000 * (0.7)^t = 7300 Divide each side by 24000 (0.7)^t = 7300/24000 (0.7)^t= 0.30416666666 Take the natural log of both sides: LN(0.7^t) = -1.190179482215518 Using the natural log identities, we have: t * LN(0.7) = -1.190179482215518 t * -0.35667494393873245= -1.190179482215518 Divide each side by -0.35667494393873245 t = 3.33687437943 [B]Rounding this up, we have t = 4[/B]

A cashier has 44 bills, all of which are $10 or $20 bills. The total value of the money is $730. How many of each type of bill does the cashier have? Let a be the amount of $10 bills and b be the amount of $20 bills. We're given two equations: [LIST=1] [*]a + b = 44 [*]10a + 20b = 730 [/LIST] We rearrange equation 1 in terms of a. We subtract b from each side and we get: [LIST=1] [*]a = 44 - b [*]10a + 20b = 730 [/LIST] Now we substitute equation (1) for a into equation (2): 10(44 - b) + 20b = 730 Multiply through to remove the parentheses: 440 - 10b + 20b = 730 Group like terms: 440 + 10b = 730 Now, to solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=440%2B10b%3D730&pl=Solve']type this equation into our search engine[/URL] and we get: b = [B]29 [/B] To get a, we take b = 29 and substitute it into equation (1) above: a = 44 - 29 a = [B]15 [/B] So we have [B]15 ten-dollar bills[/B] and [B]29 twenty-dollar bills[/B]

A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 per month. However, if you go over your 5 GB of data in a month, you have to pay an extra $10 for each GB. How many GB would be used to make both plans cost the same? Let g be the number of GB. The limited plan has a cost as follows: C = 10(g - 5) + 55 C = 10g - 50 + 55 C = 10g + 5 We want to set the limited plan equal to the unlimited plan and solve for g: 10g + 5 = 70 Solve for [I]g[/I] in the equation 10g + 5 = 70 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 5 and 70. To do that, we subtract 5 from both sides 10g + 5 - 5 = 70 - 5 [SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE] 10g = 65 [SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE] 10g/10 = 65/10 g = [B]6.5[/B] Check our work for g = 6.5: 10(6.5) + 5 65 + 5 70

A certain group of woman has a 0.69% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness? 0.69% = 0.0069. There exists a statistics theorem for an event A that states: P(A) + P(A') = 1 where A' is the event not happening In this case, A is the woman having red/green color blindness. So A' is the woman [U][B][I]not[/I][/B][/U][I] having red/green color blindness[/I] So we have: 0.0069 + P(A') = 1 Subtract 0.0069 from each side, we get: P(A') = 1 - 0.0069 P(A') = [B]0.9931[/B]

A certain Illness is spreading at a rate of 10% per hour. How long will it take to spread to 1,200 people if 3 people initially exposes? Round to the nearest hour. Let h be the number of hours. We have the equation: 3 * (1.1)^h = 1,200 Divide each side by 3: 1.1^h = 400 [URL='https://www.mathcelebrity.com/natlog.php?num=1.1%5Eh%3D400&pl=Calculate']Type this equation into our search engine [/URL]to solve for h: h = 62.86 To the nearest hour, we round up and get [B]h = 63[/B]

Let x be the number. We have: x^2 + x = 30 Subtract 30 from each side: x^2 + x - 30 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get potential solutions of: [B]x = 5 or x = -6[/B] Check 5: 5 + 5^2 = 5 + 25 = [B]30[/B] Check -6 -6 + -6^2 = -6 + 36 = [B]30[/B]

A company now has 4900 employees nationwide. It wishes to reduce the number of employees by 300 per year through retirements, until its total employment is 2560. How long will this take? Figure out how many reductions are needed 4900 - 2560 = 2340 We want 300 per year for retirements, so let x equal how many years we need to get 2340 reductions. 300x = 2340 Divide each side by 300 x = 7.8 years. If we want full years, we would do 8 full years

A computer screen has a diagonal dimension of 19 inches and a width of 15 inches. Approximately what is the height of the screen? We have a right triangle, with hypotenuse of 19, and width of 15. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=15&hypinput=19&pl=Solve+Missing+Side']Using our right triangle calculator, we get [/URL][B]height = 11.662[/B]

A construction company can remove 2/3 tons of dirt from a construction site each hour. How long will it take them to remove 30 tons of dirt from the site? Let h be the number of hours. We have the following equation: 2/3h = 30 Multiply each side by 3: 2(3)h/3 = 30 * 3 Cancel the 3 on the left side: 2h = 90 [URL='https://www.mathcelebrity.com/1unk.php?num=2h%3D90&pl=Solve']Type 2h = 90 into the search engine[/URL], we get [B]h = 45[/B].

A construction company needs to remove 24 tons of dirt from a construction site. They can remove 3/8 ton s of dirt each hour. How long will I it take to remove the dirt? Let h be the number of hours it takes, we have: 3/8h = 24 Multiply each side by 8/3 h = 24(8)/3 24/3 = 8, so we have: h = 8(8) h = [B]64 hours[/B]

A construction crew has just built a new road. They built 43.75 kilometers of road at a rate of 7 kilometers per week. How many weeks did it take them? Let w = weeks 7 kilometers per week * w = 43.75 To solve for w, we divide each side of the equation by 7: 7w/7 = 43.75/7 Cancel the 7's, we get: w = [B]6.25 [/B]

A cookie recipe uses 10 times as much flour as sugar. If the total amount of these ingredients is 8 1/4 cups, how much flour and how much sugar would it be? Let f be the number of cups of sugar. And let f be the number of cups of flour. We're given two equations: [LIST=1] [*]f = 10s [*]s + f = 8 & 1/4 [/LIST] Substitute (1) into (2): s + 10s = 8 & 1/4 11fs= 33/4 <-- 8 & 1/4 = 33/4 Cross multiply: 44s = 33 Divide each side by 44: s= 33/44 Divide top and bottom by 11 and we get s [B]= 3/4 or 0.75[/B] Now substitute this into (1): f = 10(33/44) [B]f = 330/44 or 7 & 22/44 or 7.5[/B]

A cube has an edge that is x cm long. What is the capacity of C(x)? Capacity is another word for volume, or the amount an object will hold. Given a side x, the capacity (volume) of a cube is: C(x) = [B]x^3[/B]

A cubical storage box has edges that are 2 feet 4 inches long. What is the volume of the storage box? Since 1 foot = 12 inches, we have: 2 feet 4 inches = 2(12) + 4 2 feet 4 inches = 24 + 4 2 feet 4 inches = 28 inches We type [URL='https://www.mathcelebrity.com/cube.php?num=28&pl=Side&type=side&show_All=1']cube side = 28[/URL] into our search engine to get: V = [B]21952 cubic inches[/B]

A dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getting a D on a math test and he gave another son an extra $35 for doing extra chores. Combined, the sons had $81. Figure out how much each son had. Let x, y, and z be the money each son received. To begin, x = y = z. But then, Dad took 20 from son X and gave it to son Y. So now, x = y - 20 Next, he gave Son Z an extra $35 for chores So z is now y + 35 since y and z used to be equal Combined, they all have 81. x + y + z = 181 But with the changes, it is: (y - 20) + y + (y + 35) Combine like terms: 3y - 20 + 35 = 81 3y + 15 = 81 Subtract 15 from each side: 3y = 66 Divide each side by 3 to isolate y y = 22 Since x = y - 20, x = 2 Since z = y + 35, we have z = 57 Checking our work, 2 + 22 + 57 = 81.

A dice has six sides. The dice is rolled once. What is the probability that a six will be the result. P(6) = [B]1/6[/B]

A fair six-sided die is rolled. Describe the sample space. [B]{1, 2, 3, 4, 5, 6}[/B]

A fake coin has heads on both sides, if the coin tossed once, what is the probability of getting a head? Since you always flip a head, we have: P(Head) = [B]1 or 100%[/B]

A father is K years old and his son is M years younger. The sum of their ages is 53. Father's Age = K Son's Age = K - M and we know K + (K - M) = 53 Combine like terms: 2K - M = 53 Add M to each side: 2K - M + M = 53 + M Cancel the M's on the left side, we get: 2K = 53+ M Divide each side by 2: 2K/2 = (53 + M)/2 Cancel the 2's on the left side: K = [B](53 + M)/2[/B]

a fever is generally considered to be a body temperature greater than 100 F. You friend has a temperature of 37 C. Does you friend have a fever? 37 Celsius equals 98.6 F. Since this is less than 100F, your friend does not have a fever.

A first number plus twice a second number is 6. Twice the first number plus the second totals 15. Find the numbers. Let the first number be x. Let the second number be y. We're given two equations: [LIST=1] [*]x + 2y = 6 [*]2x + y = 15 [/LIST] Multiply the first equation by -2: [LIST=1] [*]-2x - 4y = -12 [*]2x + y = 15 [/LIST] Now add them -2x + 2x - 4y + y = -12 + 15 -3y = 3 Divide each side by -3: y = 3/-3 y =[B] -1[/B] Plug this back into equation 1: x + 2(-1) = 6 x - 2 = 6 To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x-2%3D6&pl=Solve']type this equation into our search engine[/URL] and we get: x = [B]8[/B]

A first number plus twice a second number is 7. Twice the first number plus the second totals 23. Find the numbers Let the first number be a and the second number be b. We have: [LIST=1] [*]a + 2b = 7 [*]2a + b = 23 [/LIST] Rearrange (1) into (3) (3) a = 7 - 2b Substitute (3) into (2): 2(7 - 2b) + b = 23 Multiply through: 14 - 4b + b = 23 Combine like terms: 14 - 3b = 23 Subtract 14 from each side: -3b = 9 Divide each side by -3 [B]b = -3[/B] Substitute this into (3) a = 7 - 2b a = 7 - 2(-3) a = 7 + 6 [B]a = 13[/B] [B](a, b) = (13, -3)[/B]

A flower bed is to be 3 m longer than it is wide. The flower bed will an area of 108 m2 . What will its dimensions be? A flower bed has a rectangle shape, so the area is: A = lw We are given l = w + 3 Plugging in our numbers given to us, we have: 108 = w(w + 3) w^2 + 3w = 108 Subtract 108 from each side: w^2 + 3w - 108 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=w%5E2%2B3w-108%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get: w = (9, -12) Since length cannot be negative, w = 9. And l = 9 + 3 --> l = 12 So we have [B](l, w) = (12, 9)[/B] Checking our work, we have: A = (12)9 A = 108 <-- Match!

A food truck sells salads for $6.50 each and drinks for $2.00 each. The food trucks revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day? Let the number of drinks be d. Let the number of salads be s. We're given two equations: [LIST=1] [*]2d + 6.50s = 836.50 [*]d + s = 209 [/LIST] We can use substitution to solve this system of equations quickly. The question asks for the number of salads (s). Therefore, we want all expressions in terms of s. Rearrange Equation 2 by subtracting s from both sides: d + s - s = 209 - s Cancel the s's, we get: d = 209 - s So we have the following system of equations: [LIST=1] [*]2d + 6.50s = 836.50 [*]d = 209 - s [/LIST] Substitute equation (2) into equation (1) for d: 2(209 - s) + 6.50s = 836.50 Multiply through to remove the parentheses: 418 - 2s + 6.50s = 836.50 To solve this equation for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=418-2s%2B6.50s%3D836.50&pl=Solve']type it into our search engine and we get[/URL]: s = [B]93[/B]

A fraction has a value of 3/4. If 7 is added to the numerator, the resulting fraction is equal to the reciprocal of the original fraction. Find the original fraction. Let the fraction be x/y. We're given two equations: [LIST=1] [*]x/y = 3/4 [*](x + 7)/y = 4/3. [I](The reciprocal of 3/4 is found by 1/(3/4)[/I] [/LIST] Cross multiply equation 1 and equation 2: [LIST=1] [*]4x = 3y [*]3(x + 7) = 4y [/LIST] Simplifying, we get: [LIST=1] [*]4x = 3y [*]3x + 21 = 4y [/LIST] If we divide equation 1 by 4, we get: [LIST=1] [*]x = 3y/4 [*]3x + 21 = 4y [/LIST] Substitute equation (1) into equation (2) for x: 3(3y/4) + 21 = 4y 9y/4 + 21 = 4y Multiply the equation by 4 on both sides to eliminate the denominator: 9y + 84 = 16y To solve this equation for y, we type it in our math engine and we get: y = [B]12 [/B] We then substitute y = 12 into equation 1 above: x = 3 * 12/4 x = 36/4 x = [B]9 [/B] So our original fraction x/y = [B]9/12[/B]

a group of students and teachers are going on a field trip. one ninth of the group will fit on 1/3 of a school bus how many buses are needed to transport the entire group 1/9g = 1/3b We want to find g, so we multiply each side through by 9 g = 9/3b Simplify: g = 3b, so we need [B]3 buses[/B]

A helicopter rose vertically 300 m and then flew west 400 m how far was the helicopter from its starting point? The distance forms a right triangle. We want the distance of the hypotenuse. Using our [URL='http://www.mathcelebrity.com/pythag.php?side1input=300&side2input=400&hypinput=&pl=Solve+Missing+Side']right triangle calculator[/URL], we get a distance of [B]500[/B]. We also could use a shortcut on this problem. If you divide 300 and 400 by 100, you get 3 and 4. Since we want the hypotenuse, you get the famous 3-4-5 triangle ratio. So the answer is 5 * 100 = 500.

A hexagon has a total 126 dots and a equal number of dots on each side. how many dots on each side? Since it has an equal number of dots on each side, each side has: Number of dots on each side = 126 dots / 6 sides Number of dots on each side = [B]21 dots per side[/B]

A home is to be built on a rectangular plot of land with a perimeter of 800 feet. If the length is 20 feet less than 3 times the width, what are the dimensions of the rectangular plot? [U]Set up equations:[/U] (1) 2l + 2w = 800 (2) l = 3w - 20 [U]Substitute (2) into (1)[/U] 2(3w - 20) + 2w = 800 6w - 40 + 2w = 800 [U]Group the w terms[/U] 8w - 40 = 800 [U]Add 40 to each side[/U] 8w = 840 [U]Divide each side by 8[/U] [B]w = 105 [/B] [U]Substitute w = 105 into (2)[/U] l = 3(105) - 20 l = 315 - 20 [B]l = 295[/B]

A house costs 3.5 times as much as the lot. Together they sold for $135,000. Find the cost of each. Let the house cost be h, and the lot cost be l. We have the following equations: [LIST=1] [*]h = 3.5l [*]h + l = 135,000 [/LIST] Substitute (1) into (2) 3.5l + l = 135,000 Combine like terms: 4.5l = 135,000 Divide each side by 4.5 to isolate l [B]l = 30,000[/B] Substitute this back into equation (1) h = 3.5(30,000) [B]h = 105,000[/B]

A jet left Nairobi and flew east at an average speed of 231 mph. A passenger plane left four hours later and flew in the same direction but with an average speed of 385 mph. How long did the jet fly before the passenger plane caught up? Jet distance = 231t Passenger plane distance = 385(t - 4) 385(t - 4) = 231t 385t - 1540 = 231t Subtract 231t from each side 154t = 1540 [URL='https://www.mathcelebrity.com/1unk.php?num=154t%3D1540&pl=Solve']Type 154t = 1540[/URL] into the search engine, we get [B]t = 10. [/B] Check our work: Jet distance = 231(10) = 2,310 Passenger plane distance = 385(10 - 4) = 385 * 6 = 2,310

A jet plane traveling at 550 mph over takes a propeller plane traveling at 150 mph that had a 3 hours head start. How far from the starting point are the planes? Use the formula D = rt where [LIST] [*]D = distance [*]r = rate [*]t = time [/LIST] The plan traveling 150 mph for 3 hours: Time 1 = 150 Time 2 = 300 Time 3 = 450 Now at Time 3, the other plane starts Time 4 = 600 Time 5 = 750 Time 6 = 450 + 150t = 550t Subtract 150t 400t = 450 Divide each side by 400 t = 1.125 Plug this into either distance equation, and we get: 550(1.125) = [B]618.75 miles[/B]

A ladder 25 feet long is leaning against a wall. If the base of the ladder is 7 feet from the wall, how high up the wall does the ladder reach? We have a right triangle, where the ladder is the hypotenuse, and we want the measurement of one leg. Set up the pythagorean theorem with these given items using our P[URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=7&hypinput=25&pl=Solve+Missing+Side']ythagorean Theorem Calculator[/URL]. We get Side 1 = [B]24 feet.[/B]

A ladder is 25 ft long. The ladder needs to reach to a window that is 24 ft above the ground. How far away from the building should the bottom of the ladder be placed? We have a right triangle, where the ladder is the hypotenuse, and the window side is one side. Using our right triangle and the [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=24&hypinput=25&pl=Solve+Missing+Side']pythagorean theorem calculator[/URL], we get a length of [B]7 ft [/B]for the ladder bottom from the wall.

A ladder rests 2.5 m from the base of a house. If the ladder is 4 m long, how far up the side of the house will the ladder reach? We have a right triangle with the hypotenuse as 4, the one leg as 2.5 We want to solve for the other leg length. We use our [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=2.5&hypinput=4&pl=Solve+Missing+Side']right triangle solver[/URL] to get [B]3.122[/B]

A local Dunkin Donuts shop reported that its sales have increased exactly 16% per year for the last 2 years. This years sales were $80,642. What were Dunkin' Donuts' sales 2 years ago? Declare variable and convert numbers: [LIST] [*]16% = 0.16 [*]let the sales 2 years ago be s. [/LIST] s(1 + 0.16)(1 + 0.16) = 80,642 s(1.16)(1.16) = 80,642 1.3456s = 80642 Solve for [I]s[/I] in the equation 1.3456s = 80642 [SIZE=5][B]Step 1: Divide each side of the equation by 1.3456[/B][/SIZE] 1.3456s/1.3456 = 80642/1.3456 s = 59930.142687277 s = [B]59,930.14[/B]

A local shop sold 499 hamburgers and cheese burgers. There were 51 fewer cheese burgers sold. How many hamburgers were sold? Let h = number of hamburgers sold and c be the number of cheeseburgers sold. We have two equations: (1) c = h - 51 (2) c + h = 499 Substitute (1) into (2) h - 51 + h = 499 Combine like terms 2h - 51 = 499 Add 51 to both sides 2h = 550 Divide each side by 2 to isolate h [B]h = 275[/B]

A man is 5 years older than his wife, and the daughter age is half of the mother, and if you add their ages is equal 100 Let the man's age be m. Let the wife's age be w. Let the daughter's age be d. We're given: [LIST=1] [*]m = w + 5 [*]d = 0.5m [*]d + m + w = 100 [/LIST] Rearrange equation 1 in terms of w my subtracting 5 from each side: [LIST=1] [*]w = m - 5 [*]d = 0.5m [*]d + m + w = 100 [/LIST] Substitute equation (1) and equation (2) into equation (3) 0.5m + m + m - 5 = 100 We [URL='https://www.mathcelebrity.com/1unk.php?num=0.5m%2Bm%2Bm-5%3D100&pl=Solve']type this equation into our search engine[/URL] to solve for m and we get: m = [B]42 [/B] Now, substitute m = 42 into equation 2 to solve for d: d = 0.5(42) d = [B]21 [/B] Now substitute m = 42 into equation 1 to solve for w: w = 42 - 5 w = [B]37 [/B] To summarize our ages: [LIST] [*]Man (m) = 42 years old [*]Daughter (d) = 21 years old [*]Wife (w) = 37 years old [/LIST]

A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children. What was the cost of each ticket? Declare variables: [LIST] [*]Let a be the number of adult's tickets [*]Let c be the number of children's tickets [/LIST] Cost = Price * Quantity We're given two equations: [LIST=1] [*]a + c = 20 [*]15a + 10c = 225 [/LIST] Rearrange equation (1) in terms of a: [LIST=1] [*]a = 20 - c [*]15a + 10c = 225 [/LIST] Now that I have equation (1) in terms of a, we can substitute into equation (2) for a: 15(20 - c) + 10c = 225 Solve for [I]c[/I] in the equation 15(20 - c) + 10c = 225 We first need to simplify the expression removing parentheses Simplify 15(20 - c): Distribute the 15 to each term in (20-c) 15 * 20 = (15 * 20) = 300 15 * -c = (15 * -1)c = -15c Our Total expanded term is 300-15c Our updated term to work with is 300 - 15c + 10c = 225 We first need to simplify the expression removing parentheses Our updated term to work with is 300 - 15c + 10c = 225 [SIZE=5][B]Step 1: Group the c terms on the left hand side:[/B][/SIZE] (-15 + 10)c = -5c [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] -5c + 300 = + 225 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 300 and 225. To do that, we subtract 300 from both sides -5c + 300 - 300 = 225 - 300 [SIZE=5][B]Step 4: Cancel 300 on the left side:[/B][/SIZE] -5c = -75 [SIZE=5][B]Step 5: Divide each side of the equation by -5[/B][/SIZE] -5c/-5 = -75/-5 c = [B]15[/B] Recall from equation (1) that a = 20 - c. So we substitute c = 15 into this equation to solve for a: a = 20 - 15 a = [B]5[/B]

A man stands at point p, 45 metres from the base of a building that is 20 metres high. Find the angle of elevation of the top of the building from the man. Draw a right triangle ABC where Side A is from the bottom of the building to the man and Side B is the bottom of the building to the top of the building. Using right triangle calculations, we want Angle A which is the angle of elevation. [URL='http://www.mathcelebrity.com/righttriangle.php?angle_a=&a=20&angle_b=&b=45&c=&pl=Calculate+Right+Triangle']Angle of Elevation[/URL] which is [B]23.9625[/B]

A man's age (a) 10 years ago is 43. Years ago means we subtract [B]a - 10 = 43 [/B] If the problem asks you to solve for a, we type this equation into our math engine and we get: Solve for [I]a[/I] in the equation a - 10 = 43 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants -10 and 43. To do that, we add 10 to both sides a - 10 + 10 = 43 + 10 [SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE] a = [B]53[/B]

A math test is worth 100 points and has 38 problems. Each problem is worth either 5 points or 2 points. How many problems of each point value are on the test? Let's call the 5 point questions m for multiple choice. Let's call the 2 point questions t for true-false. We have two equations: [LIST=1] [*]m + t = 38 [*]5m + 2t = 100 [/LIST] Rearrange (1) to solve for m - subtract t from each side: 3. m = 38 - t Now, substitute (3) into (2) 5(38 - t) + 2t = 100 190 - 5t + 2t = 100 Combine like terms: 190 - 3t = 100 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=190-3t%3D100&pl=Solve']equation solver[/URL], we get [B]t = 30[/B]. Plugging t = 30 into (1), we get: 30 + t = 38 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=m%2B30%3D38&pl=Solve']equation solver[/URL] again, we get [B]m = 8[/B]. Check our work for (1) 8 + 30 = 38 <-- Check Check our work for (2) 5(8) + 2(30) ? 100 40 + 60 ? 100 100 = 100 <-- Check You could also use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=m+%2B+t+%3D+38&term2=5m+%2B+2t+%3D+100&pl=Cramers+Method']simultaneous equations calculator[/URL]

A medium orange has 70 calories. This is 10 calories less then 1/4 of the calories in a sugar krunchy. How many calories are in a sugar crunchy? Let s = calories in a sugar crunch. Let o = 70 be the calories in a medium orange. Set up the equation: o = 1/4s - 10 70 = 1/4s - 10 Add 10 to each side 1/4s = 80 Multiply each side by 4 [B]s = 320[/B]

A Middleweight UFC fighter weighs between 170 lbs and 185 lbs. Let w be the UFC fighter's weight: We have a compound inequality. Right side includes 185 lbs. because between means includes 185lbs. Left side includes 170 lbs. because between means includes 17lb0s [B]170 <= w <= 185[/B]

A motorboat travels 408 kilometers in 8 hours going upstream and 546 kilometers in 6 hours going downstream. What is the rate of the boat in still water and what is the rate of the current? [U]Assumptions:[/U] [LIST] [*]B = the speed of the boat in still water. [*]S = the speed of the stream [/LIST] Relative to the bank, the speeds are: [LIST] [*]Upstream is B - S. [*]Downstream is B + S. [/LIST] [U]Use the Distance equation: Rate * Time = Distance[/U] [LIST] [*]Upstream: (B-S)6 = 258 [*]Downstream: (B+S)6 = 330 [/LIST] Simplify first by dividing each equation by 6: [LIST] [*]B - S = 43 [*]B + S = 55 [/LIST] Solve this system of equations by elimination. Add the two equations together: (B + B) + (S - S) = 43 + 55 Cancelling the S's, we get: 2B = 98 Divide each side by 2: [B]B = 49 mi/hr[/B] Substitute this into either equation and solve for S. B + S = 55 49 + S = 55 To solve this, we [URL='https://www.mathcelebrity.com/1unk.php?num=49%2Bs%3D55&pl=Solve']type it in our search engine[/URL] and we get: S = [B]6 mi/hr[/B]

A movie theater has a seating capacity of 143. The theater charges $5.00 for children, $7.00 for students, and $12.00 of adults. There are half as many adults as there are children. If the total ticket sales was $ 1030, How many children, students, and adults attended? Let c be the number of children's tickets, s be the number of student's tickets, and a be the number of adult's tickets. We have 3 equations: [LIST=1] [*]a + c + s = 143 [*]a = 0.5c [*]12a + 5c + 7s =1030 [/LIST] Substitute (2) into (1) 0.5c + c + s = 143 1.5c + s = 143 Subtract 1.5c from each side 4. s = 143 - 1.5c Now, take (4) and (2), and plug it into (3) 12(0.5c) + 5c + 7(143 - 1.5c) = 1030 6c + 5c + 1001 - 10.5c = 1030 Combine like terms: 0.5c + 1001 = 1030 Use our [URL='http://www.mathcelebrity.com/1unk.php?num=0.5c%2B1001%3D1030&pl=Solve']equation calculator[/URL] to get [B]c = 58[/B]. Plug this back into (2) a = 0.5(58) [B]a = 29 [/B] Now take the a and c values, and plug it into (1) 29 + 58 + s = 143 s + 87 = 143 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=s%2B87%3D143&pl=Solve']equation calculator[/URL] again, we get [B]s = 56[/B]. To summarize, we have: [LIST] [*]29 adults [*]58 children [*]56 students [/LIST]

A national political party has a budget of $30,000,000 to spend on the inauguration of the new president. 16% of the costs will be paid to personnel, 12% of the costs will go toward food, and 10% will go to decorations. How much money will go for personnel, food, and decorations? [LIST] [*]Personnel Costs = 0.16 * 30,000,000 = $4,800,000 [*]Food Costs = 0.12 * 30,000,000 = $3,600,000 [*]Decoration Costs = 0.10 * 30,000,000 = $3,000,000 [/LIST]

A new company president is said to have caused the company "to do a 180." Before the new president, the company was losing money. What is the company most likely doing under the new president? A 180 is a completely different direction. Since 180 degrees means the other way, a half-circle, a switch in direction. This means if the company was losing money, after doing a "180", they're making money.

A non-profit organization is having a couples banquet for a fundraiser. The banquet hall will only hold 250 people. The President, Vice-President, two volunteers, and a guest speaker will be working the event. How many couples will be able to attend the banquet? We subtract the 5 people working the event to get: 250 - 5 = 245 A couple is 2 people, so we have 245/2 = 122.5 We round down to [B]122 couples[/B].

A number multiplied by 6 and divided by 5 give four more than a number? A number is represented by an arbitrary variable, let's call it x. Multiply by 6: 6x Divide by 5 6x/5 The word "gives" means equals, so we set this equal to 4 more than a number, which is x + 4. 6x/5 = x + 4 Now, multiply each side of the equation by 5, to eliminate the fraction on the left hand side: 6x(5)/5 = 5(x + 4) The 5's cancel on the left side, giving us: 6x = 5x + 20 Subtract 5x from each side [B]x = 20[/B] Check our work from our original equation: 6x/5 = x + 4 6(20)/5 ? 20 + 4 120/5 ?24 24 = 24 <-- Yes, we verified our answer

A parallelogram has a perimeter of 48 millimeters. Two of the sides are each 20 millimeters long. What is the length of each of the other two sides? 2 sides * 20 mm each is 40 mm subtract this from the perimeter of 48: 48 - 40 = 8 Since the remaining two sides equal each other, their length is: 8/2 = [B]4mm[/B]

A parallelogram has a perimeter of 54 centimeters. Two of the sides are each 17 centimeters long. What is the length of each of the other two sides? A parallelogram is a rectangle bent on it's side. So we have the perimeter formula P below: P = 2l + 2w We're given w = 17 and P = 54. So we plug this into the formula for perimeter: 2l + 2(17) = 54 2l + 34 = 54 Using our [URL='https://www.mathcelebrity.com/1unk.php?num=2l%2B34%3D54&pl=Solve']equation calculator[/URL], we get [B]l = 10[/B].

A parking meter contains 27.05 in quarters and dimes. All together there are 146 coins. How many of each coin are there? Let d = the number of dimes and q = the number of quarters. We have two equations: (1) d + q = 146 (2) 0.1d + 0.25q = 27.05 Rearrange (1) into (3) solving for d (3) d = 146 - q Substitute (3) into (2) 0.1(146 - q) + 0.25q = 27.05 14.6 - 0.1q + 0.25q = 27.05 Combine q's 0.15q + 14.6 = 27.05 Subtract 14.6 from each side 0.15q = 12.45 Divide each side by 0.15 [B]q = 83[/B] Plugging that into (3), we have: d = 146 - 83 [B]d = 63[/B]

A passenger train left station A at 6:00 P.M. Moving with the average speed 45 mph, it arrived at station B at 10:00 p.m. A transit train left from station A 1 hour later than the passenger train, but it arrived at the station B at the same time with the passenger train. What was the average speed of the transit train? [U]Passenger Train[/U] [LIST] [*]45 miles per hour and it got there in 4 hours. [/LIST] Using our formula D = rt where: [LIST] [*]D = Distance [*]r = rate [*]t = time [/LIST] [LIST] [*]D = rt [*]D = 45(4) [*]D = 180 miles from Station A to Station B [/LIST] Transit Train [LIST] [*]It has to go the same distance, 180 miles, so D = 180 [*]It made it there in 3 hours. This is r [*]We want to solve for t [/LIST] D = rt 180 = 3r Divide each side by 3 [B]r = 60 miles per hour[/B]

A penny has a diameter of 19 millimeters. What is the radius of the penny. D = 2r To solve for r, we divide each side by 2: r = D/2 Plugging in D = 19, we get: r = [B]19/2 or 9.5[/B]

A person invested 30,000 in stocks and bonds. Her investment in bonds is 2000 more than 1-third her investments in stocks. How much did she invest in stocks? How much did she invest in bonds? Let the stock investment be s, and the bond investment be b. We're given: [LIST=1] [*]b + s = 30000 [*]b = 1/3s + 2000 [/LIST] Plug in (2) to (1): 1/3s + 2000 + s = 30000 Group like terms: (1/3 + 1)s + 2000 = 30000 Since 1 = 3/3, we have: 4/3s + 2000 = 30000 Subtract 2000 from each side: 4/3s + 2000 - 2000 = 30000 - 2000 Cancel the 2000's on the left side, we get: 4/3s = 28000 [URL='https://www.mathcelebrity.com/1unk.php?num=4%2F3s%3D28000&pl=Solve']Typing this equation into our calculator[/URL], we get: s = [B]21,000[/B]

A pet supply chain called pet city has 15 hamsters and 12 gerbils for sale at its seaside location. At its livingston location there are 19 hamsters and 10 gerbils. Which location has a lower ratio of hamsters to gerbils? Seaside ratio 15/12 = 1.25 Livingston ratio 19/10 = 1.9 Since 1.25 < 1.9, Seaside has the lower ratio of hamsters to gerbils

A pile of coins, consisting of quarters and half dollars, is worth 11.75. If there are 2 more quarters than half dollars, how many of each are there? Let h be the number of half-dollars and q be the number of quarters. Set up two equations: (1) q = h + 2 (2) 0.25q + 0.5h = 11.75 [U]Substitute (1) into (2)[/U] 0.25(h + 2) + 0.5h = 11.75 0.25h + 0.5 + 0.5h = 11.75 [U]Group h terms[/U] 0.75h + 0.5 = 11.75 [U]Subtract 0.5 from each side[/U] 0.75h = 11.25 [U]Divide each side by h[/U] [B]h = 15[/B] [U]Substitute h = 15 into (1)[/U] q = 15 + 2 [B]q = 17[/B]

A population grows at 6% per year. How many years does it take to triple in size? With a starting population of P, and triple in size means 3 times the original, we want to know t for: P(1.06)^t = 3P Divide each side by P, and we have: 1.06^t = 3 Typing this equation into our search engine to solve for t, we get: t = [B]18.85 years[/B] Note: if you need an integer answer, we round up to 19 years

A private jet flies the same distance in 4 hours that a commercial jet flies in 2 hours. If the speed of the commercial jet was 154 mph less than 3 times the speed of the private jet, find the speed of each jet. Let p = private jet speed and c = commercial jet speed. We have two equations: (1) c = 3p - 154 (2) 4p =2c Plug (1) into (2): 4p = 2(3p - 154) 4p = 6p - 308 Subtract 4p from each side: 2p - 308 = 0 Add 308 to each side: 2p = 308 Divide each side by 2: [B]p = 154[/B] Substitute this into (1) c = 3(154) - 154 c = 462 - 154 [B]c = 308[/B]

A professor assumed there was a correlation between the amount of hours people were expose to sunlight and their blood vitamin D level. The null hypothesis was that the population correlation was__ a. Positive 1.0 b. Negative 1.0 c. Zero d. Positive 0.50 [B]c. Zero[/B] Reason: Since the professor wanted to assume a correlation (either positive = 1.0 or negative = -1.0), then we take the other side of that assumption for our null hypothesis and say that there is no correlation (Zero)

A quarter of the learners in a class have blond hair and two thirds have brown hair. The rest of the learners in the class have black hair. How many learners in the class if 9 of them have blonde hair? Total learners = Blond + Brown + Black Total Learners = 1/4 + 2/3 + Black Total Learners will be 1, the sum of all fractions 1/4 + 2/3 + Black = 1 Using common denominators of 12, we have: 3/12 + 8/12 + Black = 12/12 11/12 + Black = 12/12 Subtract 11/12 from each side: Black = 1/12 Let t be the total number of people in class. We are given for blondes: 1/4t = 9 Multiply each side by 4 [B]t = 36[/B] Brown Hair 2/3(36) = 24 Black Hair 1/12(36) = 3

A realtor makes an annual salary of $25000 plus a 3% commission on sales. If a realtor's salary is $67000, what was the amount of her sales? Total post-salary pay = $67,000 - $25,000 = $42,000 Let Sales be s. So 0.03s = $42,000 Divide each side by 0.03 s = $1,400,000

a recipe of 20 bread rolls requires 5 tablespoons of butter. How many tablespoons of butter are needed for 30 bread rolls? Set up a proportion of bread rolls per tablespoons of butter where t is the amount of tablespoons of butter needed for 30 bread rolls: 20/5 = 30/t Cross multiply our proportion: Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2 20t = 30 * 5 20t = 150 Divide each side of the equation by 20: 20t/20 = 150/20 Cancel the 20's on the left side and we get: t = [B]7.5[/B]

a rectangle has an area of 238 cm 2 and a perimeter of 62 cm. What are its dimensions? We know the rectangle has the following formulas: Area = lw Perimeter = 2l + 2w Given an area of 238 and a perimeter of 62, we have: [LIST=1] [*]lw = 238 [*]2(l + w) = 62 [/LIST] Divide each side of (1) by w: l = 238/w Substitute this into (2): 2(238/w + w) = 62 Divide each side by 2: 238/w + w = 31 Multiply each side by w: 238w/w + w^2 = 31w Simplify: 238 + w^2 = 31w Subtract 31w from each side: w^2 - 31w + 238 = 0 We have a quadratic. So we run this through our [URL='https://www.mathcelebrity.com/quadratic.php?num=w%5E2-31w%2B238%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL] and we get: w = (14, 17) We take the lower amount as our width and the higher amount as our length: [B]w = 14 l = 17 [/B] Check our work for Area: 14(17) = 238 <-- Check Check our work for Perimeter: 2(17 + 14) ? 62 2(31) ? 62 62 = 62 <-- Check

A rectangular football pitch has its length equal to twice its width and a perimeter of 360m. Find its length and width. The area of a rectangle (A) is: A = lw --> where l is the length and w is the width We're given l = 2w, so we substitute this into the Area equation: A = (2w)w A = 2w^2 We're given the area of the pitch is 360, so we set: 2w^2 = 360 We [URL='https://www.mathcelebrity.com/1unk.php?num=2w%5E2%3D360&pl=Solve']type this equation into our search engine[/URL], follow the links, and get: w = [B]6*sqrt(5) [/B] Now we take this, and substitute it into this equation: 6*sqrt(5)l = 360 Dividing each side by 6*sqrt(5), we get: l = [B]60/sqrt(5)[/B]

A rectangular garden is 5 ft longer than it is wide. Its area is 546 ft2. What are its dimensions? [LIST=1] [*]Area of a rectangle is lw. lw = 546ft^2 [*]We know that l = w + 5. [/LIST] Substitute (2) into (1) (w + 5)w = 546 w^2 + 5w = 546 Subtract 546 from each side w^2 + 5w - 546 = 0 Using the positive root in our [URL='http://www.mathcelebrity.com/quadratic.php?num=w%5E2%2B5w-546%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get [B]w = 21[/B]. This means l = 21 + 5. [B]l = 26[/B]

A retired couple invested $8000 in bonds. At the end of one year, they received an interest payment of $584. What was the simple interest rate of the bonds? For simple interest, we have: Balance * interest rate = Interest payment 8000i = 584 Divide each side of the equation by 8000 to isolate i: 8000i/8000 = 584/8000 Cancelling the 8000's on the left side, we get: i = 0.073 Most times, interest rates are expressed as a percentage. Percentage interest = Decimal interest * 100% Percentage interest = 0.073 * 100% Multiplying by 100 is the same as moving the decimal point two places right: Percentage interest = [B]7.3%[/B]

A Salesperson receives a weekly salary of $100 plus a 5.5% commission on sales. Her salary last week was $1090. What were her sales that week? $1,090 - 100 = $990. This is her commission. Let s = Sales. So 0.055s = $990 Divide each side by 0.055. s = $18,000

A skier is trying to decide whether or not to buy a season ski pass. A daily pass costs $75. A season ski pass costs $350. The skier would have to rent skis with either pass for $20 per day. How many days would the skier have to go skiing in order to make the season pass less expensive than the daily passes? Let d be the number of days: Daily Plan cost: 75d + 20d = 95d Season Pass: 350 + 20d We want to find d such that 350 + 20d < 95d Subtract 20d from each side 75d > 350 Divide each side by 75 d > 4.66667 [B]d = 5[/B]

a son is 1/4 of his fathers age. the difference in their ages is 30. what is the fathers age. Declare variables: [LIST] [*]Let f be the father's age [*]Let s be the son's age [/LIST] We're given two equations: [LIST=1] [*]s = f/4 [*]f - s = 30. [I]The reason why we subtract s from f is the father is older[/I] [/LIST] Using substitution, we substitute equaiton (1) into equation (2) for s: f - f/4 = 30 To remove the denominator/fraction, we multiply both sides of the equation by 4: 4f - 4f/4 = 30 *4 4f - f = 120 3f = 120 To solve for f, we divide each side of the equation by 3: 3f/3 = 120/3 Cancel the 3's on the left side and we get: f = [B]40[/B]

A spherical water tank holds 11,500ft^3 of water. What is the diameter? The tank holding amount is volume. And the volume of a sphere is: V = (4pir^3)/3 We know that radius is 1/2 of diameter: r =d/2 So we rewrite our volume function: V = 4/3(pi(d/2)^3) We're given V = 11,500 so we have: 4/3(pi(d/2)^3) = 11500 Multiply each side by 3/4 4/3(3/4)(pi(d/2)^3) = 11,500*3/4 Simplify: pi(d/2)^3 = 8625 Since pi = 3.1415926359, we divide each side by pi: (d/2)^3 = 8625/3.1415926359 (d/2)^3 = 2745.42 Take the cube root of each side: d/2 = 14.0224 Multiply through by 2: [B]d = 28.005[/B]

A square has a perimeter of 24 inches. What is the area of the square? Perimeter of a square = 4s where s = the length of a side. Therefore, we have: 4s = P 4s = 24 Using our equation solver, [URL='https://www.mathcelebrity.com/1unk.php?num=4s%3D24&pl=Solve']we type in 4s = 24[/URL] and get: s = 6 The problems asks for area of a square. It's given by A = s^2 Plugging in s = 6, we get: A = 6^2 A = 6 * 6 A = [B]36 [/B] Now if you want a shortcut in the future, type in the shape and measurement you know. Such as: [I][URL='https://www.mathcelebrity.com/square.php?num=24&pl=Perimeter&type=perimeter&show_All=1']square perimeter = 24[/URL][/I] From the link, you'll learn every other measurement about the square.

A square of an integer is the integer. Find the integer. Let the integer be n. The square means we raise n to the power of 2, so we have: n^2 = n Subtract n from each side: n^2 - n = n - n n^2 - n = 0 Factoring this, we get: n(n - 1) = 0 So n is either [B]0 or 1[/B].

a stone mason builds 7 houses in 3 days. How many days does it take to build 11 houses? The build rate of houses per days is proportional. Set up a proportion of [I]houses to days[/I] where d is the number of days it takes to build 11 houses: 7/3 = 11/d Cross multiply: Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2 7d = 11 * 3 7d = 33 Divide each side of the equation by 7: 7d/7 = 33/7 d = [B]4.7142857142857[/B]

A store owner bought 240 cartons of eggs. The owner sold 5/8 of the eggs and set aside 5 cartons. How many cartons of eggs did the owner have left to sale? If the owner sold 5/8 of the eggs, they have 1 - 5/8 left. 1 = 8/8, so we have 8/8 - 5/8 = 3/8 left 3/8 (240 cartons) = 90 cartons remaining The owner set aside 5 cartons. We're left with 90 cartons - 5 cartons = [B]85 cartons[/B]

A suitcase contains nickels, dimes and quarters. There are 2&1/2 times as many dimes as nickels and 5 times the number of quarters as the number of nickels. If the coins have a value of $24.80, how many nickels are there in the suitcase? Setup number of coins: [LIST] [*]Number of nickels = n [*]Number of dimes = 2.5n [*]Number of quarters = 5n [/LIST] Setup value of coins: [LIST] [*]Value of nickels = 0.05n [*]Value of dimes = 2.5 * 0.1n = 0.25n [*]Value of quarters = 5 * 0.25n = 1.25n [/LIST] Add them up: 0.05n + 0.25n + 1.25n = 24.80 Solve for [I]n[/I] in the equation 0.05n + 0.25n + 1.25n = 24.80 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (0.05 + 0.25 + 1.25)n = 1.55n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 1.55n = + 24.8 [SIZE=5][B]Step 3: Divide each side of the equation by 1.55[/B][/SIZE] 1.55n/1.55 = 24.80/1.55 n = [B]16[/B] [B] [URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.25n%2B1.25n%3D24.80&pl=Solve']Source[/URL][/B]

A sum of money doubles in 20 years on simple interest. It will get triple at the same rate in: a. 40 years b. 50 years c. 30 years d. 60 years e. 80 years Simple interest formula if we start with 1 dollar and double to 2 dollars: 1(1 + i(20)) = 2 1 + 20i = 2 Subtract 1 from each side: 20i = 1 Divide each side by 20 i = 0.05 Now setup the same simple interest equation, but instead of 2, we use 3: 1(1 + 0.05(t)) = 3 1 + 0.05t = 3 Subtract 1 from each side: 0.05t = 2 Divide each side by 0.05 [B]t = 40 years[/B]

A super deadly strain of bacteria is causing the zombie population to double every day. Currently, there are 25 zombies. After how many days will there be over 25,000 zombies? We set up our exponential function where n is the number of days after today: Z(n) = 25 * 2^n We want to know n where Z(n) = 25,000. 25 * 2^n = 25,000 Divide each side of the equation by 25, to isolate 2^n: 25 * 2^n / 25 = 25,000 / 25 The 25's cancel on the left side, so we have: 2^n = 1,000 Take the natural log of each side to isolate n: Ln(2^n) = Ln(1000) There exists a logarithmic identity which states: Ln(a^n) = n * Ln(a). In this case, a = 2, so we have: n * Ln(2) = Ln(1,000) 0.69315n = 6.9077 [URL='https://www.mathcelebrity.com/1unk.php?num=0.69315n%3D6.9077&pl=Solve']Type this equation into our search engine[/URL], we get: [B]n = 9.9657 days ~ 10 days[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel? [LIST] [*]Each ride will cost 1.50 + 0.8x where x is the number of miles per trip. [*]This expression must be less than 12. [/LIST] [U]Setup the inequality:[/U] 1.5 + 0.8x < 12 [U]Subtracting 1.5 from each side of the inequality[/U] 0.8x < 10.5 [U]Simplifying even more by dividing each side of the inequality by 0.8, we have:[/U] [B]x < 13.125[/B]

A taxi charges a flat rate of 1.75, plus an additional 0.65 per mile. If Erica has at most 10 to spend on the cab ride, how far could she travel? Setup an equation where x is the number of miles traveled: 0.65x + 1.75 = 10 Subtract 1.75 from each side: 0.65x = 8.25 Divide each side by 0.65 [B]x = 12.69 miles[/B] If we do full miles, we round down to 12. [MEDIA=youtube]mFqUe2mjX-w[/MEDIA]

a textbook store sold a combined total of 296 sociology and history text books in a week. the number of history textbooks sold was 42 less than the number of sociology textbooks sold. how many text books of each type were sold? Let h = history book and s = sociology books. We have 2 equations: (1) h = s - 42 (2) h + s = 296 Substitute (1) to (2) s - 42 + s = 296 Combine variables 2s - 42 = 296 Add 42 to each side 2s = 338 Divide each side by 2 s = 169 So h = 169 - 42 = 127

A tortoise is walking in the desert. It walks at a speed of 5 meters per minute for 12.5 meters. For how many minutes does it walk? Distance formula (d) for a rate (r) and time (t) is: d = rt We're given d = 12.5 and r = 5 12.5 = 5t 5t = 12.5 Solve for t. Divide each side of the equation by 5: 5t/5 = 12.5/5 Cancel the 5's on left side and we get: t = [B]2.5[/B]

A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was Marcos car towed if he received a bill for $71 Set up a cost equation C(m) where m is the number of miles: C(m) = Cost per mile * m + Service Fee Plugging in the service fee of 50 and cost per mile of 1.75, we get: C(m) = 1.75m + 50 The question asks for what m is C(m) = 71. So we set C(m) = 71 and solve for m: 1.75m + 50 = 71 Solve for [I]m[/I] in the equation 1.75m + 50 = 71 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 50 and 71. To do that, we subtract 50 from both sides 1.75m + 50 - 50 = 71 - 50 [SIZE=5][B]Step 2: Cancel 50 on the left side:[/B][/SIZE] 1.75m = 21 [SIZE=5][B]Step 3: Divide each side of the equation by 1.75[/B][/SIZE] 1.75m/1.75 = 21/1.75 m = [B]12[/B]

A trapezoid has one base that is 120% of the length of the other base. The two sides are each 1/2 the length of the smaller base. If the perimeter of the trapezoid is 54.4 inches, what is the length of the smaller base of the trapezoid? Setup measurements: [LIST] [*]Small base = n [*]Large base = 1.2n [*]sides = n/2 [*]Perimeter = n + 1.2n + 0.5n + 0.5n = 54.4 [/LIST] Solve for [I]n[/I] in the equation n + 1.2n + 0.5n + 0.5n = 54.4 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 + 1.2 + 0.5 + 0.5)n = 3.2n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 3.2n = + 54.4 [SIZE=5][B]Step 3: Divide each side of the equation by 3.2[/B][/SIZE] 3.2n/3.2 = 54.4/3.2 n = [B]17[/B] [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B1.2n%2B0.5n%2B0.5n%3D54.4&pl=Solve']Source[/URL]

A traveler is walking on a moving walkway in an airport. the traveler must walk back on the walkway to get a bag he forgot. the traveler's ground speed is 2 ft/s against the walkway and 6 ft/s with the walkway. what is the traveler's speed off the walkway? What is the speed of the moving walkway. We have two equations, where w is the speed of the walkway and t is the speed of the traveler. [LIST=1] [*]t - w = 2 [*]t + w = 6 [*]Rearrange (1) to solve for t: t = w + 2 [/LIST] Now plug (3) into (2) (w + 2) + w = 6 Combine like terms: 2w + 2 = 6 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2w%2B2%3D6&pl=Solve']equation solver[/URL], we get [B]w = 2[/B] Plug this into (1) t - 2 = 2 Add 2 to each side [B]t = 4[/B]

A trench is 40 feet long and the trencher goes 2 feet per minute. How long does it take to dig the trench? 2 feet per minute * x minutes = 40 feet Divide each side by 2 [B]x = 20 minutes[/B]

a triangle has side lengths of 12,16, and 20 centimeters. is it a right triangle? First, we see if we can simplify. So we [URL='https://www.mathcelebrity.com/gcflcm.php?num1=12&num2=16&num3=20&pl=GCF']type GCF(12,16,20) [/URL]and we get 4. We divide the 3 side lengths by 4: 12/4 = 3 16/4 = 4 20/4 = 5 And lo and behold, we get a Pythagorean Triple of 3, 4, 5. So [B]yes, this is a right triangle[/B].

A vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendor takes in a total of d dollars, how many cents does a soda cost? Let the cost of the soda be p. So the cost of a hot dog is 2p. The total cost of hot dogs: 2hp The total cost of sodas: ps The total cost of both equals d. So we set the total cost of hots dogs plus sodas equal to d: 2hp + ps = d We want to know the cost of a soda (p). So we have a literal equation. We factor out p from the left side: p(2h + s) = d Divide each side of the equation by (2h + s) p(2h + s)/(2h + s) = d/(2h + s) Cancel the (2h + s) on the left side, we get: p = [B]d/(2h + s[/B])

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and walked back to the starting point along the same path. She walks 4mph on level ground, 3 mph uphill, and 6 mph downhill. Find the distance she walked. Hint: Think about d = rt, which means that t = d/r. Think about each section of her walk, what is the distance and the rate. You know that the total time is 5 hours, so you know the sum of the times from each section must be 5. Let Level distance = L and hill distance = H. Add the times it took for each section of the walk: L/4 + H /3 + H/6 + L/4 = 5 The LCD of this is 12 from our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=6&pl=LCM']LCD Calculator[/URL] [U]Multiply each side through by our LCD of 12[/U] 3L + 4H + 2H + 3L = 60 [U]Combine like terms:[/U] 6L + 6H = 60 [U]Divide each side by 3:[/U] 2L + 2H = 20 The woman walked [B]20 miles[/B]

a writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel in x number of pages, determine a function model that will represent the accumulated writing hours to finish his novel if 3 pages = 5 hours, then we divide each side by 3 to get: 1 page = 5/3 hours per page So x pages takes: 5x/3 hours Our function for number of pages x is: [B]f(x) = 5x/3[/B]

A young dad, who was a star football player in college, set up a miniature football field for his five-year-old young daughter, who was already displaying an unusual talent for place-kicking. At each end of the mini-field, he set up goal posts so she could practice kicking extra points and field goals. He was very careful to ensure the goalposts were each straight up and down and that the crossbars were level. On each set, the crossbar was six feet long, and a string from the top of each goalpost to the midpoint between them on the ground measured five feet. How tall were the goalposts? How do you know this to be true? The center of each crossbar is 3 feet from each goalpost. We get this by taking half of 6, since midpoint means halfway. Imagine a third post midway between the two goal posts. It has the same height as the two goalposts. From the center post, the string from the top of a goalpost to the base of the center post, and half the crossbar form and right triangle with hypotenuse 5 feet and one leg 3 feet. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=3&hypinput=5&pl=Solve+Missing+Side']Using the Pythagorean Theorem[/URL], the other leg -- the height of each post -- is 4 feet.

A zoo has 15 Emperor penguins. The Emperor penguins make up 30% percent of all the penguins at the zoo. How many penguins live at the zoo? Let p be the total number penguins at the zoo. We're told: 30% of p = 15 Since 30% = 0.3, we have: 0.3p = 15 Solve for [I]p[/I] in the equation 0.3p = 15 [SIZE=5][B]Step 1: Divide each side of the equation by 0.3[/B][/SIZE] 0.3p/0.3 = 15/0.3 p = [B]50[/B]

A+B+D=255 B+15=A D+12=B A= [LIST=1] [*]A + B + D = 255 [*]B + 15 = A [*]D + 12 = B [*]A = ? [*]Rearrange (3) to solve for D by subtracting 12 from each side: D = B - 12 [/LIST] Substitute (2) and (5) into 1 (B + 15) + B + (B - 12) = 255 Combine like terms: 3B + 3 = 255 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3b%2B3%3D255&pl=Solve']equation solver[/URL], b = 84 Substitute b = 84 into equation (2): A = 84 + 15 [B]A = 99[/B]

a/m - b = c for m Add b to both sides: a/m - b + b = c + b Cancel b on both sides: a/m = c + b Multiply each side by m: am/m = m(c + b) Cancel the m's on the left side: a = m(c + b) Divide each side by (c + b) a/(c + b) = m(c + b)/(c + b) Cancel the (c + b) on the right side, and we get: m[B] = a/(c + b)[/B]

A=0.5(bh), for h Divide each side by 0.5b [B]h = A/0.5b[/B]

Multiply through: A = 2l + 2w To solve for l, subtract 2w from each side: 2l = A - 2w Divide each side by 2 l = (A - 2w)/2

Multiply through using the distributive property, so we have: A = 2l + 2w Subtract 2l from each side 2w = A - 2l Divide each side by w w = (A - 2l)/2 [MEDIA=youtube]Nm-tYD4aEY4[/MEDIA]

A=a+b+c+d4 for c Assume A and a are different variables: Cross multiply: a + b + c + d = 4A Subtract a, b, and d from each side: a + b + c + d - (a + b + d) = 4A - (a + b + d) Cancel the a + b + d on the left side [B]c = 4A - a - b - d[/B]

Aaron bought a bagel and 3 muffins for $7.25. Bea bought a bagel and 2 muffins for $6. How much is bagel and how much is a muffin? Let b be the number of bagels and m be the number of muffins. We have two equations: [LIST=1] [*]b + 3m = 7.25 [*]b + 2m = 6 [/LIST] Subtract (2) from (1) [B]m = 1.25 [/B] Plug this into (2), we have: b + 2(1.25) = 6 b + 2.5 = 6 Subtract 2.5 from each side [B]b = 3.5[/B]

ab/d + c = e for d I know this is a literal equation because we are asked to solve for a variable [U]in terms of[I] another variable [/I][/U] Subtract c from each side to isolate the d term: ab/d + c - c = e - c Cancel the c's on the left side and we get: ab/d = e - c Cross multiply: ab = d(e - c) Divide each side of the equation by (e - c): ab/(e - c)= d(e - c)/(e - c) Cancel the (e - c) on the right side, and we get: d = [B]ab/(e - c)[/B]

ab/d+c=e for d Subtract c from each side: ab/d+c - c = e - c ab/d = e - c Multiply each side by d: abd/d = d(e - c) ab = d(e - c) Divide each side by (e - c): ab/(e - c) = d(e - c)/(e - c) d =[B] ab/(e - c)[/B]

acw+cz=y for a Solve this literal equation: Subtract cz from each side: acw + cz - cz = y - cz Cancel the cz on the left side: acw = y - cz Divide each side by cw to isolate a: acw/cw = (y - cz)/cw Cancel cw on the left side: [B]a = (y - cz)/cw[/B]

Adam drove the 10 miles to school at a speed of 60 mph. On his way home, due to traffic, his speed was 30 mph. What was his average speed for the round trip to school and back? D = rt To school: 60 miles in 60 minutes = 10 miles in 10 minutes To home: 30 miles in 60 minutes = 10 miles in 20 minutes Total time: 10 + 20 = 30 minutes or 0.5 hours With a speed of s, we have: d = st 20 = 0.5s Divide each side by 2: s = [B]40 mph[/B]

admission to the school fair is $2.50 for students and $3.75 for others. if 2848 admissions were collected for a total of 10,078.75, how many students attended the fair Let the number of students be s and the others be o. We're given two equations: [LIST=1] [*]o + s = 2848 [*]3.75o + 2.50s = 10078.75 [/LIST] Since we have no coefficients for equation 1, let's solve this the fast way using substitution. Rearrange equation 1 by subtracting o from each side to isolate s [LIST=1] [*]o = 2848 - s [*]3.75o + 2.50s = 10078.75 [/LIST] Now substitute equation 1 into equation 2: 3.75(2848 - s) + 2.50s =10078.75 To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=3.75%282848-s%29%2B2.50s%3D10078.75&pl=Solve']type this equation into our search engine[/URL] and we get: s = [B]481[/B]

After a long journey, you finally arrive at the edge o a deep gorge where there are two identical bridges from which to choose your path to the other side. One bridge is safe, while the other is very dangerous and has caused the deaths of hundreds of travelers. The owner of the first bridge is a talking rat, while the owner of the second bridge is a talking frog. Friends told you before you left that one of the bridge owners always tells the truth, while the other always lies. You are allowed one question to ask of either the frog or the rat to find out which bridge is the safe bridge. What is the question that you would ask? [B]Ask the frog the following question: "If I were to ask the rat which bridge is the same bridge, which one would he point to?" [/B] If the frog is the truth teller, he would tell you that the rat would point to the dangerous bridge. If the frog is the liar, the truth telling rat would point out the safe bridge, but the lying frog would tell you he said the dangerous bridge. In both situations, the dangerous bridge would be pointed to. Take the other bridge.

Let f be the age of the father and d be the age of the daughter and s be the age of the son. We have: [LIST=1] [*]f = 3s [*]d = s - 3 [*]d - 3 + f - 3 + s - 3 = 63 [/LIST] Simplify (3) d + f + s - 9 = 63 d + f + s = 72 Now, substitute (1) and (2) into the modified (3) (s - 3) + 3s + s = 72 Combine like terms: 5s - 3 = 72 Add 3 to each side 5s = 75 Divide each side by 5 s = 15 We want f, so we substitute s = 15 into (1) f = 3(15) [B]f = 45[/B]

Also generates binomial theorem expansions and polynomial expansions with or without an outside constant multiplier.

Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age? Let a be Alvin's age and e be Elga's age. We have the following equations: [LIST=1] [*]a = e - 12 [*]a + e = 60 [/LIST] Plugging in (1) to (2), we get: (e - 12) + e = 60 Grouping like terms: 2e - 12 = 60 Add 12 to each side: 2e = 72 Divide each side by 2 [B]e = 36[/B]

Amy and ryan operate a car dealing and repair service. For a car detailing (full wash outside and inside. Amy charges 40$ and Ryan charges 50$ . In addition they charge a hourly rate. Amy charges $35/h and ryan charges $30/h. How many hours does amy and ryan have to work to make the same amount of money? Set up the cost functions C(h) where h is the number of hours. [U]Amy:[/U] C(h) = 35h + 40 [U]Ryan:[/U] C(h) = 30h + 50 To make the same amount of money, we set both C(h) functions equal to each other: 35h + 40 = 30h + 50 To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=35h%2B40%3D30h%2B50&pl=Solve']type this equation into our search engine[/URL] and we get: h = [B]2[/B]

An ancient Greek was said to have lived 1/4 of his live as a boy, 1/5 as a youth, 1/3 as a man, and spent the last 13 years as an old man. How old was he when he died? Set up his life equation per time lived as a boy, youth, man, and old man 1/4 + 1/5 + 1/3 + x = 1. Using our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=5&pl=LCM']LCM Calculator[/URL], we see the LCM of 3,4,5 is 60. This is our common denominator. So we have 15/60 + 12/60 + 20/60 + x/60 = 60/60 [U]Combine like terms[/U] x + 47/60 = 60/60 [U]Subtract 47/60 from each side:[/U] x/60 = 13/60 x = 13 out of the 60 possible years, so he was [B]60 when he died[/B].

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds, what is the maximum number of people (p) that can be on the elevator at one time? Total weight = average weight per person * Number of people Total weight = 150p We know from the problem that: 150p < 2700 We want to solve this inequality for p. Divide each side of the inequality by 150: 150p/150 < 2700/150 Cancel the 150's on the left side and we get: p < [B]18[/B]

An equilateral triangle has three sides of equal length. What is the equation for the perimeter of an equilateral triangle if P = perimeter and S = length of a side? P = s + s + s [B]P = 3s[/B]

An experienced accountant can balance the books twice as fast as a new accountant. Working together it takes the accountants 10 hours. How long would it take the experienced accountant working alone? Person A: x/2 job per hour Person B: 1/x job per hour Set up our equation: 1/x + 1/(2x) = 1/10 Multiply the first fraction by 2/2 to get common denominators; 2/(2x) + 1/(2x) = 1/10 Combine like terms 3/2x = 1/10 Cross multiply: 30 = 2x Divide each side by 2: [B]x = 15[/B]

An irregular pentagon is a five sided figure. The two longest sides of a pentagon are each three times as long as the shortest side. The remaining two sides are each 8m longer than the shortest side. The perimeter of the Pentagon is 79m. Find the length of each side of the Pentagon. Let long sides be l. Let short sides be s. Let medium sides be m. We have 3 equations: [LIST=1] [*]2l + 2m + s = 79 [*]m = s + 8 [*]l = 3s [/LIST] Substitute (2) and (3) into (1): 2(3s) + 2(s + 8) + s = 79 Multiply through and simplify: 6s + 2s + 16 + s = 79 9s + 16 = 79 [URL='https://www.mathcelebrity.com/1unk.php?num=9s%2B16%3D79&pl=Solve']Using our equation calculator[/URL], we get [B]s = 7[/B]. This means from Equation (2): m = 7 + 8 [B]m = 15 [/B] And from equation (3): l = 3(7) [B]l = 21[/B]

An orchard has 816 apple trees. The number of rows exceeds the number of trees per row by 10. How many trees are there in each row? Let the rows be r and the trees per row be t. We're given two equations: [LIST=1] [*]rt = 816 [*]r = t + 10 [/LIST] Substitute equation (2) into equation (1) for r: (t + 10)t = 816 t^2 + 10t = 816 Subtract 816 from each side of the equation: t^2 + 10t - 816 = 816 - 816 t^2 + 10t - 816 = 0 We have a quadratic equation. To solve this, we [URL='https://www.mathcelebrity.com/quadratic.php?num=t%5E2%2B10t-816%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type it in our search engine [/URL]and we get: t = (24, -34) Since the number of trees per row can't be negative, we choose [B]24[/B] as our answer

Andrea has one hour to spend training for an upcoming race she completes her training by running full speed in the distance of the race and walking back the same distance to cool down if she runs at a speed of 9 mph and walks back at a speed of 3 mph how long should she plan on spending to walk back Let r = running time. Let w = walking time We're given two equations [LIST=1] [*]r + w = 1 [*]9r = 3w [/LIST] Rearrange equation (1) by subtract r from each side: [LIST=1] [*]w = 1 - r [*]9r = 3w [/LIST] Now substitute equation (1) into equation (2): 9r = 3(1 - r) 9r = 3 - 3r To solve for r, [URL='https://www.mathcelebrity.com/1unk.php?num=9r%3D3-3r&pl=Solve']we type this equation into our search engine[/URL] and we get: r = 0.25 Plug this into modified equation (1) to solve for w, and we get: w = 1. 0.25 [B]w = 0.75[/B]

April, May and June have 90 sweets between them. May has three-quarters of the number of sweets that June has. April has two-thirds of the number of sweets that May has. How many sweets does June have? Let the April sweets be a. Let the May sweets be m. Let the June sweets be j. We're given the following equations: [LIST=1] [*]m = 3j/4 [*]a = 2m/3 [*]a + j + m = 90 [/LIST] Cross multiply #2; 3a = 2m Dividing each side by 2, we get; m = 3a/2 Since m = 3j/4 from equation #1, we have: 3j/4 = 3a/2 Cross multiply: 6j = 12a Divide each side by 12: a = j/2 So we have: [LIST=1] [*]m = 3j/4 [*]a = j/2 [*]a + j + m = 90 [/LIST] Now substitute equation 1 and 2 into equation 3: j/2 + j + 3j/4 = 90 Multiply each side by 4 to eliminate fractions: 2j + 4j + 3j = 360 To solve this equation for j, we [URL='https://www.mathcelebrity.com/1unk.php?num=2j%2B4j%2B3j%3D360&pl=Solve']type it in our search engine[/URL] and we get: j = [B]40[/B]

Arnie bought some bagels at 20 cents each. He ate 4, and sold the rest at 30 cents each. His profit was $2.40. How many bagels did he buy? Let x be the number of bagels Arnie sold. We have the following equation: 0.30(x - 4) - 0.20(4) = 2.40 Distribute and simplify: 0.30x - 1.20 - 0.8 = 2.40 Combine like terms: 0.30x - 2 = 2.40 Add 2 to each side: 0.30x = 4.40 Divide each side by 0.3 [B]x = 14.67 ~ 15[/B]

As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at least $100. What is the minimum number of sales you must make to earn at least $100? Set up the inequality where s is the amount of sales you make: 50 + 2s >= 100 We use >= because the phrase [I]at least[/I] 100 means 100 or more Subtract 50 from each side: 2s >= 50 Divide each side by 2 [B]s >= 25[/B]

At 4am the outside temperature was -28 By 4pm it rose 38 What was the temperature at 4pm -28 + 38 = [B]10 degrees (above zero)[/B]

At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same? Set up the cost functions where x is the number of aerobics classes: [LIST] [*]Members: C(x) = 10 + 3x [*]Non-members: C(x) = 5x [/LIST] Set them equal to each other 10 + 3x = 5x Subtract 3x from both sides: 2x = 10 Divide each side by 2 [B]x = 5 classes[/B]

At a local fitness center, members pay an $8 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members be equal to nonmembers? Set up two cost equations C(x): [LIST=1] [*]Members: C(x) = 8 + 3x [*]Nonmembers: C(x) = 5x [/LIST] Set the two cost equations equal to each other: 8 + 3x = 5x Subtract 3x from each side 2x = 8 Divide each side by 2 [B]x = 4[/B]

At the end of the week, Francesca had a third of her babysitting money left after spending $14.65 on a movie and popcorn and another $1.35 on a pen. How much did she earn babysitting? Let the original amount of money earned for babysitting be b. We're given: [LIST=1] [*]Start with b [*]Spending 14.65 for a movie means we subtract 14.65 from b: b - 14.65 [*]Spending 1.35 on a pen means we subtract another 1.35 from step 2: b - 14.65 - 1.35 [*]Francesca has a third of her money left. So we set step 3 equal to 1/3 of b [/LIST] b - 14.65 - 1.35 = b/3 Multiply each side of the equation by 3 to remove the fraction 3(b - 14.65 - 1.35) = 3b/3 3b - 43.95 - 4.05 = b To solve this equation for b, [URL='https://www.mathcelebrity.com/1unk.php?num=3b-43.95-4.05%3Db&pl=Solve']we type it in our search engine[/URL] and we get: b =[B] 24[/B]

At what simple interest rate will 4500$ amount to 8000$ in 5 years? Simple Interest is written as 1 + it. With t = 5, we have: 4500(1 + 5i) = 8000 Divide each side by 4500 1 + 5i = 1.77777778 Subtract 1 from each side: 5i = 0.77777778 Divide each side by 5 i = 0.15555 As a percentage we multiply by 100 to get [B]15.5%[/B]

Austin has 15 CDs, which is 3 less than his sister has. How many CDs does his sister have? Let s be the number of CD's his sister has and a be the number Austin has [LIST=1] [*]a = 15 [*]a = s - 3 [/LIST] Substitute (1) into (2) 15 = s - 3 Add 3 to each side [B]s = 18[/B]

Ava is 4 times as old as Peter. What equation can be used to find Peters age? [U]Assumptions[/U] Let a be Ava's age Let p be Peter's age We're given: a = 4p To find Peter's age, we divide each side of the equation by 4 to get: a/4 = 4p/4 p = [B]a/4[/B]

Ava set her watch 2 seconds behind, every day it sets back 1 second. How many days has it been since she last set her watch if it is 41 seconds behind? Right now: Watch is 2 seconds behind [U]Let d be the day after right now[/U] (1)d + 2 = 41 d + 2 = 41 [U]Subtract 2 from each side[/U] [B]d = 39[/B]

We are solving for x: Subtract b from each side: ax = cx - d - b Subtract cx from each side: ax - cx = -d - b Factor out x from the left side: x(a - c) = -d - b Divide each side by (a - c) x = (-d - b)/(a - c)

ax - mn = mn + bx for x Add mn to each side: ax - mn + mn = mn + bx + mn Cancel the mn terms on the left side and we get: ax = bx + 2mn Subtract bx from each side: ax - bx = bx - bx + 2mn Cancel the bx terms on the right side: ax - bx = 2mn Factor out x on the left side: x (a - b) = 2mn Divide each side of the equation by (a - b): x (a - b)/(a - b) = 2mn/(a - b) Cancel the (a - b) on the left side and we get: x = [B]2mn/(a - b)[/B]

a^2 + b^2 = c^2 for c Take the square root of each side: c = [B]sqrt(a^2 + b^2)[/B]

B+c =10/a for a Cross multiply: a(B + c) = 10 Divide each side by a [B]a = 10/(B + c)[/B]

b/3d - h = 343 for b A literal equation means we solve for one variable in terms of another variable or variables Add h to each side to isolate the b term: b/3d - h + h = 343 + h Cancel the h's on the left side, we get: b/3d = 343 + h Cross multiply: b = [B]3d(343 + h)[/B]

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money? Let w be the number of weeks that go by for saving/spending. Set up Barney's balance equation, B(w). Spending means we [U]subtract[/U] B(w) = Initial Amount - spend per week * w weeks B(w) = 450 - 3w Set up Betty's balance equation, B(w). Saving means we [U]add[/U] B(w) = Initial Amount + savings per week * w weeks B(w) = 120 + 8w The same amount of money means both of their balance equations B(w) are equal. So we set Barney's balance equal to Betty's balance and solve for w: 450 - 3w = 120 + 8w Add 3w to each side to isolate w: 450 - 3w + 3w = 120 + 8w + 3w Cancelling the 3w on the left side, we get: 450 = 120 + 11w Rewrite to have constant on the right side: 11w + 120 = 450 Subtract 120 from each side: 11w + 120 - 120 = 450 - 120 Cancelling the 120's on the left side, we get: 11w = 330 To solve for w, we divide each side by 11 11w/11 = 330/11 Cancelling the 11's on the left side, we get: w = [B]30 [MEDIA=youtube]ifG_q-utgJI[/MEDIA][/B]

Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign, then divided by 3. What was the original equation [LIST=1] [*]If we added 7 to both sides, that means we had a minus 7 (-7) to start with as a constant. Since subtraction undoes addition. [*]If we divided by 3, this means we multiplied x by 3 to begin with. Since division undoes multiplication [/LIST] So we have the start equation: 3x - 7 If the answer was x = -4, then we plug this in to get our number on the right side of the equation: 3(-4) - 7 -12 - 7 -19 This means our original equation was: [B]3x - 7 = -19[/B] And if we want to solve this to prove our answer, we [URL='https://www.mathcelebrity.com/1unk.php?num=3x-7%3D-19&pl=Solve']type the equation into our search engine [/URL]and we get: x = -4

Ben has $4.50 in quarters(Q) and dimes(D). a)Write an equation expressing the total amount of money in terms of the number of quarters and dimes. b)Rearrange the equation to isolate for the number of dimes (D) a) The equation is: [B]0.1d + 0.25q = 4.5[/B] b) Isolate the equation for d. We subtract 0.25q from each side of the equation: 0.1d + 0.25q - 0.25q = 4.5 - 0.25q Cancel the 0.25q on the left side, and we get: 0.1d = 4.5 - 0.25q Divide each side of the equation by 0.1 to isolate d: 0.1d/0.1 = (4.5 - 0.25q)/0.1 d = [B]45 - 2.5q[/B]

Besides 8 and 1, what is one factor of 8. Using our [URL='http://www.mathcelebrity.com/factoriz.php?num=8&pl=Show+Factorization']factor calculator[/URL], or entering the shortcut [B]Factor 8[/B], we get the following factors: 1, 2, 4, 8 Excluding 1 and 8, we have [B]2, 4[/B]

Beth made a trip to the train station and back. On the trip there she traveled 45 km/h and on the return trip she went 30 km/h. How long did the trip there take if the return trip took six hours? We use the distance formula: D = rt where D = distance, r = rate, and t = time. Start with the return trip: D = 45(6) D = 270 The initial trip is: 270= 30t Divide each side by 30 [B]t = 9 hours[/B]

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This utilizes the First-Outside-Inside-Last (F.O.I.L.) method.

Blake writes 4 pages per hour. How many hours will Blake have to spend writing this week in order to have written a total of 16 pages? [U]Let x = the number of hours Blake needs to write[/U] 4 pages per hour * x hours = 16 [U]Divide each side by 4[/U] [B]x = 4 hours[/B]

Brandon can shovel his sidewalk in 8 minutes, while his brother can shovel the walk in 12 minutes. If they work together, how long will it take them to shovel the sidewalk? Set up unit rates: [LIST] [*]Brandon can shovel 1/8 of a sidewalk per minute [*]His brother can shovel 1/12 of a sidewalk per minute [/LIST] Together, they can shovel: [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F8&frac2=1%2F12&pl=Add']1/8 + 1/12[/URL] = 5/24 of a sidewalk per minute 1 minute = 60 seconds 5/24 / 60 seconds = 1/x seconds 5/24 * 60 = 1/x 5/1440 = 1/x Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=5&num2=1&den1=1440&den2=x&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get: x = 288 288/60 = [B]4 minutes and 48 seconds[/B]

Brice has 1200 in the bank. He wants to save a total of 3000 by depositing 40 per week from his paycheck. How many weeks will it take until he saves 3000? Remaining Savings = 3,000 - 1,200 = 1,800 40 per week * x weeks = 1,800 40x = 1800 Divide each side of the equation by 40 [B]x = 45 weeks[/B]

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy. Write and solve and inequality to find how many on demand movies could you buy if you want your bill to be less than $150 for the month. Let x equal to the number room movie rentals per month. Our inequality is: 120 + 2.99x < 150 To solve for the number of movies, Add 120 to each side 2.99x < 30 Divide each side by 2.99 x < 10.03, which means 10 since you cannot buy a fraction of a movie

by + 2/3 = c for y Subtract 2/3 from each side of the literal equation: by + 2/3 - 2/3 = c - 2/3 Cancel the 2/3 on the left side to get: by = c - 2/3 Divide each side by b to isolate y: by/b = (c - 2/3)/b Cancel the b's on the left side to get: y = [B](c - 2/3)/b[/B]

by + 2/3 = c, for y Subtract 2/3 from each side: by = c - 2/3 Divide each side by b y = [B](c - 2/3)/b[/B]

b^2 - 6 = 5an for a Divide each side of the equation by 5n to isolate a: (b^2 - 6)/5n = 5an/5n Cancel the 5n on the right side and we get: a = [B](b^2 - 6)/5n[/B]

c/a=db/r for a Cross multiply the proportion: cr = adb Divide each side of the equation by db to isolate a: cr/db = adb/db Cancel the db terms on the left side and we get: a = [B]cr/db[/B]

c=59f-288 for f Add 288 to each side: c + 288 = 59f - 288 + 288 Cancel the 288 on the right side, we get: 59f = c + 288 Divide each side by 59 to isolate f: 59f/59 = (c + 288)/59 Cancel the 59 on the left side, we get: f = [B](c + 288)/59[/B]

calculate cos(x) given tan(x)=8/15 tan(x) = sin(x)/cos(x) sin(x)/cos(x) = 8/15 Cross multiply: 15sin(x) = 8cos(x) Divide each side by 8 [B]cos(x) = 15sin(x)/8[/B]

Let f = dollars spent on flights, h dollars spent on hotels, and p dollars spent on all other purchases. [U]Set up our equations:[/U] (1) 4f + 2h + p = 14660 (2) f + h + p = 9480 (3) f = 2h + 140 [U]First, substitute (3) into (2)[/U] (2h + 140) + h + p = 9480 3h + p + 140 = 9480 3h + p = 9340 [U]Subtract 3h to isolate p to form equation (4)[/U] (4) p = 9340 - 3h [U]Take (3) and (4), and substitute into (1)[/U] 4(2h + 140) + 2h + (9340 - h) = 14660 [U]Multiply through[/U] 8h + 560 + 2h + 9340 - 3h = 14660 [U]Combine h terms and constants[/U] (8 + 2 - 3)h + (560 + 9340) = 14660 7h + 9900 = 14660 [U]Subtract 9900 from both sides:[/U] 7h = 4760 [U]Divide each side by 7[/U] [B]h = 680[/B] [U]Substitute h = 680 into equation (3)[/U] f = 2(680) + 140 f = 1360 + 140 [B]f = 1,500[/B] [U] Substitute h = 680 and f = 1500 into equation (2)[/U] 1500 + 680 + p = 9480 p + 2180 = 9480 [U]Subtract 2180 from each side:[/U] [B]p = 7,300[/B]

Consider a paper cone, pointing down, with the height 6 cm and the radius 3 cm; there is currently 9/4 (pie) cubic cm of water in the cone, and the cone is leaking at a rate of 2 cubic centimeters of water per second. How fast is the water level changing, in cm per second?

Casey is 26 years old. Her daughter Chloe is 4 years old. In how many years will Casey be double her daughter's age Declare variables for each age: [LIST] [*]Let Casey's age be c [*]Let her daughter's age be d [*]Let n be the number of years from now where Casey will be double her daughter's age [/LIST] We're told that: 26 + n = 2(4 + n) 26 + n = 8 + 2n Solve for [I]n[/I] in the equation 26 + n = 8 + 2n [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables n and 2n. To do that, we subtract 2n from both sides n + 26 - 2n = 2n + 8 - 2n [SIZE=5][B]Step 2: Cancel 2n on the right side:[/B][/SIZE] -n + 26 = 8 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 26 and 8. To do that, we subtract 26 from both sides -n + 26 - 26 = 8 - 26 [SIZE=5][B]Step 4: Cancel 26 on the left side:[/B][/SIZE] -n = -18 [SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE] -1n/-1 = -18/-1 n = [B]18[/B] Check our work for n = 18: 26 + 18 ? 8 + 2(18) 44 ? 8 + 36 44 = 44

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Chris, Alex and Jesse are all siblings in the same family. Alex is 5 years older than chris. Jesse is 6 years older than Alex. The sum of their ages is 31 years. How old is each one of them? Set up the relational equations where a = Alex's age, c = Chris's aged and j = Jesse's age [LIST=1] [*]a = c + 5 [*]j = a + 6 [*]a + c + j = 31 [*]Rearrange (1) in terms of c: c = a - 5 [/LIST] [U]Plug in (4) and (2) into (3)[/U] a + (a - 5) + (a + 6) = 31 [U]Combine like terms:[/U] 3a + 1 = 31 [U]Subtract 1 from each side[/U] 3a = 30 [U]Divide each side by 3[/U] [B]a = 10[/B] [U]Plug in 1 = 10 into Equation (4)[/U] c = 10 - 5 [B]c = 5[/B] Now plug 1 = 10 into equation (2) j = 10 + 6 [B]j = 16[/B]

Class A has 8 pupils and class B has 10 pupils. Both classes sit the same maths test. The mean score for class A is 55. The mean score for both classes is 76. What is the mean score (rounded to 1 DP) in the maths test for class B Mean of the sum equals the sum of the means. U(A + B) = U(A) + U(B) 76 = 55 + U(B) Subtract 55 from each side, we get: [B]U(B) = 21[/B]

Colin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was the original number? Let the number be n. Divide by 8: n/8 Then add 1: n/8 + 1 The phrase [I]get an answer[/I] of means an equation, so we set n/8 + 1 equal to 2: n/8 + 1 = 2 To solve for n, we subtract 1 from each side to isolate the n term: n/8 + 1 - 1 = 2 - 1 Cancel the 1's on the left side, we get: n/8 = 1 Cross multiply: n = 8*1 n = [B]8[/B]

Company a charges $25 plus $0.10 a mile. Company b charges $20 plus $0.15 per mile. How far would you need to travel to get each charge to be the same? Let x be the number of miles traveled Company A charge: C = 25 + 0.10x Company B charge: C = 20 + 0.15x Set up an equation find out when the charges are the same. 25 + 0.10x = 20 + 0.15x Combine terms and simplify 0.05x = 5 Divide each side of the equation by 0.05 to isolate x x = [B]100[/B]

Connor runs 2 mi more each day than David. The sum of the distances they run each week is 56 mi. How far does David run each day? Let Connor's distance be c Let David's distance be d We're given two equations: [LIST=1] [*]c = d + 2 [*]7(c + d) = 56 [/LIST] Simplifying equation 2 by dividing each side by 7, we get: [LIST=1] [*]c = d + 2 [*]c + d = 8 [/LIST] Substitute equation (1) into equation (2) for c d + 2 + d = 8 To solve for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=d%2B2%2Bd%3D8&pl=Solve']type this equation into our calculation engine[/URL] and we get: d = [B]3[/B]

Consider a firm that has two assembly lines, 1 and 2, both producing calculator. Assume that you have purchased a calculator and it turns out to be defective. And the line 1 produces 60% of all calculators produced. L1: event that the calculator is produced on line 1 L2: event that the calculator is produced on line 2 Suppose that your are given the conditional information: 10% of the calculator produced on line 1 is defective 20% of the calculator produced on line 2 is defective Q: If we choose one defective, what is the probability that the defective calculator comes from Line 1 and Line2? L1 = event that the calculator is produced on line 1 = 0.6 L2 = event that the calculator is produced on line 2 = 1 - 0.6 = 0.4 D = Defective D|L1 Defective from Line 1 = 0.1 D|L2 = Defective from Line 2 = 0.20 [U]Defective from Line 1[/U] P(L1|D) = P(L1)P(D/L1) / [ P(L1)P(D/L1) + P(L2)P(D/L2)] P(L1|D) = (.60)(.10) /[(.60)(.10)+ (.40)(.20)] [B]P(L1|D) = 0.4286[/B] [U]Defective from Line 2[/U] P(L2|D) = P(L2)P(D/L2) / [ P(L1)P(D/L1) + P(L2)P(D/L2)] P(L2|D) = (.40)(.20) /[(.60)(.10)+ (.40)(.20)] [B]P(L2|D) = 0.5714[/B]

Consider a probability model consisting of randomly drawing two colored balls from a jar containing 2 red and 1 blue balls. What is the Sample Space of this experiment? (assume B= blue and R=red) The sample space is the list of all possible events [LIST] [*]RRB [*]RBR [*]BRR [/LIST]

Consider the case of a manufacturer who has an automatic machine that produces an important part. Past records indicate that at the beginning of the data the machine is set up correctly 70 percent of the time. Past experience also shows that if the machine is set up correctly it will produce good parts 90 percent of the time. If it is set up incorrectly, it will produce good parts 40 percent of the time. Since the machine will produce 60 percent bad parts, the manufacturer is considering using a testing procedure. If the machine is set up and produces a good part, what is the revised probability that it is set up correctly? [U]Determine our events:[/U] [LIST] [*]C = Correctly Set Machine = 0.7 [*]C|G = Correctly Set Machine And Good Part = 0.9 [*]I = Incorrectly Set Machine = 1 - 0.7 = 0.3 [*]I|G = Incorrectly Set Machine And Good Part = 0.4 [*]B< = BAD PARTS = 0.60 [/LIST] P[correctly set & part ok] = P(C) * P(C|G) P[correctly set & part ok] = 70% * 90% = 63% P[correctly set & part ok] = P(I) * P(I|G) P[incorrectly set & part ok] = 30% *40% = 12% P[correctly set | part ok] = P[correctly set & part ok]/(P[correctly set & part ok] + P[incorrectly set & part ok]) P[correctly set | part ok] = 63/(63+12) = [B]0.84 or 84%[/B]

Consider the following 15 numbers 1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20 - The mean of the last 10 numbers is TWICE the mean of the first 10 numbers - The sum of the last 2 numbers is FIVE times the sum of the first 3 numbers (i) Calculate the values of x and y We're given two equations: [LIST=1] [*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = 2(1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/10 [*]3x - 20 = 5(1 + 2 + y - 4) [/LIST] Let's evaluate and simplify: [LIST=1] [*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = (1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5 [*]3x - 20 = 5(y - 1) [/LIST] Simplify some more: [URL='https://www.mathcelebrity.com/polynomial.php?num=x%2B6%2B7%2B8%2By%2B9%2B10%2B12%2B3x%2B20&pl=Evaluate'](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10[/URL] = (4x + y + 72)/10 [URL='https://www.mathcelebrity.com/polynomial.php?num=1%2B2%2By-4%2B4%2B5%2Bx%2B6%2B7%2B8%2By&pl=Evaluate'](1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5[/URL] = (2y + x + 29)/5 5(y - 1) = 5y - 5 So we're left with: [LIST=1] [*](4x + y + 72)/10 = (2y + x + 29)/5 [*]3x - 20 = 5y - 5 [/LIST] Cross multiply equations in 1, we have: 5(4x + y + 72) = 10(2y + x + 29) 20x + 5y + 360 = 20y + 10x + 290 We have: [LIST=1] [*]20x + 5y + 360 = 20y + 10x + 290 [*]3x - 20 = 5y - 5 [/LIST] Combining like terms: [LIST=1] [*]10x - 15y = -70 [*]3x - 5y = 15 [/LIST] Now we have a system of equations which we can solve any of three ways: [LIST] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Substitution']Substitution Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Elimination']Elimination Method[/URL] [*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Cramers+Method']Cramer's Rule[/URL] [/LIST] No matter which method we choose, we get the same answer: (x, y) = [B](-115, -72)[/B]

Consider the formula for the area of a trapezoid: A=12h(a+b) . Is it mathematically simpler to solve for a, b, or h? Why? Solve for each of these variables to demonstrate. The variable "h" is the easiest to solve for. Because you only have one step. Let's review: Divide each side of the equation by 12(a + b) h = 12(a + b)/A Solving for "a", we two steps. Divide each side by 12h: A/12h = a + b Subtract b from each side a = A/12h - b Solving for "b" takes two steps as well. Divide each side by 12h: A/12h = a + b Subtract a from each side b = A/12h - a

Free Container Arrangements Calculator - Given a set of items inside a container, this calculates the probability that you draw certain items in the following fashion:
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How many ways can you choose m items of a, n items of b, o items of c, etc.

cot(?)=12 and ? is in Quadrant I, what is sin(?)? cot(?) = cos(?)/sin(?) 12 = cos(?)/sin(?) Cross multiply: 12sin(?) = cos(?) Divide each side by 12: sin(?) = [B]12cos(?)[/B]

Country A produces about 7 times the amount of diamonds in carats produce in Country B. If the total produced in both countries is 40,000,000 carats, find the amount produced in each country. Set up our two given equations: [LIST=1] [*]A = 7B [*]A + B = 40,000,000 [/LIST] Substitute (1) into (2) (7B) + B = 40,000,000 Combine like terms 8B = 40,000,000 Divide each side by 8 [B]B = 5,000,000[/B] Substitute this into (1) A = 7(5,000,000) [B]A = 35,000,000[/B]

Free Cube Calculator - Solves for Volume (Capacity), Lateral Area,Surface Area, and the value of a side for a cube.

cx+b/d=y for b Subtract cx from each side to isolate b/d: cx - cx + b/d = y - cx Cancel the cx terms on each side: b/d = y - cx Cross multiply: b = [B]d(y - cx)[/B]

d - f^3 = 4a for a Solve this literal equation for a: Divide each side of the equation by 4: (d - f^3)/4 = 4a/4 Cancel the 4's on the right side, and rewrite with our variable to solve for on the left side: a = [B](d - f^3)/4[/B]

Dan makes 11 an hour working at the local grocery store. Over the past year he has saved 137.50 toward a new pair of retro sneakers. If sneakers cost 240, how many hours will he need to be able to buy the sneakers? Figure out his remaining savings target: 240 - 137.50 = 102.50 Let x equal the number of remaining hours Dan needs to work 11x = 102.50 Divide each side by 11 x = 9.318 We round up for a half-hour to 9.5, or a full hour to 10.

Debra buys candy that costs 4 per pound. She will spend less than 20 on candy. What are the possible numbers of pounds she will buy? Set up an inequality using less than < and p for pounds: 4p < 20 Divide each side by 4: 4p/4 < 20/4 [B]p < 5[/B]

Free Decagon Calculator - Solves for the side, perimeter, and area of a decagon.

Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then added a mile to his run each day. By the end of the week (7 days), he had run a total of 70 miles. How many miles did Dennis run the first day? Setup distance ran for the 7 days: [LIST=1] [*]n [*]n + 1 [*]n + 2 [*]n + 3 [*]n + 4 [*]n + 5 [*]n + 6 [/LIST] Add them all up: 7n + 21 = 70 Solve for [I]n[/I] in the equation 7n + 21 = 70 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 21 and 70. To do that, we subtract 21 from both sides 7n + 21 - 21 = 70 - 21 [SIZE=5][B]Step 2: Cancel 21 on the left side:[/B][/SIZE] 7n = 49 [SIZE=5][B]Step 3: Divide each side of the equation by 7[/B][/SIZE] 7n/7 = 49/7 n =[B] 7 [URL='https://www.mathcelebrity.com/1unk.php?num=7n%2B21%3D70&pl=Solve']Source[/URL][/B]

Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up her account with a starting amount of $850 and is going to take out $65 per month. When would the two accounts have the same amount of money? We set up a balance equation B(m) where m is the number of months. [U]Set up Deon's Balance equation:[/U] Withdrawals mean we subtract from our current balance B(m) = Starting Balance - Withdrawal Amount * m B(m) = 650 - 40m [U]Set up Mai's Balance equation:[/U] Withdrawals mean we subtract from our current balance B(m) = Starting Balance - Withdrawal Amount * m B(m) = 850 - 65m When the two accounts have the same amount of money, we can set both balance equations equal to each other and solve for m: 650 - 40m = 850 - 65m Solve for [I]m[/I] in the equation 650 - 40m = 850 - 65m [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables -40m and -65m. To do that, we add 65m to both sides -40m + 650 + 65m = -65m + 850 + 65m [SIZE=5][B]Step 2: Cancel -65m on the right side:[/B][/SIZE] 25m + 650 = 850 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 650 and 850. To do that, we subtract 650 from both sides 25m + 650 - 650 = 850 - 650 [SIZE=5][B]Step 4: Cancel 650 on the left side:[/B][/SIZE] 25m = 200 [SIZE=5][B]Step 5: Divide each side of the equation by 25[/B][/SIZE] 25m/25 = 200/25 m = [B]8[/B]

Determine whether the statement is true or false. If 0 < a < b, then Ln a < Ln b We have a logarithmic property that states: ln(a) - ln(b) = ln (a / b) We're given a < b, so (a / b) < 1 Therefore: ln (a / b) < 0 And since ln(a) - ln(b) = ln (a / b) Then Ln(a) - Ln(b) < 0 Add Ln(b) to each side and we get: Ln(a) - Ln(b) + Ln(b) < 0 + Ln(b) Cancel the Ln(b) on the left side and we get: Ln(a)

Diana invested $3000 in a savings account for 3 years. She earned $450 in interest over that time period. What interest rate did she earn? Use the formula I=Prt to find your answer, where I is interest, P is principal, r is rate and t is time. Enter your solution in decimal form rounded to the nearest hundredth. For example, if your solution is 12%, you would enter 0.12. Our givens are: [LIST] [*]I = 450 [*]P = 3000 [*]t = 3 [*]We want r [/LIST] 450 = 3000(r)(3) 450 = 9000r Divide each side by 9000 [B]r = 0.05[/B]

difference between 2 positive numbers is 3 and the sum of their squares is 117 Declare variables for each of the two numbers: [LIST] [*]Let the first variable be x [*]Let the second variable be y [/LIST] We're given 2 equations: [LIST=1] [*]x - y = 3 [*]x^2 + y^2 = 117 [/LIST] Rewrite equation (1) in terms of x by adding y to each side: [LIST=1] [*]x = y + 3 [*]x^2 + y^2 = 117 [/LIST] Substitute equation (1) into equation (2) for x: (y + 3)^2 + y^2 = 117 Evaluate and simplify: y^2 + 3y + 3y + 9 + y^2 = 117 Combine like terms: 2y^2 + 6y + 9 = 117 Subtract 117 from each side: 2y^2 + 6y + 9 - 117 = 117 - 117 2y^2 + 6y - 108 = 0 This is a quadratic equation: Solve the quadratic equation 2y2+6y-108 = 0 With the standard form of ax2 + bx + c, we have our a, b, and c values: a = 2, b = 6, c = -108 Solve the quadratic equation 2y^2 + 6y - 108 = 0 The quadratic formula is denoted below: y = -b sqrt(b^2 - 4ac)/2a [U]Step 1 - calculate negative b:[/U] -b = -(6) -b = -6 [U]Step 2 - calculate the discriminant ?:[/U] ? = b2 - 4ac: ? = 62 - 4 x 2 x -108 ? = 36 - -864 ? = 900 <--- Discriminant Since ? is greater than zero, we can expect two real and unequal roots. [U]Step 3 - take the square root of the discriminant ?:[/U] ?? = ?(900) ?? = 30 [U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U] Numerator 1 = -b + ?? Numerator 1 = -6 + 30 Numerator 1 = 24 [U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U] Numerator 2 = -b - ?? Numerator 2 = -6 - 30 Numerator 2 = -36 [U]Step 6 - calculate your denominator which is 2a:[/U] Denominator = 2 * a Denominator = 2 * 2 Denominator = 4 [U]Step 7 - you have everything you need to solve. Find solutions:[/U] Solution 1 = Numerator 1/Denominator Solution 1 = 24/4 Solution 1 = 6 Solution 2 = Numerator 2/Denominator Solution 2 = -36/4 Solution 2 = -9 [U]As a solution set, our answers would be:[/U] (Solution 1, Solution 2) = (6, -9) Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Divide 73 into two parts whose product is 40 Our first part is x Our second part is 73 - x The product of the two parts is: x(73 - x) = 40 Multiplying through, we get: -x^2 + 73x = 402 Subtract 40 from each side, we get: -x^2 + 73x - 402 = 0 This is a quadratic equation. To solve this, we type it in our search engine, choose "solve Quadratic", and we get: [LIST=1] [*]x = [B]6[/B] [*]x = [B]67[/B] [/LIST]

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers business cards for $0.15 each with a setup charge of $10. What numbers of business cards cost the same from either company Declare variables: [LIST] [*]Let b be the number of business cards. [/LIST] [U]Set up the cost function C(b) for Dunder Mifflin:[/U] C(b) = Cost to print each business card * b + Setup Charge C(b) = 0.1b + 15 [U]Set up the cost function C(b) for Werham Hogg:[/U] C(b) = Cost to print each business card * b + Setup Charge C(b) = 0.15b + 10 The phrase [I]cost the same[/I] means we set both C(b)'s equal to each other and solve for b: 0.1b + 15 = 0.15b + 10 Solve for [I]b[/I] in the equation 0.1b + 15 = 0.15b + 10 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 0.1b and 0.15b. To do that, we subtract 0.15b from both sides 0.1b + 15 - 0.15b = 0.15b + 10 - 0.15b [SIZE=5][B]Step 2: Cancel 0.15b on the right side:[/B][/SIZE] -0.05b + 15 = 10 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 15 and 10. To do that, we subtract 15 from both sides -0.05b + 15 - 15 = 10 - 15 [SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE] -0.05b = -5 [SIZE=5][B]Step 5: Divide each side of the equation by -0.05[/B][/SIZE] -0.05b/-0.05 = -5/-0.05 b = [B]100[/B]

Dylan is playing darts. He hit the bullseye on 5 out of his last 20 tosses. Considering this data, how many bullseyes would you expect Dylan to get during his next 16 tosses? We have a proportion of bullseyes to tosses where b is the number of bullseyes for 16 tosses: 5/20 = b/16 [URL='https://www.mathcelebrity.com/prop.php?num1=5&num2=b&den1=20&den2=16&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into our search engine[/URL] and we get: b = [B]4[/B]

Each side of a square is lengthened by 3 inches . The area of this new, larger square is 25 square inches. Find the length of a side of the original square. area of a square is s^2 New square has sides s + 3, so the area of 25 is: (s + 3)^2 = 25 [URL='https://www.mathcelebrity.com/1unk.php?num=%28s%2B3%29%5E2%3D25&pl=Solve']Solving for s[/URL], we get: s = [B]2[/B]

Emily is three years older than twice her sister Marys age. The sum of their ages is less than 30. What is the greatest age Mary could be? Let e = Emily's age and m = Mary's age. We have the equation e = 2m + 3 and the inequality e + m < 30 Substitute the equation for e into the inequality: 2m + 3 + m < 30 Add the m terms 3m + 3 < 30 Subtract 3 from each side of the inequality 3m < 27 Divide each side of the inequality by 3 to isolate m m < 9 Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.

Equation 2y+5x=40. Interprt the intercepts Y intercept is when X = 0 2y + 5(0) = 40 2y = 40 Divide each side by 2 [B]y = 20 [/B] X intercept is when Y = 0 2(0) + 5x = 40 5x = 40 Divide each side by 5 [B]x = 8[/B]

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Erik is rolling two regular six-sided number cubes. What is the probability that he will roll an even number on one cube and a prime number on the other? P(Even on first cube) = (2,4,6) / 6 total choices P(Even on first cube) = 3/6 P(Even on first cube) = 1/2 <-- [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F6&frac2=3%2F8&pl=Simplify']Using our fraction simplify calculator[/URL] P(Prime on second cube) = (2,3,5) / 6 total choices P(Prime on second cube) = 3/6 P(Prime on second cube) = 1/2 <-- [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F6&frac2=3%2F8&pl=Simplify']Using our fraction simplify calculator[/URL] Since each event is independent, we have: P(Even on the first cube, Prime on the second cube) = P(Even on the first cube) * P(Prime on the second cube) P(Even on the first cube, Prime on the second cube) = 1/2 * 1/2 P(Even on the first cube, Prime on the second cube) = [B]1/4[/B]

Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3 times as many 3-cent stamps as 37-cent stamps and half the number of 5-cent stamps as 37-cent stamps. The value of the stamps and coins is $8.28. How many 37-cent stamps does Erin have? Number of stamps: [LIST] [*]Number of 37 cent stamps = s [*]Number of 3-cent stamps = 3s [*]Number of 5-cent stamps = 0.5s [/LIST] Value of stamps and coins: [LIST] [*]37 cent stamps = 0.37s [*]3-cent stamps = 3 * 0.03 = 0.09s [*]5-cent stamps = 0.5 * 0.05s = 0.025s [*]Quarter, 2 dime, 7 pennies = 0.52 [/LIST] Add them up: 0.37s + 0.09s + 0.025s + 0.52 = 8.28 Solve for [I]s[/I] in the equation 0.37s + 0.09s + 0.025s + 0.52 = 8.28 [SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE] (0.37 + 0.09 + 0.025)s = 0.485s [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 0.485s + 0.52 = + 8.28 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 0.52 and 8.28. To do that, we subtract 0.52 from both sides 0.485s + 0.52 - 0.52 = 8.28 - 0.52 [SIZE=5][B]Step 4: Cancel 0.52 on the left side:[/B][/SIZE] 0.485s = 7.76 [SIZE=5][B]Step 5: Divide each side of the equation by 0.485[/B][/SIZE] 0.485s/0.485 = 7.76/0.485 s = [B]16[/B] [URL='https://www.mathcelebrity.com/1unk.php?num=0.37s%2B0.09s%2B0.025s%2B0.52%3D8.28&pl=Solve']Source[/URL]

Explain the steps you would take to find an equation for the line perpendicular to 4x - 5y = 20 and sharing the same y-intercept Get this in slope-intercept form by adding 5y to each side: 4x - 5y + 5y = 5y + 20 Cancel the 5y's on the left side and we get: 5y + 20 = 4x Subtract 20 from each side 5y + 20 - 20 = 4x - 20 Cancel the 20's on the left side and we get: 5y = 4x - 20 Divide each side by 5: 5y/5 = 4x/5 - 4 y = 4x/5 - 4 So we have a slope of 4/5 to find our y-intercept, we set x = 0: y = 4(0)/5 - 4 y = 0 - 4 y = -4 If we want a line perpendicular to the line above, our slope will be the negative reciprocal: The reciprocal of 4/5 is found by flipping the fraction making the numerator the denominator and the denominator the numerator: m = 5/4 Next, we multiply this by -1: -5/4 So our slope-intercept of the perpendicular line with the same y-intercept is: [B]y = -5x/4 - 4[/B]

Express the confidence interval 0.039 < p < 0.479 in the form of p E. We find the range of this interval: Range = Upper Bound - Lower Bound Range = 0.479 - 0.039 Range = 0.44 Each piece on opposite sides of p gets: 0.44/2 = 0.22 So our expression becomes [B]p 0.22 [MEDIA=youtube]FGZcvcuWCpE[/MEDIA][/B]

ey/n + k = t for y Let's take this literal equation in pieces: Subtract k from each side: ey/n + k - k = t - k Cancel the k's on the left side, we have: ey/n = t - k Now multiply each side by n: ney/n = n(t - k) Cancel the n's on the left side, we have: ey = n(t - k) Divide each side by e: ey/e = n(t - k)/e Cancel the e's on the left side, we have: [B]y = n(t - k)/e[/B]

f - g = 1/4b for b Multiply each side of the equation by 4 to remove the 1/4 and isolate b: 4(f - g) = 4/4b 4/4 = 1, so we have: b = [B]4(f - g)[/B] [I]the key to this problem was multiplying by the reciprocal of the constant[/I]

f(x)=a(b)^x and we know that f(3)=17 and f(7)=3156. what is the value of b Set up both equations with values When x = 3, f(3) = 17, so we have a(b)^3 = 17 When x = 7, f(7) = 3156, so we have a(b)^7 = 3156 Isolate a in each equation a = 17/(b)^3 a = 3156/(b)^7 Now set them equal to each other 17/(b)^3 = 3156/(b)^7 Cross Multiply 17b^7 = 3156b^3 Divide each side by b^3 17b^4 = 3156 Divide each side by 17 b^4 = 185.6471 [B]b = 3.6912[/B]

f+g/e=r for g Subtract f from each side g/e = r - f Multiply each side by e [B]g = e(r - f)[/B]

F/B=(M-N*L)/D for L Cross multiply: DF/B = M - N*L Subtract M from each side: DF/B - M = -N*L Divide each side by -N [B]L = -DF/BN[/B]

Farmer Yumi has too many plants in her garden. If she starts out with 150 plants and is losing them at a rate of 4% each day, how long will it take for her to have 20 plants left? Round UP to the nearest day. We set up the function P(d) where d is the number of days sine she started losing plants: P(d) = Initial plants * (1 - Loss percent / 100)^d Plugging in our numbers, we get: 20 = 150 * (1 - 4/100)^d 20 = 150 * (1 - 0.04)^d Read left to right so it's easier to read: 150 * 0.96^d = 20 Divide each side by 150, and we get: 0.96^d = 0.13333333333 To solve this logarithmic equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=0.96%5Ed%3D0.13333333333&pl=Calculate']type it in our search engine[/URL] and we get: d = 49.35 The problem tells us to round up, so we round up to [B]50 days[/B]

Find a linear function f, given f(16)=-2 and f(-12)=-9. Then find f(0). We've got 2 points: (16, -2) and (-12, -9) Calculate the slope (m) of this line using: m = (y2 - y1)/(x2 - x1) m = (-9 - -2)/(-12 - 16) m = -7/-28 m = 1/4 The line equation is denoted as: y = mx + b Let's use the first point (x, y) = (16, -2) -2 = 1/4(16) + b -2 = 4 + b Subtract 4 from each side, and we get: b = -6 So our equation of the line is: y = 1/4x - 6 The questions asks for f(0): y = 1/4(0) - 6 y = 0 - 6 [B]y = -6[/B]

So x + y <=18 y = x + 1 x + x + 1 <=18 2x + 1 <= 18 Subtract 1 from both sides 2x <= 17 x<=8.5 --> 8 So we have {(0,1),(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9)}

find the difference between a mountain with an altitude of 1,684 feet above sea level and a valley 216 feet below sea level. Below sea level is the same as being on the opposite side of zero on the number line. To get the difference, we do the following: 1,684 - (-216) Since subtracting a negative is a positive, we have: 1,684 + 216 [B]1,900 feet[/B]

Find the largest of three consecutive even integers when six times the first integers is equal to five times the middle integer. Let the first of the 3 consecutive even integers be n. The second consecutive even integer is n + 2. The third (largest) consecutive even integer is n + 4. We are given 6n = 5(n + 2). Multiply through on the right side, and we get: 6n = 5n + 10 [URL='https://www.mathcelebrity.com/1unk.php?num=6n%3D5n%2B10&pl=Solve']Typing 6n = 5n + 10 into the search engine[/URL], we get n = 10. Remember, n was our smallest of 3 consecutive even integers. So the largest is: n + 4 10 + 4 [B]14[/B]

Consider the first 8 calculations of 7 to an exponent: [LIST] [*]7^1 = 7 [*]7^2 = 49 [*]7^3 = 343 [*]7^4 = 2,401 [*]7^5 = 16,807 [*]7^6 = 117,649 [*]7^7 = 823,543 [*]7^8 = 5,764,801 [/LIST] Take a look at the last digit of the first 8 calculations: 7, 9, 3, 1, 7, 9, 3, 1 The 7, 9, 3, 1 repeats through infinity. So every factor of 4, the cycle of 7, 9, 3, 1 restarts. Counting backwards from 2013, we know that 2012 is the largest number divisible by 4: 7^2013 = 7^2012 * 7^1 The cycle starts over after 2012. Which means the last digit of 7^2013 = [B]7 [MEDIA=youtube]Z157jj8R7Yc[/MEDIA][/B]

Formula for finding sides of a polygon given interior angle sum I: [B][URL='https://www.mathcelebrity.com/polygon-side-calculator.php?num=2700&pl=Calculate+Sides']Formula[/URL][/B] n = I/180 + 2 [B]Solve for n when I = 2700[/B] n = 2700/180 + 2 n = 15 + 2 n = [B]17 [MEDIA=youtube]MlhulFxUzXY[/MEDIA][/B]

Find two consecutive integers if the sum of their squares is 1513 Let the first integer be n. The next consecutive integer is (n + 1). The sum of their squares is: n^2 + (n + 1)^2 = 1513 n^2 + n^2 + 2n + 1 = 1513 2n^2 + 2n + 1 = 1513 Subtract 1513 from each side: 2n^2 + 2n - 1512 = 0 We have a quadratic equation. We [URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B2n-1512%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this into our search engine[/URL] and get: n = (-27, 28) Let's take the positive solution. The second integer is: n + 1 28 + 1 = 29

Find two consecutive positive integers such that the sum of their squares is 25. Let the first integer be x. The next consecutive positive integer is x + 1. The sum of their squares equals 25. We write this as:: x^2 + (x + 1)^2 Expanding, we get: x^2 + x^2 + 2x + 1 = 25 Group like terms: 2x^2 + 2x + 1 = 25 Subtract 25 from each side: 2x^2 + 2x - 24 = 0 Simplify by dividing each side by 2: x^2 + x - 12 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get x = 3 or x = -4. The problem asks for positive integers, so we discard -4, and use 3. This means, our next positive integer is 3 + 1 = 4. So we have [B](3, 4) [/B]as our answers. Let's check our work: 3^2 + 4^2 = 9 + 16 = 25

Find y if the line through (1,y) and (4,5) has a slope of 3. Slope formula is: m = (y2 - y1)/(x2 - x1) With m = 3, we have: 3 = (5 - y)/(4 - 1) 3 = (5 - y)/3 Cross multiply: 5 - y = 9 Subtract 5 from each side -y = 4 Multiply each side by -1 [B]y = -4[/B]

fixed cost 500 marginal cost 8 item sells for 30. Set up Cost Function C(x) where x is the number of items sold: C(x) = Marginal Cost * x + Fixed Cost C(x) = 8x + 500 Set up Revenue Function R(x) where x is the number of items sold: R(x) = Revenue per item * items sold R(x) = 30x Set up break even function (Cost Equals Revenue) C(x) = R(x) 8x + 500 = 30x Subtract 8x from each side: 22x = 500 Divide each side by 22: x = 22.727272 ~ 23 units for breakeven

Foster is centering a photo that is 9/1/2 inches wide on a scrapbook pages that is 10 inches wide. How far from each side of the pages should he put the picture? Enter your answer as a mixed number. First, determine your margins, which is the difference between the width and photo width, divided by 2. 10 - 9 & 1/2 = 1/2 1/2 / 2 = [B]1/4[/B]

FV-O/T=A for o Add O/T to each side: FV-O/T + O/T = A + O/T We have: A + O/T = FV Subtract A from each side: A - A + O/T = FV + A Cancelling the A's, e have: O/T = FV - A Cross multiply the T: [B]O = T(FV - A)[/B]

f^2+5g = 3md for d Divide each side by 3m to isolate d: (f^2+5g)/3m = 3md/3md Cancel the 3m on the right side and we get: d = [B](f^2+5g)/3m[/B]

Let g = Gary's pets and a = Abe's pets. We are given two equations: (1) g = a - 3 (2) a + g = 15 Substitute (1) into (2) a + (a - 3) = 15 Combine Like Terms: 2a - 3 = 15 Add 3 to each side: 2a = 18 Divide each side by 2 to isolate a: a = 9 --> Abe has 9 pets Substitute a = 9 into Equation (1) g = 9 - 3 g = 6 --> Gary has 6 pets

Let x equal the number of sticks he started with. We have: The equation is 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) - 0.2 = 19 Add 0.2 to each side: 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) = 19.2 Multiply each side by 5/4 (3/4 * (2/3 * (0.5x - 0.5) - 1/3) - 0.75) = 24 Multiply the inside piece first: 2/6x - 2/6 - 1/3 2/6x - 4/6 Now subtract 0.75 which is 3/4 2/6x - 4/6 - 3/4 4/6 is 8/12 and 3/4 is 9/12, so we have: 2/6x - 17/12 Now multiply by 3/4 6/24x - 51/48 = 24 Simplify: 1/4x - 17/16 = 24 Multiply through by 4 x - 17/4 = 96 Since 17/4 = 4.25, add 4.25 to each side x = 100.25 Since he did not cut up any sticks, he has a full stick to start with: So x = [B]101[/B]

Let x equal the number of sticks he started with. We have: The equation is 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) - 0.2 = 19 Add 0.2 to each side: 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) = 19.2 Multiply each side by 5/4 (3/4 * (2/3 * (0.5x - 0.5) - 1/3) - 0.75) = 24 Multiply the inside piece first: 2/6x - 2/6 - 1/3 2/6x - 4/6 Now subtract 0.75 which is 3/4 2/6x - 4/6 - 3/4 4/6 is 8/12 and 3/4 is 9/12, so we have: 2/6x - 17/12 Now multiply by 3/4 6/24x - 51/48 = 24 Simplify: 1/4x - 17/16 = 24 Multiply through by 4 x - 17/4 = 96 Since 17/4 = 4.25, add 4.25 to each side x = 100.25 Since he did not cut up any sticks, he has a full stick to start with: So x = [B]101[/B]

Gigis family left their house and drove 14 miles south to a gas station and then 48 miles east to a water park. How much shorter would their trip to the water park have been if they hadnt stopped at the gas station and had driven along the diagonal path instead? [IMG]https://mathcelebrity.com/community/data/attachments/0/pythag-diagonal.jpg[/IMG] Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=14&side2input=48&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we see the diagonal route would be: 50 miles The original trip distance was: Original Trip Distance = 14 + 48 Original Trip Distance = 62 miles Diagonal Trip was 50 miles, so the difference is: Difference = Original Trip Distance - Diagonal Distance Difference = 62 - 50 Difference = [B]12 miles[/B]

Given the rectangular prism below, if AB = 6 in., AD = 8 in. and BF = 24, find the length of FD. [IMG]http://www.mathcelebrity.com/images/math_problem_library_129.png[/IMG] If AB = 6 and AD = 8, by the Pythagorean theorem, we have BD = 10 from our [URL='http://www.mathcelebrity.com/pythag.php?side1input=6&side2input=8&hypinput=&pl=Solve+Missing+Side']Pythagorean Theorem[/URL] Calculator Using that, we have another right triangle which we can use the [URL='http://www.mathcelebrity.com/pythag.php?side1input=10&side2input=24&hypinput=&pl=Solve+Missing+Side']pythagorean theorem[/URL] calculator to get [B]FD = 26[/B]

Given: 9 - 4x = -19 Prove: x = 7 Solve for [I]x[/I] in the equation 9 - 4x = - 19 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 9 and -19. To do that, we subtract 9 from both sides -4x + 9 - 9 = -19 - 9 [SIZE=5][B]Step 2: Cancel 9 on the left side:[/B][/SIZE] -4x = -28 [SIZE=5][B]Step 3: Divide each side of the equation by -4[/B][/SIZE] -4x/-4 = -28/-4 x = [B]7[/B]

gy=-g/v+w for g Multiply each side of the equation by v to eliminate fractions: gvy = -g + vw Add g to each side: gvy + g = -g + g + vw Cancel the g's on the right side and we geT: gvy + g = vw Factor out g on the left side: g(vy + 1) = vw Divide each side of the equation by (vy + 1): g(vy + 1)/(vy + 1) = vw/(vy + 1) Cancel the (vy + 1) on the left side and we geT: g = [B]vw/(vy + 1)[/B]

Happy Paws charges $16.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $11.00 plus $2.75 per hour. Complete the equation and solve it to find for how many hours the total cost of the services is equal. Use the variable h to represent the number of hours. Happy Paws Cost: C = 16 + 1.5h Woof Watchers: C = 11 + 2.75h Setup the equation where there costs are equal 16 + 1.5h = 11 + 2.75h Subtract 11 from each side: 5 + 1.5h = 2.75h Subtract 1.5h from each side 1.25h = 5 Divide each side by 1.25 [B]h = 4[/B]

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of columns. Find the number of rows and columns. Let r be the number of rows and c be the number of columns. We have the area: rc = 324 Since rows equal columns, we have a square, and we can set r = c. c^2 = 324 Take the square root of each side: [B]c = 18[/B] Which means [B]r = 18[/B] as well. What we have is a garden of 18 x 18.

heat loss of a glass window varies jointly as the window's area and the difference between the outside and the inside temperature. a window 6 feet wide by 3 feet long loses 1,320 btu per hour when the temperature outside is 22 degree colder than the temperature inside. Find the heat loss through a glass window that is 3 feet wide by 5 feet long when the temperature outside is 9 degree cooler than the temperature inside. Find k of the equation: 6*3*22*k = 1320 396k = 1,320 k = 3.33333 [URL='https://www.mathcelebrity.com/1unk.php?num=396k%3D1320&pl=Solve']per our equation solver[/URL] Now, find the heat loss for a 3x5 window when the temperature is 9 degrees cooler than the temperature inside: 3*5*9*3.333333 = [B]450 btu per hour[/B]

Free Heptagon Calculator - Solves for side length, perimeter, and area of a heptagon.

Free Hexagon Calculator - This calculator solves for side length (s), Area (A), and Perimeter (P) of a hexagon given one of the 3 entries.

How many cubic inches are in a cubic foot? Volume of a cube with 12 inch (1 foot sides) = 12 * 12 * 12 = [B]1728 cubic inches[/B]

How much do 10 pieces of candy cost if 1000 pieces cost 100.00? Set up a proportion of pieces to cost 10/x = 1000/100 Divide the right side by 100 on top and bottom 10/x = 10/1 [B]x = 1[/B]

How much would you need to deposit in an account now in order to have $6000 in the account in 10 years? Assume the account earns 6% interest compounded monthly. We start with a balance of B. We want to know: B(1.06)^10 = 6000 B(1.79084769654) = 6000 Divide each side of the equation by 1.79084769654 to solve for B B = [B]3,350.37[/B]

How much would you need to deposit in an account now in order to have $6000 in the account in 15 years? Assume the account earns 8% interest compounded monthly. 8% compounded monthly = 8/12 = 0.6667% per month. 15 years = 15*12 = 180 months We want to know an initial balance B such that: B(1.00667)^180 = $6,000 3.306921B = $6,000 Divide each side by 3.306921 [B]B = $1,814.38[/B]

I only own blue blankets and red blankets. 8 out of every 15 blankets I have are red. If have i 45 blankets, how many are blue? If 8 out of 15 blankets are red, then 15 - 8 = 7 are blue So 7 out of every 15 blankets are blue. Set up a proportion of blue blankets to total blankets where b is the number of blue blankets in 45 blankets 7/15 = b/45 Cross multiply: If 2 proportions are equal, then we can do the following: Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2 15b = 45 * 7 15b = 315 To solve for b, divide each side of the equation by 15: 15b/15 = 315/15 Cancel the 15's on the left side and we get: b = [B]21[/B]

If $9000 grows to $9720 in 2 years find the simple interest rate. Simple interest formula is Initial Balance * (1 + tn) = Current Balance We have [LIST] [*]Initial Balance = 9000 [*]Current Balance = 9720 [*]n = 2 [/LIST] Plugging in these values, we get: 9000 * (1 + 2t) = 9720 Divide each side by 9000 1 + 2t = 1.08 Subtract 1 from each sdie 2t = 0.08 Divide each side by 2 t = [B]0.04 or 4%[/B]

If (a - b)/b = 3/7, which of the following must also be true? A) a/b = -4/7 B) a/b = 10/7 C) (a + b)/b = 10/7 D) (a - 2b)/b = -11/7 We can rewrite (a - b)/b as: a/b - b/b = 3/7 Since b/b = 1, we have: a/b - 1 = 3/7 Since -1 = -7/7, we have: a/b - 7/7 = 3/7 Add 7/7 to each side: a/b - 7/7 + 7/7 = 3/7 + 7/7 Cancel the 7/7 on the left side, we get: [B]a/b = 10/7 or Answer B [MEDIA=youtube]PKjLuwoso1U[/MEDIA][/B]

If 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners and 2 walkers? [U]Set up a joint variation equation, for the 100 runners, 4 bicyclists, and 5 walkers:[/U] 100 = 4 * 5 * k 100 = 20k [U]Divide each side by 20[/U] k = 5 <-- Coefficient of Variation [U]Now, take scenario 2 to determine the bicyclists with 20 runners and 2 walkers[/U] 20 = 2 * 5 * b 20 = 10b [U]Divide each side by 10[/U] [B]b = 2[/B]

If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numerator and denominator it become 4/5. Find the fractions. Convert 2 to a fraction with a denominator of 10: 20/2 = 10, so we multiply 2 by 10/10: 2*10/10 = 20/10 Add 2 to the numerator and denominator: (n + 2)/(d + 2) = 9/10 Cross multiply and simplify: 10(n + 2) = 9(d + 2) 10n + 20 = 9d + 18 Move constants to right side by subtracting 20 from each side and subtracting 9d: 10n - 9d = -2 Subtract 3 from the numerator and denominator: (n - 3)/(d - 3) = 4/5 Cross multiply and simplify: 5(n - 3) = 4(d - 3) 5n - 15 = 4d - 12 Move constants to right side by adding 15 to each side and subtracting 4d: 5n - 4d = 3 Build our system of equations: [LIST=1] [*]10n - 9d = -2 [*]5n - 4d = 3 [/LIST] Multiply equation (2) by -2: [LIST=1] [*]10n - 9d = -2 [*]-10n + 8d = -6 [/LIST] Now add equation (1) to equation (2) (10 -10)n (-9 + 8)d = -2 - 6 The n's cancel, so we have: -d = -8 Multiply through by -1: d = 8 Now bring back our first equation from before, and plug in d = 8 into it to solve for n: 10n - 9d = -2 10n - 9(8) = -2 10n - 72 = -2 To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=10n-72%3D-2&pl=Solve']plug this equation into our search engine[/URL] and we get: n = 7 So our fraction, n/d = [B]7/8[/B]

If 2x + y = 7 and y + 2z = 23, what is the average of x, y, and z? A. 5 B. 7.5 C. 15 D. 12.25 Add both equations to get all variables together: 2x + y + y + 2z = 23 + 7 2x + 2y + 2z = 30 We can divide both sides by 2 to simplify: (2x + 2y + 2z)/2= 30/2 x + y + z = 15 Notice: the average of x, y, and z is: (x + y + z)/3 But x + y + z = 15, so we have: 15/3 = [B]5, answer A[/B] [MEDIA=youtube]tOCAhhfMCLI[/MEDIA]

If 3(c + d) = 5, what is the value of c + d? A) 3/5 B) 5/3 C) 3 D) 5 Divide each side of the equation by 3 to [U]isolate[/U] c + d 3(c + d)/3 = 5/3 Cancel the 3's on the left side, we get: c + d = [B]5/3, or answer B[/B]

If 4x+7=xy-6, then what is the value of x, in terms of y Subtract xy from each side: 4x + 7 - xy = -6 Add 7 to each side: 4x - xy = -6 - 7 4x - xy = -13 Factor out x: x(4 - y) = -13 Divide each side of the equation by (4 - y) [B]x = -13/(4 - y)[/B]

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integer. [LIST] [*]Let the integer be "x". [*]Square the integer: x^2 [*]7 times the square: 7x^2 [*]5 times the integer: 5x [*]Add them together: 7x^2 + 5x [*][I]The result is[/I] means an equation, so we set 7x^2 + 5x equal to 2 [/LIST] 7x^2 + 5x = 2 [U]This is a quadratic equation. To get it into standard form, we subtract 2 from each side:[/U] 7x^2 + 5x - 2 = 2 - 2 7x^2 + 5x - 2 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=7x%5E2%2B5x-2%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get two solutions: [LIST=1] [*]x = 2/7 [*]x= -1 [/LIST] The problem asks for an integer, so our answer is x[B] = -1[/B]. [U]Let's check our work by plugging x = -1 into the quadratic:[/U] 7x^2 + 5x - 2 = 0 7(-1)^2 + 5(-1) - 2 ? 0 7(1) - 5 - 2 ? 0 0 = 0 So we verified our answer, [B]x = -1[/B].

If 800 feet of fencing is available, find the maximum area that can be enclosed. Perimeter of a rectangle is: 2l + 2w = P However, we're given one side (length) is bordered by the river and the fence length is 800, so we have: So we have l + 2w = 800 Rearranging in terms of l, we have: l = 800 - 2w The Area of a rectangle is: A = lw Plug in the value for l in the perimeter into this: A = (800 - 2w)w A = 800w - 2w^2 Take the [URL='https://www.mathcelebrity.com/dfii.php?term1=800w+-+2w%5E2&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']first derivative[/URL]: A' = 800 - 4w Now set this equal to 0 for maximum points: 4w = 800 [URL='https://www.mathcelebrity.com/1unk.php?num=4w%3D800&pl=Solve']Typing this equation into the search engine[/URL], we get: w = 200 Now plug this into our perimeter equation: l = 800 - 2(200) l = 800 - 400 l = 400 The maximum area to be enclosed is; A = lw A = 400(200) A = [B]80,000 square feet[/B]

if a+b=2 and a2-b2=-4, what is the value of a-b? a^2 - b^2 = -4 Factor this: (a + b)(a - b) = -4 We know from above, (a +b) = 2, so substitute: 2(a - b) = -4 Divide each side by 2 [B](a - b) = -2[/B]

if a number is added to its square, the result is 72. find the number. Let the number be n. We're given: n + n^2 = 72 Subtract 72 from each side, we get: n^2 + n - 72 = 0 This is a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this equation into our search engine[/URL], and we get: [B]n = 8 and n = -9[/B]

If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and distance? Divide each side by S to isolate T: D/S = S x T/S Cancel the S's on the right side: [B]T = D/S[/B]

If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG? By segment addition, we know that: EF + FG = EG Substituting in our values for the 3 segments, we get: 9x - 17 + 17x - 14 = 20x + 17 Group like terms and simplify: (9 + 17)x + (-17 - 14) = 20x - 17 26x - 31 = 20x - 17 Solve for [I]x[/I] in the equation 26x - 31 = 20x - 17 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 26x and 20x. To do that, we subtract 20x from both sides 26x - 31 - 20x = 20x - 17 - 20x [SIZE=5][B]Step 2: Cancel 20x on the right side:[/B][/SIZE] 6x - 31 = -17 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -31 and -17. To do that, we add 31 to both sides 6x - 31 + 31 = -17 + 31 [SIZE=5][B]Step 4: Cancel 31 on the left side:[/B][/SIZE] 6x = 14 [SIZE=5][B]Step 5: Divide each side of the equation by 6[/B][/SIZE] 6x/6 = 14/6 x = [B]2.3333333333333[/B]

If f(x) = 3x + 1 and g(x) = x^2 + 2x, find x when f(g(x)) = 10 [U]Evaluate f(g(x))[/U] f(g(x)) = 3(x^2 + 2x) + 1 f(g(x)) = 3x^2 + 6x + 1 [U]When f(g(x)) = 10, we have[/U] 10 = 3x^2 + 6x + 1 [U]Subtract 10 from each side:[/U] 3x^2 + 6x - 9 = 0 Divide each side of the equation by 3 x^2 + 2x - 3 = 0 Factor, we have: (x + 3)(x - 1) = 0 So x is either [B]1 or -3[/B]

If f(x) = ax^2 + bx + c and f(0) = 1 and f(-1) = 3, what is a - b Evaluate f(0) f(0) = a(0^2) + b(0) + c f(0) = a(0) + b(0) + c f(0) = c Since f(0) = 1, we have c = 1 Evaluate f(-1) f(-1) = a(-1^2) + b(-1) + c f(-1) = a(1) - b + c f(-1) = a - b + c Since f(-1) = 3, we have: a - b + c = -3 We learned above that f(0) = 1, so c = 1. Plug c = 1 into f(-1) a - b + 1 = -3 Subtract 1 from each side: a - b + 1 - 1 = -3 - 1 Cancel the 1's on the left side and we get: a - b = [B]-4[/B]

If it is raining, I will get wet if I go outside. I went outside and got wet. Do I know it was raining when I went outside? [B]No. [/B] It could have been sunny and a sprinkler or a hose was on.

If Mr hernandez has 5 times as many students as Mr daniels and together they have 150 students how many students do each have? Let h = Mr. Hernandez's students and d = Mr. Daniels students. We are given two equations: (1) h = 5d (2) d + h = 150 Substitute equation (1) into equation (2) d + (5d) = 150 Combine like terms: 6d = 150 Divide each side of the equation by 6 to isolate d d = 25 <-- Mr. Daniels Students Now, plug the value for d into equation (1) h = 5(25) h = 125 <-- Mr. Hernandez students

if n(A) = 6, n(A intersect B) = 2 and n(A union B) = 11, find n(B). n(A union B) = n(A) + n(B) - n(A intersect B) Plugging in our given values, we have: 11 = 6 + n(B) - 2 11 = 4 + n(B) Subtract 4 from each side: [B]n(B) = 7[/B]

If p+4=2 and q-3=2, what is the value of qp? Isolate p by subtracting 4 from each side using our [URL='http://www.mathcelebrity.com/1unk.php?num=p%2B4%3D2&pl=Solve']equation calculator[/URL] p = -2 Isolate q by adding 3 to each side using our [URL='http://www.mathcelebrity.com/1unk.php?num=q-3%3D2&pl=Solve']equation calculator[/URL]: q = 5 pq = (-2)(5) [B]pq = -10[/B]

If tanx = 3/4 ,what is cosx? tan(x) = sin(x)/cos(x), so we have: sin(x)/cos(x) = 3/4 cross multiply: 4sin(x) = 3cos(x) Divide each side by 3 to isolate cos(x): cos(x) = [B]4sin(x)/3 [/B]

If the circumference of a circular rug is 16? feet, then what is the area of the rug in terms of pi C = 2pir, so we have: C = 16? 16? = 2?r Divide each side by 2?: r = 16?/2? r = 8 Now, the area of a circle A is denoted below: A = ?r^2 Given r = 8 from above, we have: A = ?(8)^2 A = [B]64?[/B]

Let a be the cost of the ball and b be the cost of the bat: We're given 2 equations: [LIST=1] [*]a + b = 1.10 [*]b = a + 1 [/LIST] Substitute equation (2) into equation (1) for b: a + a + 1 = 1.10 Combine like terms: 2a + 1 = 1.10 Subtract 1 from each side: 2a + 1 - 1 = 1.10 - 1 2a = 0.10 Divide each side by 2: 2a/2 = 0.10/2 a = [B]0.05[/B] [MEDIA=youtube]79q346Hy7R8[/MEDIA]

If the equation of a line passes through the points (1, 3) and (0, 0), which form would be used to write the equation of the line? [URL='https://www.mathcelebrity.com/slope.php?xone=1&yone=3&slope=+&xtwo=0&ytwo=0&bvalue=+&pl=You+entered+2+points']Typing (1,3),(0,0) into the search engine[/URL], we get a point-slope form: [B]y - 3 = 3(x - 1)[/B] If we want mx + b form, we have: y - 3 = 3x - 3 Add 3 to each side: [B]y = 3x[/B]

If the perimeter of a rectangular field is 120 feet and the length of one side is 25 feet, how wide must the field be? The perimeter of a rectangle P, is denoted as: P = 2l + 2w We're given l = 25, and P = 120, so we have 2(25) + 2w = 120 Simplify: 2w + 50 = 120 [URL='https://www.mathcelebrity.com/1unk.php?num=2w%2B50%3D120&pl=Solve']Typing this equation into our search engine[/URL], we get: [B]w = 35[/B]

if the point (.53,y) is on the unit circle in quadrant 1, what is the value of y? Unit circle equation: x^2 + y^2 = 1 Plugging in x = 0.53, we get (0.53)^2 + y^2 = 1 0.2809 + y^2 = 1 Subtract 0.2809 from each side: y^2 = 0.7191 y = [B]0.848[/B]

if the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y? Set up our given ratio: 2x/5y = 3/4 Cross multiply: 2x * 4 = 5y * 3 8x = 15y Divide each side by 8: 8x/8 = 15y/8 x = 15y/8 Now divide each side by y to find x/y: x/y = 15y/8y x/y =[B] 15/8[/B]

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find the time taken by aeroplane to cover 1200m initially. We know from the distance formula (d) using rate (r) and time (t) that: d = rt Regular speed: 1200 = rt Divide each side by t, we get: r = 1200/t Reduced speed. 20 minutes = 60/20 = 1/3 of an hour. So we multiply 1,200 by 3 3600 = (r - 40)(t + 1/3) If we multiply 3 by (t + 1/3), we get: 3t + 1 So we have: 3600 = (r - 40)(3t + 1) Substitute r = 1200/t into the reduced speed equation: 3600 = (1200/t - 40)(3t + 1) Multiply through and we get: 3600 = 3600 - 120t + 1200/t - 40 Subtract 3,600 from each side 3600 - 3600 = 3600 - 3600 - 120t + 1200/t - 40 The 3600's cancel, so we get: - 120t + 1200/t - 40 = 0 Multiply each side by t: -120t^2 - 40t + 1200 = 0 We've got a quadratic equation. To solve for t, [URL='https://www.mathcelebrity.com/quadratic.php?num=-120t%5E2-40t%2B1200%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type this in our search engine[/URL] and we get: t = -10/3 or t = 3. Since time [I]cannot[/I] be negative, our final answer is: [B]t = 3[/B]

If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two numbers? Let the smaller number be n. The next consecutive even number is n + 2. Add them together to equal 226: n + n + 2 = 226 Solve for [I]n[/I] in the equation n + n + 2 = 226 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 + 1)n = 2n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2n + 2 = + 226 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 2 and 226. To do that, we subtract 2 from both sides 2n + 2 - 2 = 226 - 2 [SIZE=5][B]Step 4: Cancel 2 on the left side:[/B][/SIZE] 2n = 224 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2n/2 = 224/2 n = [B]112 [URL='https://www.mathcelebrity.com/1unk.php?num=n%2Bn%2B2%3D226&pl=Solve']Source[/URL][/B]

If V is the volume of a cube whose side is s, express s in terms of V: We know the Volume (V) of a cube with side length s is: V = s^3 Take the cube root of each side: V^1/3 = (s^3)^1/3 s = [B]V^1/3[/B]

If x/2y = 3/4, what is the value of y/x? Cross multiply this proportion: 4x = 3(2y) 4x = 6y Divide each side by x: 4x/x = 6y/x The x's cancel, and we have: 6y/x = 4 Divide each side by 6: 6y/6x = 4/6 The 6's on the left cancel, we have: y/x = 4/6 We can simplify this. [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F6&frac2=3%2F8&pl=Simplify']Type in Simplify 4/6 into the search engine[/URL], and we get 2/3. y/x = [B]2/3[/B]

If x2 is added to x, the sum is 42. x^2 + x = 42 Subtract 42 from both sides: x^2 + x - 42 = 0 We have a quadratic equation. Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-42%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation solver[/URL], we get: [B]x = 6 and x = -7 [/B] Since the problem does not state positive number solutions, they are both answers.

if x^2=y^3, for what value of z does x^{3z}= y^9 y^9 = y^3 * y^3, so if we square the right side, we must square the left side for equivalence: x^2 * x^2 = x^4 Therefore, x^4 = y^9 Going back to our problem, x^{3z}= y^9, so 3z = 4 Divide each side by 3 to isolate z, and we have: 3z/3 = 4/3 z = [B]4/3[/B]

If you can buy 1?3 of a box of chocolates for 6 dollars, how much can you purchase for 4 dollars? Write your answer as a fraction of a box. Set up a proportion of dollars to boxes where b is the number of boxes for $4: 6/1/3 = 4/b Cross multiply: 6b = 4/3 Multiply each side by 1/6 to isolate b: b = 4/18 [URL='https://www.mathcelebrity.com/gcflcm.php?num1=4&num2=18&num3=&pl=GCF+and+LCM']Type in GCF(4,18) into the search engine[/URL]. We get a greatest common factor of 2. Divide 4 and 18 in the fraction by 2. We get the reduced fraction of: [B]b = 2/9[/B]

If you divide my brother's age by 3 and then add 20, you will get my age, which is 31. What is my brothers age? Let b be the brother's age. We're given the following relationship for the brother's age and my age: b/3 + 20 = 31 Subtract 20 from each side: b/3 + 20 - 20 = 31 - 20 Cancel the 20's on the left side and we get: b/3 = 11 Cross multiply, and we get: b = 3 * 11 b = [B]33 [/B] Check our work using b = 33 for b/3 + 20 = 31: 33/3 + 20 ? 31 11 + 20 ? 31 31 = 31

If you triple a number and then add 10, you get one-half of the original number. What is the number? Let the number be n. We have: 3n + 10 = 0.5n Subtract 0.5n from each side 2.5n + 10 = 0 Subtract 10 from each side: 2.5n = -10 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2.5n%3D-10&pl=Solve']equation calculator,[/URL] we get: [B]n = -4[/B]

In 16 years, Ben will be 3 times as old as he is right now. Let Ben's age right now be b. We have, in 16 years, Ben's age will be 3 times what his age is now: b + 16 = 3b Subtract b from each side: 2b = 16 Divide each side by 2 [B]b = 8[/B] Check our work: 16 years from now, Ben's age is 8 + 16 = 24 And, 8 x 3 = 24

In 45 years, Gabriela will be 4 times as old as she is right now. Let a be Gabriela's age. we have: a + 45 = 4a Subtract a from each side: 3a = 45 Divide each side by a [B]a = 15[/B]

In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there in the class? We start by declaring variables for boys and girls: [LIST] [*]Let b be the number of boys [*]Let g be the number of girls [/LIST] We're given two equations: [LIST=1] [*]b = g + 5 [*]b + g = 13 [/LIST] Substitute equation (1) for b into equation (2): g + 5 + g = 13 Grouping like terms, we get: 2g + 5 = 13 Subtract 5 from each side: 2g + 5 - 5 = 13 - 5 Cancel the 5's on the left side and we get: 2g = 8 Divide each side of the equation by 2 to isolate g: 2g/2 = 8/2 Cancel the 2's on the left side and we get: g = 4 Substitute g = 4 into equation (1) to solve for b: b = 4 + 5 b = [B]9[/B]

In a given year, Houston has good air quality 48% of the days, moderate air quality 41% of the days, and unhealthy air quality 4% of the days. How many days per year do residents have unhealthy air quality? 4% of 365 days in a year = [B]14.6 days. If we are talking full days, we have 14.[/B]

In a newspaper, it was reported that yearly robberies in Springfield were up 40% to 77 in 2012 from 2011. How many robberies were there in Springfield in 2011? Let r be the number of robberies in 2011. We have: Robberies in 2012 = Robberies in 2011 * 1.4 77 = r * 1.4 Divide each side by 1.4 [B]r = 55[/B]

In a presidential election ohio had 20 electoral votes. This is 14 less than Texas had. How many electoral votes did Texas have? Let 0 = Ohio votes and t = Texas votes. We have: [LIST=1] [*]o = 20 [*]0 = t - 14 [/LIST] [U]Substitute (1) into (2)[/U] 20 = t - 14 [U]Add 14 to each side[/U] [B]t = 34[/B]

In a video game, Shar has to build a pen shaped like a right triangle for her animals. If she needs 16 feet of fence for the shortest side and 20 feet of fence for the longest side, how many feet of fencing is needed for the entire animal pen? Using our [URL='https://www.mathcelebrity.com/righttriangle.php?angle_a=&a=&angle_b=&b=16&c=20&pl=Calculate+Right+Triangle']right triangle calculator[/URL]: Remaining side = 12 Total fencing needed is 16 + 20 + 12 = [B]48 feet of fencing[/B]

In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game? In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game? Let w be the winning team's points, and l be the losing team's points. We have two equations: [LIST=1] [*]w + l = 41 [*]w - l = 27 [/LIST] Add the two equations: 2w = 68 Divide each side by 2 [B]w = 34[/B] Substitute this into (1) 34 + l = 41 Subtract 34 from each side [B]l = 7[/B] Check your work: [LIST=1] [*]34 + 7 = 41 <-- check [*]34 - 7 = 27 <-- check [/LIST] The final score of the game was [B]34 to 7[/B]. You could also use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=w+%2B+l+%3D+41&term2=w+-+l+%3D+27&pl=Cramers+Method']simultaneous equation solver[/URL].

In the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 Ric, Nancy, and Michael ages added up to 78 years. How old was Ric in 1980? Age in 1980: [LIST] [*]Let Michael's age be m [*]Nancy's age is 2m [*]Rick's age is 2 * 2m = 4m [/LIST] Age in 1992: [LIST] [*]Michael's age = m + 12 [*]Nancy's age is 2m + 12 [*]Rick's age is 2 * 2m = 4m + 12 [/LIST] Total them up: m + 12 + 2m + 12 + 4m + 12 = 78 Solve for [I]m[/I] in the equation m + 12 + 2m + 12 + 4m + 12 = 78 [SIZE=5][B]Step 1: Group the m terms on the left hand side:[/B][/SIZE] (1 + 2 + 4)m = 7m [SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE] 12 + 12 + 12 = 36 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 7m + 36 = + 78 [SIZE=5][B]Step 4: Group constants:[/B][/SIZE] We need to group our constants 36 and 78. To do that, we subtract 36 from both sides 7m + 36 - 36 = 78 - 36 [SIZE=5][B]Step 5: Cancel 36 on the left side:[/B][/SIZE] 7m = 42 [SIZE=5][B]Step 6: Divide each side of the equation by 7[/B][/SIZE] 7m/7 = 42/7 m = 6 Rick's age = 6 * 4 = [B]24 [URL='https://www.mathcelebrity.com/1unk.php?num=m%2B12%2B2m%2B12%2B4m%2B12%3D78&pl=Solve']Source[/URL] [/B]

In this class of 4/5 students are right handed. if there are 20 right handed students, what is the total number of students in this class? Let x be the total number of students in the class. We have: 4/5x = 20 Cross multiplying or using our [URL='http://www.mathcelebrity.com/1unk.php?num=4x%3D100&pl=Solve']equation calculator[/URL], we get: 4x = 100 Divide each side by 4 [B]x = 25[/B]

In x years time, Peter will be 23 years old. How old is he now? Let Peter's current age be a. In x years time means we add x to a, so we're given: a + x = 23 We want to find a, s we subtract x from each side to get: a + x - x = 23 - x Cancel the x terms on the left side and we get: a = [B]23 - x[/B]

Is 3 standard deviations above the means considered an outlier? [B]Yes.[/B] Using the empirical rule, we know that: [LIST] [*]68% of the values lie within one standard deviation of the mean [*]95% of the values lie within two standard deviations of the mean [/LIST] Anything out side of two standard deviations is considered an outlier.

Isabel will run less than 36 minutes today. So far, she has run 22 minutes. What are the possible numbers of additional minutes she will run? Set up our inequality. If she ran 22 minutes, we need to find an expression to find out the remaining minutes x + 22 < 36 Subtract 22 from each side: x < 14 Remember, she cannot run negative minutes, so our lower bound is 0, so we have: [B]0 < x < 14 [/B]

It takes 60 minutes for 7 people to paint 5 walls. How many minutes does it take 10 people to paint 10 walls? Rate * Time = Output Let "Rate" (r) be the rate at which [B]one person[/B] works. So we have: 7r * 60 = 5 Multiply through and simplify: 420r = 5 Divide each side by 5 to isolate r: r = 1/84 So now we want to find out how many minutes it takes 10 people to paint 10 walls using this rate: 10rt = 10 With r = 1/84, we have: 10t/84 = 10 Cross multiply: 10t = 840 To solve for t, we t[URL='https://www.mathcelebrity.com/1unk.php?num=10t%3D840&pl=Solve']ype this equation into our search engine[/URL] and we get: t = [B]84 minutes[/B]

j - m/4 = 4k for m Multiply each side by 4: 4j - 4m/4 = 4(4k) Simplify: 4j - m = 16k Add m to each side: 4j - m + m = 16k + m The m's cancel on the left side, so we have: 4j = 16k + m Subtract 16k from each side: 4j - 16k = 16k - 16k + m The 16k cancels on the right side, so we're left with: [B]m = 4j - 16k or 4(j - 4k)[/B]

Jack bought 7 tickets for a movie. He paid $7 for each adult ticket and $4 for each child ticket. Jack spent $40 for the tickets Let a = Number of adult tickets and c be the number of child tickets. [LIST=1] [*]7a + 4c = 40 [*]a + c = 7 [*]Rearrange (2): a = 7 - c [/LIST] Now substitute a in (3) into (1): 7(7 - c) + 4c = 40 49 - 7c + 4c = 40 49 - 3c = 40 Add 3c to each side and subtract 40: 3c = 9 Divide each side by 3: [B]c = 3 [/B] Substitute c = 3 into Equation (2) a + 3 = 7 Subtract 3 from each side: [B]a = 4[/B]

Jake used 5 boxes to pack 43.5 kg of books. If the boxes each weighed the same and held 8 books, what did each book weigh? [U]Set up equations were w is the weight of each book:[/U] [LIST=1] [*]5 boxes * 8 books * w = 43.5 [*]40w = 43.5 [/LIST] [U]Divide each side by 40[/U] [B]w = 1.0875 kg[/B]

Jamie spent $15.36 on several items at the store. he spent an equal amount on each item. if jamie spent $1.92 on each item, how many items did he buy? Let x equal the number of items Jamie bought. We have: 1.92x = 15.36 Divide each side by 1.92 [B]x = 8[/B]

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy. Let s be the number of sodas. Cost for the day is: Price per soda * s + Admission Price 4.25s + 42 We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55 [B]4.25s + 42 <= 55[/B] [B][/B] If the problems asks you to solve for s, we type it in our math engine and we get: Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 42 and 55. To do that, we subtract 42 from both sides 4.25s + 42 - 42 ? 55 - 42 [SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE] 4.25s ? 13 [SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE] 4.25s/4.25 ? 13/4.25 [B]s ? 3.06[/B]

Jane is twice a old as Joel. If their ages total 63, how old is Joel? Joel = j Jane = 2j j + 2j = 63 3j = 63 Divide each side by 3: 3j/3 = 63/j Cancel the 3's on the left side: j = [B]21[/B]

Jane received 183 more votes than jack. If jack received n votes, how many votes did Jane receive? Let j = jane's votes. We have, j + 183 = n Subtract 183 from each side: j = n - 183

Janet drove 395 kilometers and the trip took 5 hours. How fast was Janet traveling? Distance = Rate * Time We're given D = 395 and t = 5 We want Rate. We divide each side of the equation by time: Distance / Time = Rate * Time / Time Cancel the Time's on each side and we get: Rate = Distance / Time Plugging our numbers in, we get: Rate = 395/5 Rate = [B]79 kilometers[/B]

Janice says that the sum of the measures of the interior angles of an octagon is 900. Is Janice correct? Why or why not? She's [B]incorrect. [/B] The interior angle sum for a polygon is found with this formula: Interior Angle Sum = (sides - 2) x 180 Since an octagon has 8 sides, we have: Interior Angle Sum = (8 - 2) x 180 Interior Angle Sum = 6 x 180 Interior Angle sum = 1080

Jason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. How many nickels does Jason have? Let the number of nickels be n Let the number of dimes be d We're given two equations: [LIST=1] [*]d = n [*]0.05n + 0.1d = 2.25 [/LIST] Substitute equation (1) for d into equation (2): 0.05n + 0.1n = 2.25 Solve for [I]n[/I] in the equation 0.05n + 0.1n = 2.25 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (0.05 + 0.1)n = 0.15n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 0.15n = + 2.25 [SIZE=5][B]Step 3: Divide each side of the equation by 0.15[/B][/SIZE] 0.15n/0.15 = 2.25/0.15 n = [B]15[/B] [URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.1n%3D2.25&pl=Solve']Source[/URL]

[SIZE=6]Jason is 9 miles ahead of Joe running at 5.5 miles per hour and Joe is running at the speed of 7 miles per hour. How long does it take Joe to catch Jason? A. 3 hours B. 4 hours C. 6 hours D. 8 hours Distance formula is d = rt Jason's formula (Add 9 since he's ahead 9 miles): d = 5.5t + 9 Joe's formula: d = 7t Set both distance formulas equal to each other: 5.5t + 9 = 7t Subtract 5.5t from each side: 5.5t - 5.5t + 9 = 7t - 5.5t 1.5t = 9 Divide each side by 1.5: 1.5t/1.5 = 9/1.5 t = [B]6 hours[/B] [U]Check our work with t = 6[/U] Joe = 7(6) = 42 Jason = 5.5(6) + 9= 33 + 9 = 42 [MEDIA=youtube]qae3WCq9wzM[/MEDIA] [/SIZE]

Jay purchased tickets for a concert. To place the order, a handling charge of $7 per ticket was charged. A sales tax of 4% was also charged on the ticket price and the handling charges. If the total charge for four tickets was $407.68, what was the ticket price? Round to the nearest dollar. with a ticket price of t, we have the total cost written as: 1.04 * (7*4 + 4t)= 407.68 Divide each side by 1.04 1.04 * (7*4 + 4t)/1.04= 407.68/1.04 Cancel the 1.04 on the left side and we get: 7*4 + 4t = 392 28 + 4t = 392 To solve this equation for t, we [URL='https://www.mathcelebrity.com/1unk.php?num=28%2B4t%3D392&pl=Solve']type it in our math engine[/URL] and we get: t = [B]91[/B]

Jenny went shoe shopping. Now she has 5 more pairs than her brother. Together they have 25 pairs. How many pairs does Jenny have and how many pairs does her brother have? [U]Let j be the number of shoes Jenny has and b be the number of s hoes her brother has. Set up 2 equations:[/U] (1) b + j = 25 (2) j = b + 5 [U]Substitute (2) into (1)[/U] b + (b + 5) = 25 [U]Group the b terms[/U] 2b + 5 = 25 [U]Subtract 5 from each side[/U] 2b = 20 [U]Divide each side by b[/U] [B]b = 10 [/B] [U]Substitute b = 10 into (2)[/U] j = 10 + 5 [B]j = 15[/B]

Jill made 122 muffins. She put them into 3 boxes and has two muffins left. How many are in each box if they all contain the same amount of muffins? Let m equal the number of muffins per box. We're told that we have 3 boxes and 2 muffins left after filling up all 3 boxes. 3m + 2 = 122 To solve for m, we subtract 2 from each side: 3m + 2 - 2 = 122 - 2 Cancel the 2's on the left side and we get: 3m = 120 Divide each side by 3 to isolate m: 3m/3 = 120/3 Cancel the 3's on the left side and we get: m = [B]40[/B]

Jim was thinking of a number. Jim adds 20 to it, then doubles it and gets an answer of 99.2. What was the original number? Start with x. Add 20 to it x + 20 Double it 2(x + 20) Set this equal to 99.2 2(x + 20) = 99.2 Divide each side by 2: x + 20 = 49.6 Subtract 20 from each side: x = [B]29.6[/B]

John Adams was born in 1732 and became president in 1797. Harry S. Truman was born in 1884 and became President in 1945. Who was older when he became president? Adams: [URL='http://www.mathcelebrity.com/longdiv.php?num1=1797&num2=1732&pl=Subtract']1797 - 1732[/URL] = 65 Truman: [URL='http://www.mathcelebrity.com/longdiv.php?num1=1945&num2=1884&pl=Subtract']1945 - 1884[/URL] = 61 Adams was older.

John read the first 114 pages of a novel, which was 3 pages less than 1/3 Set up the equation for the number of pages (p) in the novel 1/3p - 3 = 114 Add 3 to each side 1/3p = 117 Multiply each side by 3 [B]p = 351[/B]

John read the first 114 pages of a novel, which was 3 pages less than 1/3 of the novel. Let n be the number of pages in the novel. We have: 1/3n - 3 = 114 Multiply each side by 3: n - 9 = 342 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=n-9%3D342&pl=Solve']equation solver[/URL], we get [B]n = 351[/B].

Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15% of his sales each month. How much will Jonathan and Raymond have to sell in order to earn the same amount each month? [U]Step 1: Set up Jonathan's sales equation S(m) where m is the amount of sales made each month:[/U] S(m) = Commission percentage * m + Base Salary 10% written as a decimal is 0.1. We want decimals to solve equations easier. S(m) = 0.1m + 1500 [U]Step 2: Set up Raymond's sales equation S(m) where m is the amount of sales made each month:[/U] S(m) = Commission percentage * m + Base Salary 15% written as a decimal is 0.15. We want decimals to solve equations easier. S(m) = 0.15m + 1200 [U]The question asks what is m when both S(m)'s equal each other[/U]: The phrase [I]earn the same amount [/I]means we set Jonathan's and Raymond's sales equations equal to each other 0.1m + 1500 = 0.15m + 1200 We want to isolate m terms on one side of the equation. Subtract 1200 from each side: 0.1m + 1500 - 1200 = 0.15m + 1200 - 1200 Cancel the 1200's on the right side and we get: 0.1m - 300 = 0.15m Next, we subtract 0.1m from each side of the equation to isolate m 0.1m - 0.1m + 300 = 0.15m - 0.1m Cancel the 0.1m terms on the left side and we get: 300 = 0.05m Flip the statement since it's an equal sign to get the variable on the left side: 0.05m = 300 To solve for m, we divide each side of the equation by 0.05: 0.05m/0.05 = 300/0.05 Cancelling the 0.05 on the left side, we get: m = [B]6000[/B]

Juan runs out of gas in a city. He walks 30yards west and then 16 yards south looking for a gas station. How far is he from his starting point? Juan is located on a right triangle. We calculate the hypotenuse: 30^2 + 16^2 = Hypotenuse^2 900 + 256 = Hypotenuse^2 Hypotenuse^2 = 1156 Take the square root of each side: [B]Hypotenuse = 34 yards[/B]

Juan spent at most $2.50 on apples and oranges. He bought 5 apples at $0.36 each. What is the most he spent on oranges? Let a be spending apples and o be spending on oranges, we have: [LIST=1] [*]a + o <= 2.36 <-- At most means less than or equal to [*]a = 5 * 0.36 = 1.8 [/LIST] Substitute (2) into (1) 1.8 + o <= 2.36 Subtract 1.8 from each side [B]o <= 0.56[/B]

Julia has a bucket of water that weighs 10lbs. The total weight is 99% water. She leaves the bucket outside overnight and some of the water evaporates, in the morning the water is only 98% of the total weight. What is the new weight? Setup the proportion: 0.99/10 = 0.98/w Using our [URL='http://www.mathcelebrity.com/prop.php?num1=0.99&num2=0.98&den1=10&den2=w&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get [B]w = 9.899 lbs[/B].

Julia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages and .10 cents for additional texts. How many texts did she send out? Let m be the number of messages. We have a cost function of: C(m) = 9 + 0.1(m - 600) We are given C(m) = 18.20 18.20 = 9 + 0.1(m - 600) 18.20 = 9 + 0.1m - 60 Combine like terms: 18.20 = 0.1m - 51 Add 51 to each side 0.1m = 69.20 Divide each side by 0.1 [B]m = 692[/B]

Julie is making a documentary about how Boxerville residents see their town. If she talks to 7 people, and each interview lasts 4 minutes, how long will the film be? 7 people * 4 minutes each = [B]28 minutes[/B]

k+m/-7=n for k Add m/7 to both sides [B]k = n + m/7[/B]

k=g-a/5 for g Add a/5 to each side; k + a/5 = g - a/5 + a/5 Cancel the a/5 terms on the right side, and we get: g = [B]k + a/5[/B]

Kamara has a square fence kennel area for her dogs in the backyard. The area of the kennel is 64 ft squared. What are the dimensions of the kennel? How many feet of fencing did she use? Explain. Area of a square with side length (s) is: A = s^2 Given A = 64, we have: s^2 = 64 [URL='https://www.mathcelebrity.com/radex.php?num=sqrt(64%2F1)&pl=Simplify+Radical+Expression']Typing this equation into our math engine[/URL], we get: s = 8 Which means the dimensions of the kennel are [B]8 x 8[/B]. How much fencing she used means perimeter. The perimeter P of a square with side length s is: P = 4s [URL='https://www.mathcelebrity.com/square.php?num=8&pl=Side&type=side&show_All=1']Given s = 8, we have[/URL]: P = 4 * 8 P = [B]32[/B]

Karin has 3 to spend in the arcade. The game she likes costs 50c per play. What are the possible numbers of times that she can play? [U]Let x = the number of games Karin can play with her money[/U] 0.5x = 3 [U]Divide each side by 0.5[/U] [B]x = 6[/B]

Let k = Katie's age and m = Mara's age. We have 2 equations: (1) k = 2m (2) k + m = 24 Substitute (1) into (2) (2m) + m = 24 Combine like terms: 3m = 24 Divide each side of the equation by 3 to isolate m m = 8 If m = 8, substituting into (1) or (2), we get k = 16. [MEDIA=youtube]Cu7gSgNkQPg[/MEDIA]

Kelly took clothes to the cleaners 3 times last month. First, she brought 4 shirts and 1 pair of slacks and paid11.45. Then she brought 5 shirts, 3 pairs of slacks, and 1 sports coat and paid 27.41. Finally, she brought 5 shirts and 1 sports coat and paid 16.94. How much was she charged for each shirt, each pair of slacks, and each sports coat? Let s be the cost of shirts, p be the cost of slacks, and c be the cost of sports coats. We're given: [LIST=1] [*]4s + p = 11.45 [*]5s + 3p + c = 27.41 [*]5s + c = 16.94 [/LIST] Rearrange (1) by subtracting 4s from each side: p = 11.45 - 4s Rearrange (3)by subtracting 5s from each side: c = 16.94 - 5s Take those rearranged equations, and plug them into (2): 5s + 3(11.45 - 4s) + (16.94 - 5s) = 27.41 Multiply through: 5s + 34.35 - 12s + 16.94 - 5s = 27.41 [URL='https://www.mathcelebrity.com/1unk.php?num=5s%2B34.35-12s%2B16.94-5s%3D27.41&pl=Solve']Group like terms using our equation calculator [/URL]and we get: [B]s = 1.99 [/B] <-- Shirt Cost Plug s = 1.99 into modified equation (1): p = 11.45 - 4(1.99) p = 11.45 - 7.96 [B]p = 3.49[/B] <-- Slacks Cost Plug s = 1.99 into modified equation (3): c = 16.94 - 5(1.99) c = 16.94 - 9.95 [B]c = 6.99[/B] <-- Sports Coat cost

Kendra is half as old as Morgan and 3 years younger than Lizzie. The total of their ages is 39. How old are they? Let k be Kendra's age, m be Morgan's age, and l be Lizzie's age. We're given: [LIST=1] [*]k = 0.5m [*]k = l - 3 [*]k + l + m = 39 [/LIST] Rearranging (1) by multiplying each side by 2, we have: m = 2k Rearranging (2) by adding 3 to each side, we have: l = k + 3 Substituting these new values into (3), we have: k + (k + 3) + (2k) = 39 Group like terms: (k + k + 2k) + 3 = 39 4k + 3 = 39 [URL='https://www.mathcelebrity.com/1unk.php?num=4k%2B3%3D39&pl=Solve']Type this equation into the search engine[/URL], and we get: [B]k = 9 [/B] Substitute this back into (1), we have: m = 2(9) [B]m = 18 [/B] Substitute this back into (2), we have: l = (9) + 3 [B][B]l = 12[/B][/B]

Kerry asked a bank teller to cash 390 check using 20 bills and 50 bills. If the teller gave her a total of 15 bills, how many of each type of bill did she receive? Let t = number of 20 bills and f = number of 50 bills. We have two equations. (1) 20t + 50f = 390 (2) t + f = 15 [U]Rearrange (2) into (3) for t, by subtracting f from each side:[/U] (3) t = 15 - f [U]Now substitute (3) into (1)[/U] 20(15 - f) + 50f = 390 300 - 20f + 50f = 390 [U]Combine f terms[/U] 300 + 30f = 390 [U]Subtract 300 from each side[/U] 30f = 90 [U]Divide each side by 30[/U] [B]f = 3[/B] [U]Substitute f = 3 into (3)[/U] t = 15 - 3 [B]t = 12[/B]

Kevin and randy have a jar containing 41 coins, all of which are either quarters or nickels. The total value of the jar is $7.85. How many of each type? Let d be dimes and q be quarters. Set up two equations from our givens: [LIST=1] [*]d + q = 41 [*]0.1d + 0.25q = 7.85 [/LIST] [U]Rearrange (1) by subtracting q from each side:[/U] (3) d = 41 - q [U]Now, substitute (3) into (2)[/U] 0.1(41 - q) + 0.25q = 7.85 4.1 - 0.1q + 0.25q = 7.85 [U]Combine q terms[/U] 0.15q + 4.1 = 7.85 [U]Using our [URL='http://www.mathcelebrity.com/1unk.php?num=0.15q%2B4.1%3D7.85&pl=Solve']equation calculator[/URL], we get:[/U] [B]q = 25[/B] [U]Substitute q = 25 into (3)[/U] d = 41 - 25 [B]d = 16[/B]

Kevin is 4 times old as Daniel and is also 6 years older than Daniel. Let k be Kevin's age and d be Daniel's age. We have 2 equations: [LIST=1] [*]k = 4d [*]k = d + 6 [/LIST] Plug (1) into (2): 4d = d + 6 Subtract d from each side: 4d - d = d - d + 6 Cancel the d terms on the right side and simplify: 3d = 6 Divide each side by 3: 3d/3 = 6/3 Cancel the 3 on the left side: d = 2 Plug this back into equation (1): k = 4(2) k = 8 So Daniel is 2 years old and Kevin is 8 years old

Kimberly wants to become a member of the desert squad at a big catering company very badly, but she must pass three difficult tests to do so. On the first Terrifying Tiramisu test she scored a 68. On the second the challenging Chocalate-Sprinkled Creme Brulee she scored a 72. If kimberly needs an average of 60 on all three tests to become a member on the squad what is the lowest score she can make on her third and final test This is a missing average problem. Given 2 scores of 68, 72, what should be score number 3 in order to attain an average score of 60? [SIZE=5][B]Setup Average Equation:[/B][/SIZE] Average = (Sum of our 2 numbers + unknown score of [I]x)/[/I]Total Numbers 60 = (68 + 72 + x)/3 [SIZE=5][B]Cross Multiply[/B][/SIZE] 68 + 72 + x = 60 x 3 x + 140 = 180 [SIZE=5][B]Subtract 140 from both sides of the equation to isolate x:[/B][/SIZE] x + 140 - 140 = 180 - 140 x = [B]40[/B]

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional amounts she will spend? The key word in this problem is [I]less than[/I]. So we know this is an inequality. Let s be Kira's possible spend. We have: s + 12 < 27 To solve for s in this inequality, we subtract 12 from each side: s + 12 - 12 < 27 - 12 Cancel the 12's on the left side, and we get: [B]s < 15 [/B] [I]The summary here is Kira can spend anything up to [U]but not including[/U] 15[/I]

Free Kites Calculator - This calculates perimeter and/or area of a kite given certain inputs such as short and long side, short and long diagonal, or angle between short and long side

Kris wants to fence in her square garden that is 40 feet on each side. If she places posts every 10 feet, how many posts will she need? Perimeter (P) of a square with side s: P = 4s Given s = 40, we have: P = 4(40) P = 160 feet 160 feet / 10 foot spaces = [B]16 posts[/B]

larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2 numbers Declare Variables for each number: [LIST] [*]Let l be the larger number [*]Let s be the smaller number [/LIST] We're given two equations: [LIST=1] [*]l = s + 12 [*]l + s = 74 [/LIST] Equation (1) already has l solved for. Substitute equation (1) into equation (2) for l: s + 12 + s = 74 Solve for [I]s[/I] in the equation s + 12 + s = 74 [SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE] (1 + 1)s = 2s [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2s + 12 = + 74 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 12 and 74. To do that, we subtract 12 from both sides 2s + 12 - 12 = 74 - 12 [SIZE=5][B]Step 4: Cancel 12 on the left side:[/B][/SIZE] 2s = 62 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2s/2 = 62/2 s = [B]31[/B] To solve for l, we substitute in s = 31 into equation (1): l = 31 + 12 l = [B]43[/B]

larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number? Declare variables for the 2 numbers: [LIST] [*]Let l be the larger number [*]Let s be the smaller number [/LIST] We're given two equations: [LIST=1] [*]l = s + 4 [*]l + s = 40 [/LIST] To get this problem in terms of the larger number l, we rearrange equation (1) in terms of l. Subtract 4 from each side in equation (1) l - 4 = s + 4 - 4 Cancel the 4's and we get: s = l - 4 Our given equations are now: [LIST=1] [*]s = l - 4 [*]l + s = 40 [/LIST] Substitute equation (1) into equation (2) for s: l + l - 4 = 40 Grouping like terms for l, we get: 2l - 4 = 40 Add 4 to each side: 2l - 4 + 4 = 40 + 4 Cancelling the 4's on the left side, we get 2l = 44 Divide each side of the equation by 2 to isolate l: 2l/2 = 44/2 Cancel the 2's on the left side and we get: l = [B]22[/B]

Larry Mitchell invested part of his $31,000 advance at 6% annual simple interest and the rest at 7% annual simple interest. If the total yearly interest from both accounts was $2,090, find the amount invested at each rate. Let x be the amount invested at 6%. Then 31000 - x is invested at 7%. We have the following equation: 0.06x + (31000 - x)0.07 = 2090 Simplify: 0.06x + 2170 - 0.07x = 2090 Combine like Terms -0.01x + 2170 = 2090 Subtract 2170 from each side -0.01x = -80 Divide each side by -0.01 x = [B]8000 [/B]at 6% Which means at 7%, we have: 31000 - 8000 = [B]23,000[/B]

Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden or a round garden with the fencing. Laura did some calculations and found that a circular garden would give her 1380 more square feet than a square garden. How many feet of fencing were in the roll that Laura found? (Round to the nearest foot.) Feet of fencing = n Perimeter of square garden = n Each side of square = n/4 Square garden's area = (n/4)^2 = n^2/16 Area of circle garden with circumference = n is: Circumference = pi * d n = pi * d Divide body tissues by pi: d = n/pi Radius = n/2pi Area = pi * n/2pi * n/2pi Area = pin^2/4pi^2 Reduce by canceling pi: n^2/4pi n^2/4 * 3.14 n^2/12.56 The problem says that the difference between the square's area and the circle's area is equal to 1380 square feet. Area of Circle - Area of Square = 1380 n^2/12.56 - n^2/16 = 1380 Common denominator = 200.96 (16n^2 - 12.56n^2)/200.96 = 1380 3.44n^2/200.96 = 1380 Cross multiply: 3.44n^2 = 277,324.8 n^2 = 80,617.7 n = 283.9 Nearest foot = [B]284[/B]

Laura spent half of her money on a necklace. She spent 14.60 of what was left having dinner with carolyn. if she had 3.90 left, how much money did she start out with? Let x equal Laura's starting money 1/2x = 14.60 + 3.90 1/2x = 18.5 Divide each side by 1/2 [B]x = $37[/B]

Laura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pounds. How much does Laura weigh? Let Laura weigh l and her dog weigh d. WE have: [LIST=1] [*]l = d + 45 [*]d + l = 85 [/LIST] Substitute equation (1) into Equation (2) for l: d + d + 45 = 85 Solve for [I]d[/I] in the equation d + d + 45 = 85 [SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE] (1 + 1)d = 2d [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2d + 45 = + 85 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 45 and 85. To do that, we subtract 45 from both sides 2d + 45 - 45 = 85 - 45 [SIZE=5][B]Step 4: Cancel 45 on the left side:[/B][/SIZE] 2d = 40 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2d/2 = 40/2 d = 20 From equation (1), we substitute d = 20: l = d + 45 l = 20 + 45 l = [B]65 pounds [URL='https://www.mathcelebrity.com/1unk.php?num=d%2Bd%2B45%3D85&pl=Solve']Source[/URL][/B]

Lauren wrote a total of 6 pages over 2 hours. How many hours will Lauren have to spend writing this week in order to have written a total of 9 pages? Solve using unit rates. 6 pages per 2 hours equals 6/2 = 3 pages per hour as a unit rate Set up equation using h hours: 3h = 9 Divide each side by 3 [B]h = 3[/B]

Let m = the number of pencils Michelle has. So, Leslie has m - 9 = 8. Add 9 to both sides: m = 17. So Michelle has 17 pencils, and Leslie has 8, which is 9 fewer than 17

Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What could N be? Is there more than one answer? For example, for 23 P(23) = 6 and S(23) = 5, but 23 could not be the N that we want since 23 <> 5 + 6 Let t = tens digit and o = ones digit P(n) = to S(n) = t + o P(n) + S(n) = to + t + o N = 10t + o Set them equal to each other N = P(N) + S(N) 10t + o = to + t + o o's cancel, so we have 10t = to + t Subtract t from each side, we have 9t = to Divide each side by t o = 9 So any two-digit number with 9 as the ones digit will work: [B]{19,29,39,49,59,69,79,89,99}[/B]

Linda takes classes at both Westside Community College and Pinewood Community College. At Westside, class fees are $98 per credit hour, and at Pinewood, class fees are $115 per credit hour. Linda is taking a combined total of 18 credit hours at the two schools. Suppose that she is taking w credit hours at Westside. Write an expression for the combined total dollar amount she paid for her class fees. Let p be the number of credit hours at Pinewood. We have two equations: [LIST] [*]98w for Westside [*]115p at Pinewood [*]w + p = 18 [*]Total fees: [B]98w + 115p[/B] [/LIST]

Liz harold has a jar in her office that contains 47 coins. Some are pennies and the rest are dimes. If the total value of the coins is 2.18, how many of each denomination does she have? [U]Set up two equations where p is the number of pennies and d is the number of dimes:[/U] (1) d + p = 47 (2) 0.1d + 0.01p = 2.18 [U]Rearrange (1) into (3) by solving for d[/U] (3) d = 47 - p [U]Substitute (3) into (2)[/U] 0.1(47 - p) + 0.01p = 2.18 4.7 - 0.1p + 0.01p = 2.18 [U]Group p terms[/U] 4.7 - 0.09p = 2.18 [U]Add 0.09p to both sides[/U] 0.09p + 2.18 = 4.7 [U]Subtract 2.18 from both sides[/U] 0.09p = 2.52 [U]Divide each side by 0.09[/U] [B]p = 28[/B] [U]Now substitute that back into (3)[/U] d =47 - 28 [B]d = 19[/B]

Lucas has nickels,dimes,and quarters in the ratio 1:3:2. If 10 of Lucas coins are quarters, how many nickels and dimes does Lucas have? 1 + 3 + 2 = 6. Quarters account for 2/6 which is 1/3 of the total coin count. Let x be the total number of coins. We have: 1/3x = 10 Multiply each side by 3 x = 30 We have the following ratios and totals: [LIST] [*]Nickels: 1/6 * 30 = [B]5 nickels[/B] [*]Dimes: 3/6 * 30 = [B]15 dimes[/B] [*]Quarters: 2/6 * 30 = [B]10 quarters[/B] [/LIST]

Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt? Let Dan's debt be d. Let Luke's debt be l. We're given two equations: [LIST=1] [*]d + l = 72 [*]l = 3d [/LIST] Substitute equation (2) for l into equation (1): d + 3d = 72 Solve for [I]d[/I] in the equation d + 3d = 72 [SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE] (1 + 3)d = 4d [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 4d = + 72 [SIZE=5][B]Step 3: Divide each side of the equation by 4[/B][/SIZE] 4d/4 = 72/4 d = [B]18[/B]

M/n = p-6 for m Solve this literal equation by multiplying each side by n to isolate M: Mn/n = n(p - 6) Cancelling the n terms on the left side, we get: [B]M = n(p - 6)[/B]

m/x = k-6 for m To solve this literal equation, multiply each side by x: x(m/x) = x(k - 6) The x's cancel on the left side, so we get: m = [B]x(k - 6)[/B]

m=u/k-r/k for k Multiply both sides by k to eliminate the k denominator: km = uk/k - rk/k Cancel the k's on the right side and we get km = u - r Divide each side by m: km/m = (u - r)/m Cancel the m on the left side: [B]k = (u - r)/m[/B]

m=y-b/x-t for y Add b/x + t to each side: m + b/x + t = y - b/x - t + b/x + t Cancel b/x terms and t terms on the right side to get: y = [B]m + b/x + t[/B]

Manuel can pay for his car insurance on a monthly basis, but if he pays an entire year's insurance in advance, he'll receive a $40 discount. His discounted bill for the year would then be $632. What is the monthly fee for his insurance? His full bill F, is denoted as: F - 40 = 632 [URL='https://www.mathcelebrity.com/1unk.php?num=f-40%3D632&pl=Solve']If we add 40 to each side[/URL], we get: F = [B]$672[/B]

Marcela is having a presidential debate watching party with all of her friends, She will be making chicken wings and hot dogs. Each chicken wing costs $2 to make and each hot dog costs $3. She needs to spend at least $500. Marcela knows that she will make more than 50 chicken wings and hot dogs combined. She also knows that she will make less than 120 chicken wings and less that 100 hot dogs. What are her inequalities? Let c be the number of chicken wings and h be the number of hot dogs. Set up the given inequalities: [LIST=1] [*]c + h > 50 [I]Marcela knows that she will make more than 50 chicken wings and hot dogs combined.[/I] [*]2c + 3h >= 500 [I]She needs to spend at least $500[/I] [*]c < 120 [I]She also knows that she will make less than 120 chicken wings[/I] [*]h < 100 [I]and less that 100 hot dogs[/I] [/LIST]

Maria bought 7 boxes. A week later half of all her boxes were destroyed in a fire. There are now only 22 boxes left. With how many did she start? [U]Let x be the starting box number. We have:[/U] (x + 7)/2 = 22 [U]Cross multiply[/U] x + 7 = 44 [U]Subtract 7 from each side[/U] [B]x = 37[/B]

Let n be the number of nickels and q be the number of quarters. We have two equations: (1) n + q = 24 (2) 0.05n + 0.25q = 3 Rearrange (1) to solve for n in terms of q for another equation (3) (3) n = 24 - q Plug (3) into (2) 0.05(24 - q) + 0.25q = 3 Multiply through: 1.2 - 0.05q + 0.25q = 3 Combine q terms 0.2q + 1.2 = 3 Subtract 1.2 from each side: 0.2q = 1.8 Divide each side by 0.2 [B]q = 9[/B]

Marty is 3 years younger than 6 times his friend Warrens age. The sum of their ages is greater than 11. What is the youngest age Warren can be? Let m be Marty's age and w be Warren's age. We have two equations: (1) m = 6w - 3 (2) m + w > 11 Plug (1) into (2) 6w - 3 + w > 11 Combine w terms 7w - 3 > 11 Add 3 to each side 7w > 14 Divide each side by 7 w > 2 which means [B]w = 3[/B] as the youngest age.

Volume of rectangular prism is: V = lwh Plugging in the numbers you gave: 195 = (6)(5)h 195 = 30h Divide each side by 30 h = 6.5 6.5 feet is 6 feet, 6 inches. This is 2 inches more than your actor, so [B]yes[/B], he will fit in the box standing up.

Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour babysitting. Write and solve and inequality to find how many hours she will need to babysit to buy the dress. Subtract remaining amount needed after savings: 112 - 40 = 72 Let h be her hourly wages for babysitting. We have the equation: [B]9h = 72[/B] Divide each side by 9 [B]h = 8[/B]

Matthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If the weight of all three pets is 35 pounds, ow much does his hamster weigh? Setup weights and relations: [LIST] [*]Hamster weight: w [*]Cat weight: w + 10 [*]Dog weight:w + 10 [/LIST] Add all the weights up: w + w + 10 + w + 10 = 35 Solve for [I]w[/I] in the equation w + w + 10 + w + 10 = 35 [SIZE=5][B]Step 1: Group the w terms on the left hand side:[/B][/SIZE] (1 + 1 + 1)w = 3w [SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE] 10 + 10 = 20 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 3w + 20 = + 35 [SIZE=5][B]Step 4: Group constants:[/B][/SIZE] We need to group our constants 20 and 35. To do that, we subtract 20 from both sides 3w + 20 - 20 = 35 - 20 [SIZE=5][B]Step 5: Cancel 20 on the left side:[/B][/SIZE] 3w = 15 [SIZE=5][B]Step 6: Divide each side of the equation by 3[/B][/SIZE] 3w/3 = 15/3 w =[B] 5[/B] [B] [URL='https://www.mathcelebrity.com/1unk.php?num=w%2Bw%2B10%2Bw%2B10%3D35&pl=Solve']Source[/URL][/B]

Max and Bob went to the store. Max bought 2 burgers and 2 drinks for $5.00. Bob bought 3 burgers and 1 drink for $5.50. How much is each burger and drink? [U]Set up the givens where b is the cost of a burger and d is the cost of a drink:[/U] Max: 2b + 2d = 5 Bob: 3b + d = 5.50 [U]Rearrange Bob's equation by subtracting 3b from each side[/U] (3) d = 5.50 - 3b [U]Now substitute that d equation back into Max's Equation[/U] 2b + 2(5.50 - 3b) = 5 2b + 11 - 6b = 5 [U]Combine b terms:[/U] -4b + 11 = 5 [U]Subtract 11 from each side[/U] -4b = -6 [U]Divide each side by -4[/U] b = 3/2 [B]b = $1.50[/B] [U]Now plug that back into equation (3):[/U] d = 5.50 - 3(1.50) d = 5.50 - 4.50 [B]d = $1.00[/B]

Michelle and Julie sold 65 cupcakes. If Julie sold 9 more cupcakes than Michelle, how many cupcakes did each of them sell? Let m = Michelle's cupcakes and j = Julie's cupcakes. We have two equations: m + j = 65 j = m + 9 Substituting, we get: m + (m + 9) = 65 Combine like terms, we get: 2m + 9 = 65 Subtract 9 from each side: 2m = 56 Divide each side by 2 to isolate m m = 28 If m = 28, then j = 28 + 9 = 37 So (m, j) = (28, 37)

mike went to canalside with $40 to spend. he rented skates for $10 and paid $3 per hour to skate.what is the greatest number of hours Mike could have skated? Let h be the number of hours of skating. We have the cost function C(h): C(h) = Hourly skating rate * h + rental fee C(h) = 3h + 10 The problem asks for h when C(h) = 40: 3h + 10 = 40 To solve this equation for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=3h%2B10%3D40&pl=Solve']type it in our search engine[/URL] and we get: h = [B]10[/B]

Mindy and troy combined ate 9 pieces of the wedding cake. Mindy ate 3 pieces of cake and troy had 1/4 of the total cake. Write an equation to determine how many pieces of cake (c) that were in total Let c be the total number of pieces of cake. Let m be the number of pieces Mindy ate. Let t be the number of pieces Troy ate. We have the following given equations: [LIST] [*]m + t = 9 [*]m = 3 [*]t = 1/4c [/LIST] Combining (2) and (3) into (1), we have: 3 + 1/4c = 9 Subtract 3 from each side: 1/4c = 6 Cross multiply: [B]c = 24 [MEDIA=youtube]aeqWQXr5f_Y[/MEDIA][/B]

Mr. Jimenez has a pool behind his house that needs to be fenced in. The backyard is an odd quadrilateral shape and the pool encompasses the entire backyard. The four sides are 1818a, 77b, 1111a, and 1919b in length. How much fencing? (the length of the perimeter) would he need to enclose the pool? The perimeter P is found by adding all 4 sides: P = 1818a + 77b + 1111a + 1919b Group the a and b terms P = (1818 + 1111)a + (77 + 1919b) [B]P = 2929a + 1996b[/B]

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use p to represent the other money he can spend there. 2 kids and Mr. Smith = 3 people. Total Ticket Cost is 3 people * 7 per ticket = 21 If he has 125 to spend, we have the following inequality using less than or equal to (<=) since he can spend up to or less than 125: p + 21 <= 125 Subtract 21 from each side: [B]p <= 104[/B]

Mr. Winkle downloaded 34 more songs than Mrs. Winkle downloaded. Together they downloaded 220 songs. How many songs did each download? Let x = Mr. Winkle downloads and y = Mrs. Winkle downloads. We then have x = y + 34 and x + y = 220. Substitute equation 1 into equation 2, we have: (y + 34) + y = 220 2y + 34 = 220 Subtract 34 from each side: 2y = 186 Divide each side by 2: y = 93 (Mrs. Winkle) x = 93 + 34 x = 127 (Mr. Winkle)

mx=ac/np for n Cross multiply: mnpx = ac Divide each side by mpx: mnpx/mpx = ac/mpx Cancel the mpx on the right side: n = [B]ac/mpx[/B]

n + .07n = $90.95 Group like terms: 1.07n = $90.95 Solve for [I]n[/I] in the equation 1.07n = 90.95 [SIZE=5][B]Step 1: Divide each side of the equation by 1.07[/B][/SIZE] 1.07n/1.07 = 90.95/1.07 n = [B]85 [URL='https://www.mathcelebrity.com/1unk.php?num=1.07n%3D90.95&pl=Solve']Source[/URL][/B]

n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6 Solve for [I]n[/I] in the equation n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 + 2 + 3 + 4)n = 10n [SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE] 2 + 3 + 4 + 5 + 6 = 20 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 10n = + 20 [SIZE=5][B]Step 4: Divide each side of the equation by 10[/B][/SIZE] 10n/10 = 20/10 n = [B]2[/B]

n + 9n - 8 - 5 = 2n + 3 Solve for [I]n[/I] in the equation n + 9n - 8 - 5 = 2n + 3 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 + 9)n = 10n [SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE] -8 - 5 = -13 [SIZE=5][B]Step 3: Form modified equation[/B][/SIZE] 10n - 13 = 2n + 3 [SIZE=5][B]Step 4: Group variables:[/B][/SIZE] We need to group our variables 10n and 2n. To do that, we subtract 2n from both sides 10n - 13 - 2n = 2n + 3 - 2n [SIZE=5][B]Step 5: Cancel 2n on the right side:[/B][/SIZE] 8n - 13 = 3 [SIZE=5][B]Step 6: Group constants:[/B][/SIZE] We need to group our constants -13 and 3. To do that, we add 13 to both sides 8n - 13 + 13 = 3 + 13 [SIZE=5][B]Step 7: Cancel 13 on the left side:[/B][/SIZE] 8n = 16 [SIZE=5][B]Step 8: Divide each side of the equation by 8[/B][/SIZE] 8n/8 = 16/8 n = [B]2[/B]

n + 9n - 90 = 0 Solve for [I]n[/I] in the equation n + 9n - 90 = 0 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 + 9)n = 10n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 10n - 90 = [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -90 and 0. To do that, we add 90 to both sides 10n - 90 + 90 = 0 + 90 [SIZE=5][B]Step 4: Cancel 90 on the left side:[/B][/SIZE] 10n = 90 [SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE] 10n/10 = 90/10 n = [B]9[/B]

n + n/2 + n/4 + n/8 + n/16 = 19,375 Convert to like fractions with a denominator of 16: 16n/16 + 8n/16 + 4n/16 + +2n/16 + n/16 = 19,375 31n/16 = 19,375 Cross multiply: 31n = 19,375 * 16 31n = 310000 Divide each side by 1: 31n/31 = 310000/31 n = [B]10,000[/B]

n - n = 10 - n Solve for [I]n[/I] in the equation n - n = 10 - n [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (1 - 1)n = 0n = 0 [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] = - n + 10 [SIZE=5][B]Step 3: Group variables:[/B][/SIZE] We need to group our variables and -n. To do that, we add n to both sides + n = -n + 10 + n [SIZE=5][B]Step 4: Cancel -n on the right side:[/B][/SIZE] n = [B]10[/B]

n = 3n - 1/2 Solve for [I]n[/I] in the equation n = 3n - 1/2 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables n and 3n. To do that, we subtract 3n from both sides n - 3n = 3n - 0.5 - 3n [SIZE=5][B]Step 2: Cancel 3n on the right side:[/B][/SIZE] -2n = -0.5 [SIZE=5][B]Step 3: Divide each side of the equation by -2[/B][/SIZE] -2n/-2 = -0.5/-2 n = [B]0.25 or 1/4[/B]

n = 5m^2d for d Divide each side by 5m^2 to isolate d: n/5m^2 = 5m^2d/5m^2 Cancel the 5m^2 on the right side and we get: d = [B]n/5m^2[/B]

n = b + d^2a for a Let's start by isolating the one term with the a variable. Subtract b from each side: n - b = b - b + d^2a Cancel the b terms on the right side and we get: n - b = d^2a With the a term isolated, let's divide each side of the equation by d^2: (n - b)/d^2 = d^2a/d^2 Cancel the d^2 on the right side, and we'll display this with the variable to solve on the left side: a = [B](n - b)/d^2 [MEDIA=youtube]BCEVsZmoKoQ[/MEDIA][/B]

n=i*x+y for i This is a literal equation. Subtract y from each side of the equation: n - y = i*x + y - y The y's cancel on the right side, so we have: n - y = ix Divide each side of the equation by x, to isolate i (n - y)/x = ix/x The x's cancel on the right side, so we have: i = [B](n - y)/x[/B]

Nancy is 10 years less than 3 times her daughters age. If Nancy is 41 years old, how old is her daughter? Declare variables for each age: [LIST] [*]Let Nancy's age be n [*]Let her daughter's age be d [/LIST] We're given two equations: [LIST=1] [*]n = 3d - 10 [*]n = 41 [/LIST] We set 3d - 10 = 41 and solve for d: Solve for [I]d[/I] in the equation 3d - 10 = 41 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants -10 and 41. To do that, we add 10 to both sides 3d - 10 + 10 = 41 + 10 [SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE] 3d = 51 [SIZE=5][B]Step 3: Divide each side of the equation by 3[/B][/SIZE] 3d/3 = 51/3 d = [B]17[/B]

Nate jars 2 liters of jam everyday. How many days did Nate spend making jam if he jarred 8 liters of jam? 2 liters per 1 day and 8 liters per x days. Set up a proportion: 2/1 = 8/x Cross multiply: 2x = 8 Divide each side by 2 x = [B]4 days[/B].

Consider the recurrence relation T(n) =2 if n = 1, T(n?1) + 4n?2 if n > 1 (i) Derive the closed form expression f(n) for this recurrence relation. (ii) Prove that T(n) = f(n),?n ?N

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Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10? Let n be the number of notebooks you can purchase. We have the following inequality: 1.39n <= 10 Divide each side by 1.39 n <= 7.194 We want whole notebooks, we cannot buy fractions of notebooks, so we have: n <= 7 The question asks for the possible numbers of notebooks we can buy. This implies we buy at least 1, but our inequality says not more than 7. So our number set is: [B]N = {1, 2, 3, 4, 5, 6, 7}[/B]

n^2 + 9 = 34 Subtract 9 from each side: n^2 + 9 - 9 = 34 - 9 n^2 = 25 Take the square root of each side: n = [B]5[/B]

n^2 - 1 = -99/100 Add 1 (100/100) to each side: n^2 - 1 + 1 = -99/100 + 100/100 Cancel the 1's on the left side: n^2 = 1/100 Take the square root of both sides: n = [B]1/10 or -1/10[/B]

n^2 = 1/4 Take the square root of each side: n = [B]1/2[/B]

n^2 = 6&1/4 [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%261%2F4&frac2=3%2F8&pl=Simplify']6&1/4[/URL] = 25/4 n^2 = 25/4 Take the square root of each side: n = [B]5/2 or -5/2[/B]

n^2 = 64 Take the square root of each side: sqrt(n^2) = sqt(64) n = [B]8[/B]

N^2=5qd for d Divide each side by 5q to isolate d: N^2/5q = 5qd/5q Cancel 5q on the right side and we get: d = [B]N^2/5q[/B]

Oceanside Bike Rental Shop charges $15.00 plus $9.00 per hour for renting a bike. Dan paid $51.00 to rent a bike. How many hours was he hiking for? Set up the cost equation C(h) where h is the number of hours needed to rent the bike: C(h) = Cost per hour * h + rental charge Using our given numbers in the problem, we have: C(h) = 9h + 15 The problem asks for h, when C(h) = 51. 9h + 15 = 51 To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=9h%2B15%3D51&pl=Solve']plug this equation into our search engine[/URL] and we get: h = [B]4[/B]

Oceanside Bike Rental Shop charges 16 dollars plus 6 dollars an hour for renting a bike. Mary paid 58 dollars to rent a bike. How many hours did she pay to have the bike checked out ? Set up the cost function C(h) where h is the number of hours you rent the bike: C(h) = Hourly rental cost * h + initial rental charge C(h) = 6h + 16 Now the problem asks for h when C(h) = 58, so we set C(h) = 58: 6h + 16 = 58 To solve this equation for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=6h%2B16%3D58&pl=Solve']type it in our math engine[/URL] and we get: h = [B]7 hours[/B]

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Let p be the novel pages. We have 1/3p - 3 = 114 Add 3 to each side 1/3p = 117 Multiply each side by 3 p = 351

Oliver invests $1,000 at a fixed rate of 7% compounded monthly, when will his account reach $10,000? 7% monthly is: 0.07/12 = .00583 So we have: 1000(1 + .00583)^m = 10000 divide each side by 1000; (1.00583)^m = 10 Take the natural log of both sides; LN (1.00583)^m = LN(10) Use the identity for natural logs and exponents: m * LN (1.00583) = 2.30258509299 0.00252458479m = 2.30258509299 m = 912.064867899 Round up to [B]913 months[/B]

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650 Let the number be n. Then the square is n^2. We're given: n^2 + n = 650 Subtract 650 from each side: n^2 + n - 650 = 0 We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get: n = 25 and n = -26 Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution. the second solution is 25^2 = [B]625[/B]

One third of the bagels in a bakery are sesame bagels. There are 72 sesame bagels. Set up our equation where b is the number of total bagels 72 = b/3 Multiply each side by 3 [B]b = 216[/B]

p = i^2r for r Divide each side of the equation by i^2 to isolate r: p/i^2 = i^2/ri^2 Cancel the i^2 on the right side and we get: r = [B]p/i^2[/B]

p/q = f/q- f for f Isolate f in this literal equation. Factor out f on the right side: p/q = f(1/q - 1) Rewriting the term in parentheses, we get: p/q = f(1 - q)/q Cross multiply: f = pq/q(1 - q) Cancelling the q/q on the right side, we get: f = [B]p/(1 - q)[/B]

p/q=f/q-f for f To solve this literal equation for f, let's factor out f on the right side: p/q=f(1/q-1) Divide each side by (1/q - 1) p/(q(1/q - 1)) = f(1/q-1)/(1/q - 1) Cancelling the (1/q - 1) on the right side, we get: f = p/(1/q - 1) Rewriting this since (1/q -1) = (1 - q)/q since q/q = 1 we have: f = [B]pq/(1 - q)[/B]

P/v=nr/t for r Cross multiply to solve this literal equation: Pt = nrv Divide each side of the equation by nv: Pt/nv = nrv/nv Cancel the nv's on the right side, we get: r = [B]Pt/nv[/B]

Subtract 15 from each side: 5d/11 = P - 15 Multiply each side by 11 5d = 11p - 165 Divide each side of the equation by d: d = (11p - 165) ------------ 5

P=ab/c, for c Cross multiply: cP = ab Divide each side by P [B]c = (ab)/P[/B]

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Perimeter of a rectangle is 372 yards. If the length is 99 yards, what is the width? The perimeter P of a rectangle with length l and width w is: 2l + 2w = P We're given P = 372 and l = 99, so we have: 2(99) + 2w = 372 2w + 198 = 372 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 198 and 372. To do that, we subtract 198 from both sides 2w + 198 - 198 = 372 - 198 [SIZE=5][B]Step 2: Cancel 198 on the left side:[/B][/SIZE] 2w = 174 [SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE] 2w/2 = 174/2 w = [B]87[/B]

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5225, how many bags does it need to sell? 5.5ob >= $5,225 Divide each side of the inequality by $5.50 b >=9.5 bags, so round up to a whole number of 10 bags.

Pleasantburg has a population growth model of P(t)=at^2+bt+P0 where P0 is the initial population. Suppose that the future population of Pleasantburg t years after January 1, 2012, is described by the quadratic model P(t)=0.7t^2+6t+15,000. In what month and year will the population reach 19,200? Set P(t) = 19,200 0.7t^2+6t+15,000 = 19,200 Subtract 19,200 from each side: 0.7t^2+6t+4200 = 0 The Quadratic has irrational roots. So I set up a table below to run through the values. At t = 74, we pass 19,200. Which means we add 74 years to 2012: 2012 + 74 = [B]2086[/B] t 0.7t^2 6t Add 15000 Total 1 0.7 6 15000 15006.7 2 2.8 12 15000 15014.8 3 6.3 18 15000 15024.3 4 11.2 24 15000 15035.2 5 17.5 30 15000 15047.5 6 25.2 36 15000 15061.2 7 34.3 42 15000 15076.3 8 44.8 48 15000 15092.8 9 56.7 54 15000 15110.7 10 70 60 15000 15130 11 84.7 66 15000 15150.7 12 100.8 72 15000 15172.8 13 118.3 78 15000 15196.3 14 137.2 84 15000 15221.2 15 157.5 90 15000 15247.5 16 179.2 96 15000 15275.2 17 202.3 102 15000 15304.3 18 226.8 108 15000 15334.8 19 252.7 114 15000 15366.7 20 280 120 15000 15400 21 308.7 126 15000 15434.7 22 338.8 132 15000 15470.8 23 370.3 138 15000 15508.3 24 403.2 144 15000 15547.2 25 437.5 150 15000 15587.5 26 473.2 156 15000 15629.2 27 510.3 162 15000 15672.3 28 548.8 168 15000 15716.8 29 588.7 174 15000 15762.7 30 630 180 15000 15810 31 672.7 186 15000 15858.7 32 716.8 192 15000 15908.8 33 762.3 198 15000 15960.3 34 809.2 204 15000 16013.2 35 857.5 210 15000 16067.5 36 907.2 216 15000 16123.2 37 958.3 222 15000 16180.3 38 1010.8 228 15000 16238.8 39 1064.7 234 15000 16298.7 40 1120 240 15000 16360 41 1176.7 246 15000 16422.7 42 1234.8 252 15000 16486.8 43 1294.3 258 15000 16552.3 44 1355.2 264 15000 16619.2 45 1417.5 270 15000 16687.5 46 1481.2 276 15000 16757.2 47 1546.3 282 15000 16828.3 48 1612.8 288 15000 16900.8 49 1680.7 294 15000 16974.7 50 1750 300 15000 17050 51 1820.7 306 15000 17126.7 52 1892.8 312 15000 17204.8 53 1966.3 318 15000 17284.3 54 2041.2 324 15000 17365.2 55 2117.5 330 15000 17447.5 56 2195.2 336 15000 17531.2 57 2274.3 342 15000 17616.3 58 2354.8 348 15000 17702.8 59 2436.7 354 15000 17790.7 60 2520 360 15000 17880 61 2604.7 366 15000 17970.7 62 2690.8 372 15000 18062.8 63 2778.3 378 15000 18156.3 64 2867.2 384 15000 18251.2 65 2957.5 390 15000 18347.5 66 3049.2 396 15000 18445.2 67 3142.3 402 15000 18544.3 68 3236.8 408 15000 18644.8 69 3332.7 414 15000 18746.7 70 3430 420 15000 18850 71 3528.7 426 15000 18954.7 72 3628.8 432 15000 19060.8 73 3730.3 438 15000 19168.3 74 3833.2 444 15000 19277.2

Distance = Rate x Time 6.4 meters = 4 meters/minute * t Divide each side by 4 [B]t = 1.6 minutes[/B]

Time 1, distance apart is 105 + 85 = 190 So every hour, the distance between them is 190 * t where t is the number of hours. Set up our distance function: D(t) = 190t We want D(t) = 494 190t = 494 Divide each side by 190 [B]t = 2.6 hours[/B]

Figure 1, we have a cone, cylinder, and cube. Let's get the volume of each Cone volume = pir^2h/3 radius = s/2 h = s Cone Volume = pi(s/2)^2(s)/3 Cone Volume = pis^3/12 Volume of cube = s^3 Volume of cylinder = pir^2h Volume of cylinder = pi(s/2)^2s Volume of cylinder = pis^3/2 But Figure 2 has no sizes? For sides, height, etc. So I cannot answer the question until I have that.

Find what was used: Used Money = Prepaid original cost - Remaining Credit Used Money = 20 - 17.47 Used Money = 2.53 Using (m) as the number of minutes, we have the following cost equation: C(m) = 0.11m C(m) = 2.53, so we have: 0.11m = 2.53 Divide each side by 0.11 [B]m = 23[/B]

A Web music store offers two versions of a popular song. The size of the standard version is 2.7 megabytes (MB). The size of the high-quality version is 4.7 MB. Yesterday, the high-quality version was downloaded three times as often as the standard version. The total size downloaded for the two versions was 4200 MB. How many downloads of the standard version were there? Let s be the standard version downloads and h be the high quality downloads. We have two equations: [LIST=1] [*]h = 3s [*]2.7s + 4.7h = 4200 [/LIST] Substitute (1) into (2) 2.7s + 4.7(3s) = 4200 2.7s + 14.1s = 4200 Combine like terms: 16.8s = 4200 Divide each side by 16.8 [B]s = 250[/B]

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Free Polygons Calculator - Using various input scenarios of a polygon such as side length, number of sides, apothem, and radius, this calculator determines Perimeter or a polygon and Area of the polygon. This also determines interior angles of a polygon and diagonals of a polygon as well as the total number of 1 vertex diagonals.

pr=xf/y for r So for this literal equation, we divide each side of the equation by p to isolate r. pr/p = xf/yp Cancel the p's on the left side and we get: r = [B]xf/yp [MEDIA=youtube]6ekuN4H3mM4[/MEDIA][/B]

Prove P(A) = 1 - P(A) The sample space S contains an Event A and everything not A, called A' We know P(S) = 1 P(S) = P(A U A') P(A U A') = 1 P(A) + P(A') = 1 subtract P(A) from each side: P(A) = 1 - P(A) [MEDIA=youtube]dNLl_8vejyE[/MEDIA]

Use proof by contradiction. Assume sqrt(2) is rational. This means that sqrt(2) = p/q for some integers p and q, with q <>0. We assume p and q are in lowest terms. Square both side and we get: 2 = p^2/q^2 p^2 = 2q^2 This means p^2 must be an even number which means p is also even since the square of an odd number is odd. So we have p = 2k for some integer k. From this, it follows that: 2q^2 = p^2 = (2k)^2 = 4k^2 2q^2 = 4k^2 q^2 = 2k^2 q^2 is also even, therefore q must be even. So both p and q are even. This contradicts are assumption that p and q were in lowest terms. So sqrt(2) [B]cannot be rational. [MEDIA=youtube]tXoo9-8Ewq8[/MEDIA][/B]

Let us take an integer x which is both even [I]and[/I] odd. [LIST] [*]As an even integer, we write x in the form 2m for some integer m [*]As an odd integer, we write x in the form 2n + 1 for some integer n [/LIST] Since both the even and odd integers are the same number, we set them equal to each other 2m = 2n + 1 Subtract 2n from each side: 2m - 2n = 1 Factor out a 2 on the left side: 2(m - n) = 1 By definition of divisibility, this means that 2 divides 1. But we know that the only two numbers which divide 1 are 1 and -1. Therefore, our original assumption that x was both even and odd must be false. [MEDIA=youtube]SMM9ubEVcLE[/MEDIA]

Put the number 123456789 exactly ones in the bubble so that each edge adds up to say number [B] Each side adds up to 17 [IMG]https://www.mathcelebrity.com/images/triangle_sum_17.png[/IMG] [/B]

pv/t = ab/c for c Cross multiply: cpv = abt Divide each side of the equation by pv to isolate c: cpv/pv = abt/pv Cancel the pv terms on the left side and we get: c = [B]abt/pv[/B]

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Q is a point on segment PR. If PQ = 2.7 and PR = 6.1, what is QR? From segment addition, we know that: PQ + QR = PR Plugging our given numbers in, we get: 2.7 + QR = 6.1 Subtract 2.7 from each side, and we get: 2.7 - 2.7 + QR = 6.1 - 2.7 Cancelling the 2.7 on the left side, we get: QR = [B]3.4[/B]

q=c+d/5 for d Subtract c from each side to solve this literal equation: q - c = c - c + d/5 Cancel the c's on the right side, we get d/5 = q - c Multiply each side by 5: 5d/5 = 5(q - c) Cancel the 5's on the left side, we get: [B]d = 5(q - c)[/B]

q=rs/2-p;p Add p to each side: q + p = rs/2 Subtract q from each side: [B]p = rs/2 - q[/B]

r=l^2w/2 for w Solve this literal equation by isolating w. Cross multiply: 2r = l^2w Divide each side by l^2 w = [B]2r/l^2[/B]

Rachel buys some scarves that cost $10 each and 2 purses that cost $16 each. The cost of Rachel's total purchase is $62. What equation can be used to find n, the number of scarves that Rebecca buys Scarves Cost + Purses Cost = Total Cost [U]Calculate Scarves Cost[/U] Scarves cost = Cost per scarf * number of scarves Scarves cost = 10n [U]Calculate Purses Cost[/U] Purses cost = Cost per purse * number of purses Purses cost = 16 * 2 Purses cost = 32 Total Cost = 62. Plug in our numbers and values to the Total Cost equation : 10n + 32 = 62 Solve for [I]n[/I] in the equation 10n + 32 = 62 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 32 and 62. To do that, we subtract 32 from both sides 10n + 32 - 32 = 62 - 32 [SIZE=5][B]Step 2: Cancel 32 on the left side:[/B][/SIZE] 10n = 30 [SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE] 10n/10 = 30/10 n = [B]3[/B]

Rearrange the following equation to make x the subject, and select the correct rearrangement from the list below 3x + 2y 1 -------- = --- 4x + y 3 [LIST] [*]x = 7y/13 [*]x = 7y/5 [*]x = -7y [*]x = -3y [*]x = 3y/5 [*]x = -5y/13 [*]x = -y [/LIST] Cross multiply: 3(3x - 2y) = 4x + y Multiply the left side through 9x - 6y = 4x + y Subtract 4x from each side and add 6y to each side 5x = 7y Divide each side by 5 to isolate x, the subject of an equation is the variable to the left [B]x = 7y/5[/B]

rectangle abcd prove: triangle adc is congruent to triangle bcd 1. Given: ABCD is a rectangle 2. AB = CD since opposite sides of rectangle are congruent 3. BC = AD since opposite sides of rectangle are congruent 4. AC = AC by the Reflexive Property of Equality 5. triangle ADC = triangle CBA by the Side-Side-Side (SSS) Property

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Let a be Rick's age We have a + 24 = 69 Subtract 24 from each side [B]a = 45[/B]

Let x be the price of one t-shirt. Set up an equation: 6 times the number of t-shirts plus 7 dollars left over get him to a total of 45 6x = 45 - 7 6x = 38 Divide each side by 6 [B]x = 6.33[/B]

rs+h^2=1 for h Subtract rs from each side to isolate h: rs - rs + h^2 = 1 - rs Cancel the rs on the left side: h^2 = 1 - rs Take the square root of each side: sqrt(h^2) = sqrt(1 - rs) [B]h = +- sqrt(1 -rs)[/B]

Running from the top of a flagpole to a hook in the ground there is a rope that is 9 meters long. If the hook is 4 meters from the base of the flagpole, how tall is the flagpole? We have a right triangle, with hypotenuse of 9 and side of 4. [URL='https://www.mathcelebrity.com/pythag.php?side1input=&side2input=4&hypinput=9&pl=Solve+Missing+Side']Using our Pythagorean Theorem calculator[/URL], we get a flagpole height of [B]8.063[/B].

s = tu^2 for u Divide each side by t u^2 = s/t Take the square root of each side [LIST] [*]u = sqrt(s/t) [*]u = -sqrt(s/t) [/LIST] We have two answers due to negative number squared is positive

s=u^2t for t Divide each side by u^2 to isolate t: u^2t/u^2 = s/u^2 Cancel the u^2 on the left side, we get: t = [B]s/u^2[/B]

s=w-10e/m for w Add 10e/m to each side to isolate w: s + 10e/m = w - 10e/m + 10e/m Cancel the 10e/m on the right side, and we get: w = [B]s + 10e/m[/B]

Sally and Adam works a different job. Sally makes $5 per hour and Adam makes $4 per hour. They each earn the same amount per week but Adam works 2 more hours. How many hours a week does Adam work? [LIST] [*]Let [I]s[/I] be the number of hours Sally works every week. [*]Let [I]a[/I] be the number of hours Adam works every week. [*]We are given: a = s + 2 [/LIST] Sally's weekly earnings: 5s Adam's weekly earnings: 4a Since they both earn the same amount each week, we set Sally's earnings equal to Adam's earnings: 5s = 4a But remember, we're given a = s + 2, so we substitute this into Adam's earnings: 5s = 4(s + 2) Multiply through on the right side: 5s = 4s + 8 <-- [URL='https://www.mathcelebrity.com/expand.php?term1=4%28s%2B2%29&pl=Expand']multiplying 4(s + 2)[/URL] [URL='https://www.mathcelebrity.com/1unk.php?num=5s%3D4s%2B8&pl=Solve']Typing this equation into the search engine[/URL], we get s = 8. The problem asks for Adam's earnings (a). We plug s = 8 into Adam's weekly hours: a = s + 2 a = 8 + 2 [B]a = 10[/B]

Salma purchased a prepaid phone card for 30. Long distance calls cost 9 cents a minute using this card. Salma used her card only once to make a long distance call. If the remaining credit on her card is 28.38, how many minutes did her call last? [U]Set up the equation where m is the number of minutes used:[/U] 0.09m = 30 - 28.38 0.09m = 1.62 [U]Divide each side by 0.09[/U] [B]m = 18[/B]

Sam and Jeremy have ages that are consecutive odd integers. The product of their ages is 783. Which equation could be used to find Jeremy's age, j, if he is the younger man. Let Sam's age be s. Let' Jeremy's age be j. We're given: [LIST=1] [*]s = j + 2 <-- consecutive odd integers [*]sj = 783 [/LIST] Substitute (1) into (2): (j + 2)j = 783 j^2 + 2j = 783 Subtract 783 from each side: j^2 + 2j - 783 = 0 <-- This is the equation to find Jeremy's age. To solve this, [URL='https://www.mathcelebrity.com/quadratic.php?num=j%5E2%2B2j-783%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type this quadratic equation into the search engine[/URL] and get: j = 27, j = -29. Since ages cannot be negative, we have: [B]j = 27[/B]

Sam leaves school to go home. He walks 10 blocks North and then 8 blocks west. How far is John from the school? Sam walked at a right angle. His distance from home to school is the hypotenuse. Using our [URL='https://www.mathcelebrity.com/pythag.php?side1input=8&side2input=10&hypinput=&pl=Solve+Missing+Side']Pythagorean theorem calculator[/URL], we get: [B]12.806 blocks[/B]

Shalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself how many plums did she have at first? Let p be the number of plums Shalini started with. We have: [LIST] [*]0.4 given to her brother [*]20% which is 0.2 given away to her sister [*]What this means is she kept 1 - (0.4 + 0.2) = 1 - 0.6 = 0.4 for herself [/LIST] 0.4p = 16 Divide each side by 0.4 [B]p = 40[/B]

Shalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself. How many plums did she have first? Let's convert everything to decimals. 20% = 0.2 So Shalini gave 0.4 + 0.2 = 0.6 of the plums away. Which means she has 1 = 0.6 = 0.4 or 40% left over. 40% represents 16 plums So our equation is 0.4p = 16 where p is the number of plums to start with Divide each side by 0.4 [B]p = 40[/B]

She earns $20 per hour as a carpenter and $25 per hour as a blacksmith, last week Giselle worked both jobs for a total of 30 hours, and a total of $690. How long did Giselle work as a carpenter and how long did she work as a blacksmith? Assumptions: [LIST] [*]Let b be the number of hours Giselle worked as a blacksmith [*]Let c be the number of hours Giselle worked as a carpenter [/LIST] Givens: [LIST=1] [*]b + c = 30 [*]25b + 20c = 690 [/LIST] Rearrange equation (1) to solve for b by subtracting c from each side: [LIST=1] [*]b = 30 - c [*]25b + 20c = 690 [/LIST] Substitute equation (1) into equation (2) for b 25(30 - c) + 20c = 690 Multiply through: 750 - 25c + 20c = 690 To solve for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=750-25c%2B20c%3D690&pl=Solve']type this equation into our search engine[/URL] and we get: c = [B]12 [/B] Now, we plug in c = 12 into modified equation (1) to solve for b: b = 30 - 12 b = [B]18[/B]

She ordered 6 large pizzas. Luckily, she had a coupon for 3 off each pizza. If the bill came to 38.94, what was the price for a large pizza? [U]Determine additional amount the pizzas would have cost without the coupon[/U] 6 pizzas * 3 per pizza = 18 [U]Add 18 to our discount price of 38.94[/U] Full price for 6 large pizzas = 38.94 + 18 Full price for 6 large pizzas = 56.94 Let x = full price per pizza before the discount. Set up our equation: 6x = 56.94 Divide each side by 6 [B]x = $9.49[/B]

Sheila wants build a rectangular play space for her dog. She has 100 feet of fencing and she wants it to be 5 times as long as it is wide. What dimensions should the play area be? Sheila wants: [LIST=1] [*]l =5w [*]2l + 2w = 100 <-- Perimeter [/LIST] Substitute (1) into (2) 2(5w) + 2w = 100 10w + 2w = 100 12w = 100 Divide each side by 12 [B]w = 8.3333[/B] Which means l = 5(8.3333) -->[B] l = 41.6667[/B]

7sqrt(3) is broken down. sqrt(12) is not broken down. Let's find all the factors of 12 and see if we stumble on a perfect square: [LIST] [*]1 * 12 [*]2 * 6 [*]3 * 4 [/LIST] 4 is a perfect square, since sqrt(4) = 2. So sqrt(12) = sqrt(3 * 4) We pull the sqrt(4) = 2 outside the radical and rewrite our problem as: 7sqrt(3) - 2sqrt(3) These are like terms, so we have: (7 - 2)sqrt(3) [B]5sqrt(3) [/B] [MEDIA=youtube]ljXVXWnKiWY[/MEDIA]

We know from the pythagorean theorem: [SIZE=5][B]sin^2(x) + cos^2(x) = 1[/B] [B]Subtract sin^2(x) from each side and we get:[/B] [B][B]cos^2(x) = 1 - [B]sin^2(x)[/B][/B][/B] [B][B][B]We can rewrite our original expression as:[/B][/B][/B] [B][B][B]sin^2(x)/cos^2(x)[/B][/B][/B] [/SIZE] [B][B][B][SIZE=5]But this expression [/SIZE][SIZE=4]equals[/SIZE][SIZE=5] tan^2(x)[/SIZE][/B][/B][/B] [MEDIA=youtube]zqYg0VRq5Ak[/MEDIA]

Solve 11 - 1/2y = 3 + 6x for y Subtract 11 from each side so we can isolate the y term: 11 -11 - 1/2y = 3 + 6x - 11 Cancelling the 11's on the left side, we get: -1/2y = 6x - 8 <-- Since 3 - 11 = -8 Multiply both sides of the equation by -2 to remove the -1/2 on the left side: -2(-1/2)y = -2(6x - 8) Simplifying, we get: y = [B]-12x + 16 [MEDIA=youtube]38uwIaj88Lw[/MEDIA][/B]

Solve a= (a + b + c + d)/4 for c Cross multiply: 4a = a + b + c + d Subtract a + b+ d from each side to isolate c: 4a - a - b - d = a + b + c + d - a - b - d Canceling the a, b, and d from the right side, we get: c = [B]3a - b - d [/B]

Solve for h. rs + h^2 = l [U]Subtract rs from each side to isolate h:[/U] rs - rs + h^2 = l - rs [U]Cancel the rs terms on the left side, and we get:[/U] h^2 = l - rs [U]Take the square root of each side:[/U] h = [B]sqrt(l - rs)[/B]

Expand the right side: 1/3x + 1/2 = 6/4x - 10 Simplify as 6/4 is 3/2 x/3 + 1/2 = 3x/2 - 10 Common denominator of 2 and 3 is 6. So we have: 2x/6 + 1/2 = 9x/6 -10 Subtract 2x/6 from each side 7x/6 - 10 = 1/2 Add 10 to each side. 10 is 20/2 7x/6 = 21/2 Using our [URL='http://www.mathcelebrity.com/prop.php?num1=7x&num2=21&den1=6&den2=2&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get: [B]x = 9[/B]

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v 1/2(2/5) = 1/5 since the 2's cancel r^2/r^2 = 1 So we simplify, and get: mgh=1/2mv^2+1/5(mv^2) for v Divide each side by m, so m's cancel in each term on the left and right side: gh = 1/2v^2 + 1/5(v^2) Combine like terms for v^2 on the right side: 1/2 + 1/5 = 7/10 from our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction calculator[/URL] So we have: gh = 7v^2/10 Multiply each side by 10: 10gh = 7v^2 Now divide each side by 7 10gh/7 = v^2 Take the square root of each side: [B]v = sqrt(10gh/7)[/B]

Some hot dogs come in packages of 8. Why would a baker of hot dog buns package 7 hot dog buns to a package For customers that like to have matching hot dogs and buns, consider this scenario. For the first round, you have one extra hot dog. Now you buy a hot dog buns package. You're over 6 buns. This continues... We want to see when packaging and hot dogs math. Find the least common multiple of 7 and 8 so that packages match. [URL='https://www.mathcelebrity.com/gcflcm.php?num1=7&num2=8&num3=&pl=LCM']LCM(7, 8[/URL][I][URL='https://www.mathcelebrity.com/gcflcm.php?num1=7&num2=8&num3=&pl=LCM']) [/URL]= 56[/I]

Sonia visited a park in California that had redwood trees. When Sonia asked how tall a certain large redwood tree was, the ranger said that he wouldn't tell its height, but would give Sonia a clue. How tall is the redwood tree Sonia asked about? Sonia said the tree is 64 times my height. The tree is also 112 feet taller than the tree next to it. The two trees plus my height total 597.5 feet. [LIST] [*]Rangers's height = n [*]Tree height = 64n [*]Smaller tree height = 64n - 112 [*]Total height = 64n - 112 + 64n = 597.5 [/LIST] Solve for [I]n[/I] in the equation 64n - 112 + 64n = 597.5 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (64 + 64)n = 128n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 128n - 112 = + 597.5 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -112 and 597.5. To do that, we add 112 to both sides 128n - 112 + 112 = 597.5 + 112 [SIZE=5][B]Step 4: Cancel 112 on the left side:[/B][/SIZE] 128n = 709.5 [SIZE=5][B]Step 5: Divide each side of the equation by 128[/B][/SIZE] 128n/128 = 709.5/128 n = 5.54296875 Tree height = 64 * ranger height Tree height = 64 * 5.54296875 Tree height = [B]354.75 feet[/B]

Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is running at 5 feet per second and Claire is running at 3 feet per second. i. After how many seconds will Sophie catch Claire? ii. If the race is 500 feet, who wins? i. Sophie's distance formula is given as D = 5s Claire's distance formula is given as D = 3s + 100 Set them equal to each other 5s = 3s + 100 Subtract 3s from both sides: 2s = 100 Divide each side by 2 [B]s = 50[/B] ii. [B]Sophie since after 50 seconds, she takes the lead and never gives it back.[/B]

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Squaring a number equals 5 times that number. The phrase [I]a number[/I] means an arbitrary variable, let's call it x. Squaring this number: x^2 5 times this number means we multiply by 5: 5x The phrase [I]equals[/I] means we set both expressions equal to each other: [B]x^2 = 5x [/B] <-- This is our algebraic expression If you want to solve for x, then we subtract 5x from each side: x^2 - 5x = 5x - 5x Cancel the 5x on the right side, leaving us with 0: x^2 - 5x = 0 Factor out x: x(x - 5) So we get x = 0 or [B]x = 5[/B]

Stacy sells art prints for $12 each. Her expenses are $2.50 per print, plus $38 for equipment. How many prints must she sell for her revenue to equal her expenses? Let the art prints be p Cost function is 38 + 2p Revenue function is 12p Set cost equal to revenue 12p = 38 + 2p Subtract 2p from each side 10p = 38 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=10p%3D38&pl=Solve']equation calculator[/URL] gives us [B]p = 3.8[/B]

Standard Error (margin of Error) = Standard Deviation / sqrt(n) 128 = 545/sqrt(n) Cross multiply: 128sqrt(n) = 545 Divide by 128 sqrt(n) = 4.2578125 Square both sides: [B]n = 18.1289672852 But we need an integer, so the answer is 19[/B]

Stanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the ruler, what was the cost of the yardstick? Let r be the cost of the ruler Let y be the cost of the yardstick We're given 2 equations: [LIST=1] [*]r + y = 1.25 [*]y = r + 0.45 [/LIST] Substitute equation (2) into equation (1) for y r + r + 0.45 = 1.25 Solve for [I]r[/I] in the equation r + r + 0.45 = 1.25 [SIZE=5][B]Step 1: Group the r terms on the left hand side:[/B][/SIZE] (1 + 1)r = 2r [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2r + 0.45 = + 1.25 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 0.45 and 1.25. To do that, we subtract 0.45 from both sides 2r + 0.45 - 0.45 = 1.25 - 0.45 [SIZE=5][B]Step 4: Cancel 0.45 on the left side:[/B][/SIZE] 2r = 0.8 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2r/2 = 0.8/2 r = 0.4 Substitute r = 0.4 into equation (2) above: y = r + 0.45 y = 0.4 + 0.45 r = [B]0.85 [URL='https://www.mathcelebrity.com/1unk.php?num=r%2Br%2B0.45%3D1.25&pl=Solve']Source[/URL][/B]

Steve woke up and it was -12 Fahrenheit outside the weatherman said it was supposed to warm up to 20 degrees. how many degrees will the temperature increase We start with a temperature of -12F Warming up means we [U][B]add[/B][/U] degrees to the original temperature. -12 + 20 = [B]+8F[/B]

Steven has some money. If he spends $9, then he will have 3/5 of the amount he started with. Let the amount Steven started with be s. We're given: s - 9 = 3s/5 Multiply each side through by 5 to eliminate the fraction: 5(s - 9) = 5(3s/5) Cancel the 5's on the right side and we get: 5s - 45 = 3s To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=5s-45%3D3s&pl=Solve']type this equation into our search engine[/URL] and we get: s = [B]22.5[/B]

sum of 3 consecutive odd integers equals 1 hundred 17 The sum of 3 consecutive odd numbers equals 117. What are the 3 odd numbers? 1) Set up an equation where our [I]odd numbers[/I] are n, n + 2, n + 4 2) We increment by 2 for each number since we have [I]odd numbers[/I]. 3) We set this sum of consecutive [I]odd numbers[/I] equal to 117 n + (n + 2) + (n + 4) = 117 [SIZE=5][B]Simplify this equation by grouping variables and constants together:[/B][/SIZE] (n + n + n) + 2 + 4 = 117 3n + 6 = 117 [SIZE=5][B]Subtract 6 from each side to isolate 3n:[/B][/SIZE] 3n + 6 - 6 = 117 - 6 [SIZE=5][B]Cancel the 6 on the left side and we get:[/B][/SIZE] 3n + [S]6[/S] - [S]6[/S] = 117 - 6 3n = 111 [SIZE=5][B]Divide each side of the equation by 3 to isolate n:[/B][/SIZE] 3n/3 = 111/3 [SIZE=5][B]Cancel the 3 on the left side:[/B][/SIZE] [S]3[/S]n/[S]3 [/S]= 111/3 n = 37 Call this n1, so we find our other 2 numbers n2 = n1 + 2 n2 = 37 + 2 n2 = 39 n3 = n2 + 2 n3 = 39 + 2 n3 = 41 [SIZE=5][B]List out the 3 consecutive odd numbers[/B][/SIZE] ([B]37, 39, 41[/B]) 37 ? 1st number, or the Smallest, Minimum, Least Value 39 ? 2nd number 41 ? 3rd or the Largest, Maximum, Highest Value

Sum of a number and it's reciprocal is 6. What is the number? Let the number be n. The reciprocal is 1/n. The word [I]is[/I] means an equation, so we set n + 1/n equal to 6 n + 1/n = 6 Multiply each side by n to remove the fraction: n^2 + 1 = 6n Subtract 6n from each side: [B]n^2 - 6n + 1 = 0 [/B]<-- This is our algebraic expression If the problem asks you to solve for n, then you [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-6n%2B1%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this quadratic equation into our search engine[/URL].

Super Snack, a convenience store, charges $4.35 for a large chicken sandwich and two large colas. For a large chicken sandwich and a large cola, they charge $4.00. How much are the Super Snack large chicken sandwiches? The difference between the orders is $0.35 and 1 large cola. Therefore, 1 large cola = $0.35. And if we use the first order of one large chicken sandwich and one large cola, we get: Large Chicken Sandwich + 0.35 = 4.35 Subtract 0.35 from each side, and we get: Large Chicken Sandwich = $[B]4.00[/B]

Suppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin does she have? Set up two equations where d is the number of dimes and q is the number of quarters: (1) d + q = 10 (2) 0.1d + 0.25q = 1.45 Rearrange (1) into (3) to solve for d (3) d = 10 - q Now plug (3) into (2) 0.1(10 - q) + 0.25q = 1.45 Multiply through: 1 - 0.1q + 0.25q = 1.45 Combine q terms 0.15q + 1 = 1.45 Subtract 1 from each side 0.15q = 0.45 Divide each side by 0.15 [B]q = 3[/B] Plug our q = 3 value into (3) d = 10 - 3 [B]d = 7[/B]

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 20 gallons of fuel, the airplane weighs 2012 pounds. When carrying 55 gallons of fuel, it weighs 2208 pounds. How much does the airplane weigh if it is carrying 65 gallons of fuel? Linear functions are written in the form of one dependent variable and one independent variable. Using g as the number of gallons and W(g) as the weight, we have: W(g) = gx + c where c is a constant We are given: [LIST] [*]W(20) = 2012 [*]W(55) = 2208 [/LIST] We want to know W(65) Using our givens, we have: W(20) = 20x + c = 2012 W(55) = 55x + c = 2208 Rearranging both equations, we have: c = 2012 - 20x c = 2208 - 55x Set them both equal to each other: 2012 - 20x = 2208 - 55x Add 55x to each side: 35x + 2012 = 2208 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=35x%2B2012%3D2208&pl=Solve']equation solver[/URL], we see that x is 5.6 Plugging x = 5.6 back into the first equation, we get: c = 2012 - 20(5.6) c = 2012 - 112 c = 2900 Now that we have all our pieces, find W(65) W(65) = 65(5.6) + 2900 W(65) = 264 + 2900 W(65) = [B]3264[/B]

T = mg - mf for f Subtract mg from each side: T - mg = mg - mg - mf Cancel the mg on the right side and we get: T - mg = -mf Multiply each side by -1: -(T - mg) = -(-mf) mg - T = mf Now Divide each side by m to isolate f: (mg - T)/m = mf/m Cancel the m on the right side and we get: f = [B](mg - T)/m[/B]

tammy earns $18000 salary with 4% comission on sales. How much should she sell to earn $55,000 total We have a commission equation below: Sales * Commission percent = Salary We're given 4% commission percent and 55,000 salary. With 4% as 0.04, we have: Sales * 0.04 = 55,000 Divide each side of the equation by 0.04, and we get: Sales = [B]1,375,000[/B]

Let x equal "a number". Double the number is 2x. The average is (x + 2x)/2 Combine the terms in the numerator: 3x/2 The word [I]is[/I] means equal to, so we set 3x/2 equal to 25.5 3x/2 = 25.5 Cross multiply the 2: 3x = 51 Divide each side by 3 [B]x = 17[/B]

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of the triangle is less than 10 units. Write and solve an inequality that represents the possible area A of the triangle We're given: b=2/7A We're also told that b is less than 10. So we have: 2/7A < 10 2A/7 < 10 Cross multiply: 2A < 7 * 10 2A < 70 Divide each side of the inequality by 2 to isolate A 2A/2 < 70/2 Cancel the 2's on the left side and we get: A < [B]35[/B]

The club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer blender t years after its purchase. How long will it take the blender to depreciate completely? Complete depreciation means the salvage value is 0. So S(t) = 0. We need to find t to make S(t) = 0 -4,500t + 54,000 = 0 Subtract 54,000 from each side -4,500t = -54,000 Divide each side by -4,500 [B]t = 12[/B]

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. How long can a person have the rototiller if the cost must be less than $95? Setup the inequality: $19.50 + $7.95x < $95 Subtract 19.50 from both sides: 7.95x < 75.50 Divide each side of the inequality by 7.95 to isolate x x < 9.5 The next lowest integer is 9. So we take 9 + the first hour of renting to get [B]10 total hours[/B]. Check our work: $7.95 * 9.5 + $19.50 $71.55 + $19.50 = $91.05

Draw this rectangle and you'll see we have a pythagorean theorem equation. a^2 + b^2= c^2 b = 8 and c= 10 a^2 + 8^2 = 10^2 a^2 + 64 = 100 Subtract 64 from each side: a^2 = 36 a= 6 Therefore, perimeter P is: P = 2l + 2w P = 2(6) + 2(8) P = 12 + 16 P = [B]28[/B] [MEDIA=youtube]8lcpRet3r18[/MEDIA]

The difference between 2 numbers is 108. 6 times the smaller is equal to 2 more than the larger. What are the numbers? Let the smaller number be x. Let the larger number be y. We're given: [LIST=1] [*]y - x = 108 [*]6x = y + 2 [/LIST] Rearrange (1) by adding x to each side: [LIST=1] [*]y = x + 108 [/LIST] Substitute this into (2): 6x = x + 108 + 2 Combine like terms 6x = x +110 Subtract x from each side: 5x = 110 [URL='https://www.mathcelebrity.com/1unk.php?num=5x%3D110&pl=Solve']Plugging this equation into our search engine[/URL], we get: x = [B]22[/B]

The difference between the squares of two consecutive numbers is 141. Find the numbers Take two consecutive numbers: n- 1 and n Given a difference (d) between the squares of two consecutive numbers, the shortcut for this is: 2n - 1 = d Proof of this: n^2- (n - 1)^2 = d n^2 - (n^2 - 2n + 1) = d n^2 - n^2 + 2n - 1 = d 2n - 1 = d Given d = 141, we have 2n - 1 = 141 Add 1 to each side: 2n = 142 Divide each side by 2: 2n/2 = 142/2 n = [B]71[/B] Therefore, n - 1 = [B]70 Our two consecutive numbers are (70, 71)[/B] Check your work 70^2 = 4900 71^2 = 5041 Difference = 5041 - 4900 Difference = 141 [MEDIA=youtube]vZJtZyYWIFQ[/MEDIA]

The difference between two numbers is 25. The smaller number is 1/6th of the larger number. What is the value of the smaller number Let the smaller number be s. Let the larger number be l. We're given two equations: [LIST=1] [*]l - s = 25 [*]s = l/6 [/LIST] Plug in equation (2) into equation (1): l - l/6 = 25 Multiply each side of the equation by 6 to remove the fraction: 6l - l = 150 To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=6l-l%3D150&pl=Solve']type this equation into our search engine[/URL] and we get: l = 30 To solve for s, we plug in l = 30 into equation (2) above: s = 30/6 [B]s = 5[/B]

The difference between two positive numbers is 5 and the square of their sum is 169. Let the two positive numbers be a and b. We have the following equations: [LIST=1] [*]a - b = 5 [*](a + b)^2 = 169 [*]Rearrange (1) by adding b to each side. We have a = b + 5 [/LIST] Now substitute (3) into (2): (b + 5 + b)^2 = 169 (2b + 5)^2 = 169 [URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get: 4b^2 + 20b + 25 Set this equal to 169 above: 4b^2 + 20b + 25 = 169 [URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get: b = (-9, 4) But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions. Substitute b = 4 into equation (1) above, and we get: a - [I]b[/I] = 5 [URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL] [B]a = 9 [/B] Therefore, we have [B](a, b) = (9, 4)[/B]

The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other is 4. Find the numbers. Let the numbers be x and y. We have: [LIST] [*]x - y = 54 [*]x/y = 4 [*]Cross multiply x/y = 4 to get x = 4y [*]Now substitute x = 4y into the first equation [*](4y) - y = 54 [*]3y = 54 [*]Divide each side by 3 [*][B]y = 18[/B] [*]If x = 4y, then x = 4(18) [*][B]x = 72[/B] [/LIST]

The difference of five and five y 5 - 5y eight and two y 8 + 2y The phrase [I]is the same as[/I] means equal to. Set 5 - 5y equal to 8 + 2y for our final algebraic expression [B]5 - 5y = 8 + 2y[/B] [B][/B] If the problem asks you to solve for y: Add 5y to each side: 5 = 8 + 7y Subtract 8 from each side: 7y = -3 Divide each side by 7: [B]y= -3/7[/B]

The difference of two numbers is 12 and their mean is 15. Find the two numbers. Let the two numbers be x and y. We're given: [LIST=1] [*]x - y = 12 [*](x + y)/2 = 15. <-- Mean is an average [/LIST] Rearrange equation 1 by adding y to each side: x - y + y = y + 12 Cancelling the y's on the left side, we get: x = y + 12 Now substitute this into equation 2: (y + 12 + y)/2 = 15 Cross multiply: y + 12 + y = 30 Group like terms for y: 2y + 12 = 30 [URL='https://www.mathcelebrity.com/1unk.php?num=2y%2B12%3D30&pl=Solve']Typing this equation into our search engine[/URL], we get: [B]y = 9[/B] Now substitute this into modified equation 1: x = y + 12 x = 9 + 12 [B]x = 21[/B]

The distance between consecutive bases is 90 feet. An outfielder catches the ball on the third base line about 40 feet behind third base. How far would the outfielder have to throw the ball to first base? We have a right triangle. From home base to third base is 90 feet. We add another 40 feet to the outfielder behind third base to get: 90 + 40 = 130 The distance from home to first is 90 feet. Our hypotenuse is the distance from the outfielder to first base. [URL='https://www.mathcelebrity.com/pythag.php?side1input=130&side2input=90&hypinput=&pl=Solve+Missing+Side']Using our Pythagorean theorem calculator[/URL], we get: d = [B]158.11 feet[/B]

The fraction has a value of 3/5. The sum of the numerator and the denominator was 40. What was the fraction? We're given two equations with a fraction with numerator (n) and denominator (d): [LIST=1] [*]n + d = 40 [*]n/d = 3/5 [/LIST] Cross multiply equation 2, we get: 5n = 3d Divide each side by 5: 5n/5 = 3d/5 n = 3d/5 Substitute this into equation 1: 3d/5 + d = 40 Multiply through both sides of the equation by 5: 5(3d/5) = 5d = 40 * 5 3d + 5d =200 To solve this equation for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=3d%2B5d%3D200&pl=Solve']type it in our search engine and we get[/URL]: d = [B]25 [/B] Now substitute that back into equation 1: n + 25 = 40 Using [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B25%3D40&pl=Solve']our equation solver again[/URL], we get: n = [B]15[/B]

The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ? f'(x) = 3x^2 - 48 Set this equal to 0: 3x^2 - 48 = 0 Add 48 to each side: 3x^2 = 48 Divide each side by 3: x^2 = 16 Therefore, x = -4, 4 Test f(4) f(4) = 4^3 - 48(4) f(4) = 64 - 192 f(4) = [B]-128 <-- Local minimum[/B] Test f(-4) f(-4) = -4^3 - 48(-4) f(-4) = -64 + 192 f(-4) = [B]128 <-- Local maximum[/B]

The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost? Take the [URL='http://www.mathcelebrity.com/dfii.php?term1=-30x%5E2+%2B+360x+%2B+785&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']derivative of the profit function[/URL]: P'(x) = -60x + 360 We find the maximum when we set the profit derivative equal to 0 -60x + 360 = 0 Subtract 360 from both sides: -60x = -360 Divide each side by -60 [B]x = 6 <-- This is the ticket price to maximize profit[/B] Substitute x = 6 into the profit equation: P(6) = -30(6)^2 + 360(6) + 785 P(6) = -1080 + 2160 + 785 [B]P(6) = 1865[/B]

The graph of a polynomial f(x) = (2x - 3)(x - 4)(x + 3) has x-intercepts at 3 values. What are they? A few things to note: [LIST] [*]X-intercepts are found when y (or f(x)) is 0. [*]On the right side, we have 3 monomials. [*]Therefore, y or f(x) could be 0 when [U]any[/U] of these monomials is 0 [/LIST] The 3 monomials are: [LIST=1] [*]2x - 3 = 0 [*]x - 4 = 0 [*]x + 3 = 0 [/LIST] Find all 3 x-intercepts: [LIST=1] [*]2x - 3 = 0. [URL='https://www.mathcelebrity.com/1unk.php?num=2x-3%3D0&pl=Solve']Using our equation calculator[/URL], we see that x = [B]3/2 or 1.5[/B] [*]x - 4 = 0 [URL='https://www.mathcelebrity.com/1unk.php?num=x-4%3D0&pl=Solve']Using our equation calculator[/URL], we see that x = [B]4[/B] [*]x + 3 = 0 [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B3%3D0&pl=Solve']Using our equation calculator[/URL], we see that x = [B]-3[/B] [/LIST] So our 3 x-intercepts are: x = [B]{-3, 3/2, 4}[/B]

The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +300. Find the average velocity of the object during the first 3 seconds? (b) Use the Mean value Theorem to verify that at some time during the first 3 seconds of the fall the instantaneous velocity equals the average velocity. Find that time. Average Velocity: [ f(3) - f(0) ] / ( 3 - 0 ) Calculate f(3): f(3) = -4.9(3^2) + 300 f(3) = -4.9(9) + 300 f(3) = -44.1 + 300 f(3) = 255.9 Calculate f(0): f(0) = -4.9(0^2) + 300 f(0) = 0 + 300 f(0) = 300 So we have average velocity: Average velocity = (255.9 - 300)/(3 - 0) Average velocity = -44.1/3 Average velocity = -[B]14.7 [/B] Velocity is the first derivative of position s(t)=-4.9t^2 +300 s'(t) = -9.8t So we set velocity equal to average velocity: -9.8t = -14.7 Divide each side by -9.8 to solve for t, we get [B]t = 1.5[/B]

The length of a rectangle is three times its width.If the perimeter is 80 feet, what are the dimensions? We're given 2 equations: [LIST=1] [*]l = 3w [*]P = 80 = 2l + 2w = 80 [/LIST] Substitute (1) into (2) for l: 2(3w) + 2w = 80 6w + 2w = 80 8w = 80 Divide each side by 8: 8w/8 = 80/8 w = [B]10 [/B] Substitute w = 10 into (1) l = 3(10) l = [B]30[/B]

The length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 feet. Find the width and length of the building. P = 2l + 2w Since P = 120, we have: (1) 2l + 2w = 120 We are also given: (2) l = 3w - 6 Substitute equation (2) into equation (1) 2(3w - 6) + 2w = 120 Multiply through: 6w - 12 + 2w = 120 Combine like terms: 8w - 12 = 120 Add 12 to each side: 8w = 132 Divide each side by 8 to isolate w: w =16.5 Now substitute w into equation (2) l = 3(16.5) - 6 l = 49.5 - 6 l = 43.5 So (l, w) = (43.5, 16.5)

The length of a train car is 50.6 feet. This is 5.8 feet less than 6 times the width. What is the width? 5.8 feet less than 6 times the width is an algebraic expression: 6w - 5.8 We set this equal to the length of 50.6 6w - 5.8 = 50.6 Add 5.8 to each side: 6w - 5.8 + 5.8 = 50.6 + 5.8 Cancel the 5.8 on the left side: 6w = 56.4 Divide each side by 6: 6w/6 = 56.4/6 [URL='http://www.mathcelebrity.com/1unk.php?num=6w-5.8%3D50.6&pl=Solve']Typing this problem into the search engine[/URL], we get [B]w = 9.4[/B]. [MEDIA=youtube]gfM-d_Ej728[/MEDIA]

The length of the flag is 2 cm less than 7 times the width. The perimeter is 60cm. Find the length and width. A flag is a rectangle shape. So we have the following equations Since P = 2l + 2w, we have 2l + 2w = 60 l = 7w - 2 Substitute Equation 1 into Equation 2: 2(7w -2) + 2w = 60 14w - 4 + 2w = 60 16w - 4 = 60 Add 4 to each side 16w = 64 Divide each side by 16 to isolate w w = 4 Which means l = 7(4) - 2 = 28 - 2 = 26

The perimeter of a rectangle is 400 meters. The length is 15 meters less than 4 times the width. Find the length and the width of the rectangle. l = 4w - 15 Perimeter = 2l + 2w Substitute, we get: 400 = 2(4w - 15) + 2w 400 = 8w - 30 + 2w 10w - 30 = 400 Add 30 to each side 10w = 370 Divide each side by 10 to isolate w w = 37 Plug that back into our original equation to find l l = 4(37) - 15 l = 148 - 15 l = 133 So we have (l, w) = (37, 133)

The perimeter of a square with side a Perimeter of a square is 4s where s is the side length. With s = a, we have: P = [B]4a[/B]

The perpendicular height of a right-angled triangle is 70 mm longer than the base. Find the perimeter of the triangle if its area is 3000. [LIST] [*]h = b + 70 [*]A = 1/2bh = 3000 [/LIST] Substitute the height equation into the area equation 1/2b(b + 70) = 3000 Multiply each side by 2 b^2 + 70b = 6000 Subtract 6000 from each side: b^2 + 70b - 6000 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=b%5E2%2B70b-6000%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get: b = 50 and b = -120 Since the base cannot be negative, we use b = 50. If b = 50, then h = 50 + 70 = 120 The perimeter is b + h + hypotenuse Using the [URL='http://www.mathcelebrity.com/righttriangle.php?angle_a=&a=70&angle_b=&b=50&c=&pl=Calculate+Right+Triangle']right-triangle calculator[/URL], we get hypotenuse = 86.02 Adding up all 3 for the perimeter: 50 + 70 + 86.02 = [B]206.02[/B]

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company called Rachel and offered her long distance service for 1/2 cent per minute, but will charge a $46 monthly fee. How many minutes per month must Rachel talk on the phone to make the discount a better deal? Minutes Rachel talks = m Current plan cost = 0.12m New plan cost = 0.005m + 46 Set new plan equal to current plan: 0.005m + 46 = 0.12m Solve for [I]m[/I] in the equation 0.005m + 46 = 0.12m [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 0.005m and 0.12m. To do that, we subtract 0.12m from both sides 0.005m + 46 - 0.12m = 0.12m - 0.12m [SIZE=5][B]Step 2: Cancel 0.12m on the right side:[/B][/SIZE] -0.115m + 46 = 0 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 46 and 0. To do that, we subtract 46 from both sides -0.115m + 46 - 46 = 0 - 46 [SIZE=5][B]Step 4: Cancel 46 on the left side:[/B][/SIZE] -0.115m = -46 [SIZE=5][B]Step 5: Divide each side of the equation by -0.115[/B][/SIZE] -0.115m/-0.115 = -46/-0.115 m = [B]400 She must talk over 400 minutes for the new plan to be a better deal [URL='https://www.mathcelebrity.com/1unk.php?num=0.005m%2B46%3D0.12m&pl=Solve']Source[/URL][/B]

The points -5, -24 and 5,r lie on a line with slope 4. Find the missing coordinate r. Slope = (y2 - y1)/(x2 - x1) Plugging in our numbers, we get: 4 = (r - -24)/(5 - -5) 4 = (r +24)/10 Cross multiply: r + 24 = 40 Subtract 24 from each side: [B]r = 16[/B]

The points 6,4 and 9,r lie on a line with slope 3. Find the missing coordinate r. Slope = (y2 - y1)/(x2 - x1) Plugging in our numbers, we get: 3 = (r - 4)/(9 - 6) 3 = (r - 4)/3 Cross multiply: r - 4 = 9 Add 4 to each side: [B]r = 13[/B]

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack? backpack cost = b Cheap backpack = b - 15 The total of both items equals 75: b + b - 15 = 75 Solve for [I]b[/I] in the equation b + b - 15 = 75 [SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE] (1 + 1)b = 2b [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 2b - 15 = + 75 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants -15 and 75. To do that, we add 15 to both sides 2b - 15 + 15 = 75 + 15 [SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE] 2b = 90 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2b/2 = 90/2 b = 45 Cheap backpack = 45 - 15 = [B]30 [URL='https://www.mathcelebrity.com/1unk.php?num=b%2Bb-15%3D75&pl=Solve']Source[/URL][/B]

the product of a number and 15 is not less than 15 The phrase [I]a number[/I] means an arbitrary variable. Let's call it x. x the product of a number and 15 means we multiply x by 15 15x The phrase [I]not less than[/I] means greater than or equal to. We set 15x greater than prequel to 15 [B]15x >= 15 <-- This is our algebraic expression [/B] [U]If the problem asks you to solve for x:[/U] Divide each side by 15: 15x/15 >= 15/15 [B]x >= 1[/B]

The product of two consecutive integers is greater than 100 Take an integer x. Next consecutive integer is x + 1 The product of those integers is: x(x + 1) This product is greater than 100 which gives us the algebraic expression of: x(x + 1) > 100 IF we want to solve for x: x^2 + x > 100 Subtract 100 from each side: x^2 + x - 100 > 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-100%3E0&pl=All&hintnum=+0']Solve this quadratic:[/URL]

The product of two positive numbers is 96. Determine the two numbers if one is 4 more than the other. Let the 2 numbers be x and y. We have: [LIST=1] [*]xy = 96 [*]x = y - 4 [/LIST] [U]Substitute (2) into (1)[/U] (y - 4)y = 96 y^2 - 4y = 96 [U]Subtract 96 from both sides:[/U] y^2 - 4y - 96 = 0 [U]Factoring using our quadratic calculator, we get:[/U] (y - 12)(y + 8) So y = 12 and y = -8. Since the problem states positive numbers, we use [B]y = 12[/B]. Substituting y = 12 into (2), we get: x = 12 - 4 [B]x = 8[/B] [B]We have (x, y) = (8, 12)[/B]

The ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largest angle in degrees? Let the smallest angle be x. Then we have 3 angles based on the ratio: x, 2x, 3x We know the sum of the angles of a triangle equals 180. So we have: x + 2x + 3x = 180 6x = 180 Divide each side by 6: 6x/6 = 180/6 x = 30 The largest angle is 3(30) = [B]90 [MEDIA=youtube]l8Lc6YtK9dg[/MEDIA][/B]

The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no votes was 6 to 5. If there were 4570 no votes, what was the total number of votes? Set up a proportion where y is the number of yes votes to 4570 no votes 6/5 = y/4570 Using our [URL='http://www.mathcelebrity.com/prop.php?num1=6&num2=y&den1=5&den2=4570&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get: [B]y = 5484[/B]

The residents of a city voted on whether to raise property taxes. The ratio of yes votes to no votes was 4 to 3 . If there were 2958 no votes, what was the total number of votes? Set up a ratio of yes to no votes 4/3 = x/2958 Using our [URL='http://www.mathcelebrity.com/prop.php?num1=4&num2=x&den1=3&den2=2958&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get x = 3,944 for yes votes. Adding yes votes and no votes together to get total votes, we get: Total Votes = Yes Votes + No Votes Total Votes = 3,944 + 2,958 Total Votes = [B]6,902[/B]

The sales price of a new compact disc player is $210 at a local discount store. At the store where the sale is going on, each new cd is on sale for $11. If Kyle purchases a player and some cds for $243 how many cds did he purchase? If Kyle bought the player, he has 243 - 210 = 33 left over. Each cd is 11, so set up an equation to see how many CDs he bought: 11x = 33 Divide each side by 11 [B]x = 3[/B]

The sales tax on a computer was $33.60. If the sales tax rate is 7%, how much did the computer cost without tax? Let the cost of the computer be c. We have: 0.07c = 33.60 Solve for [I]c[/I] in the equation 0.07c = 33.60 [SIZE=5][B]Step 1: Divide each side of the equation by 0.07[/B][/SIZE] 0.07c/0.07 = 33.60/0.07 c = $[B]480[/B] [URL='https://www.mathcelebrity.com/1unk.php?num=0.07c%3D33.60&pl=Solve']Source[/URL]

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be equal to 3 miles? Set up a proportion of scale to actual distance 1/2 / 3/4 = x/3 4/3 = x/3 Cross multiply: 3x = 12 Divide each side by 3: 3x/3 = 12/3 x = [B]4 (1/2 inch sections) or 2 inches[/B]

The science club charges 4.50 per car at their car wash. Write and solve and inequality to find how many cars they have to wash to earn at least 300 Let x be the number of cars they wash. Set up our inequality. Note, at least 300 means 300 or greater, so we use greater than or equal to. [U]Inequality:[/U] [B]4.50x >= 300 [/B] [U]So solve for x, divide each side by 4[/U] [B]x >= 66.67[/B]

The sides of a triangle are consecutive numbers. If the perimeter of the triangle is 240 m, find the length of each side Let the first side be n. Next side which is consecutive is n + 1 Next side which is consecutive is n + 1 + 1 = n + 2 So we have the sum of 3 consecutive numbers is 240. We type in [I][URL='https://www.mathcelebrity.com/sum-of-consecutive-numbers.php?num=sumof3consecutivenumbersis240&pl=Calculate']sum of 3 consecutive numbers is 240[/URL][/I] into our search engine and we get: [B]79, 80, 81[/B]

The singular form of the word "dice" is "die". Tom was throwing a six-sided die. The first time he threw, he got a three; the second time he threw, he got a three again. What's the probability of getting a three at the third time? Since all trials are independent: 1/6 * 1/6 * 1/6 = [B]1/216[/B]

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer Let the integer be x. [LIST] [*]The sum of the integer and 12 is written as x + 12. [*]The square of a positive integer is written as x^2. [/LIST] We set these equal to each other: x^2 = x + 12 Subtract x + 12 from each side: x^2 - x - 12 = 0 We have a quadratic function. [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-x-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Run it through our search engine[/URL] and we get x = 3 and x = -4. The problem asks for a positive integer, so we have [B]x = 3[/B]

The square of a positive integer minus twice its consecutive integer is equal to 22. Find the integers. Let x = the original positive integer. We have: [LIST] [*]Consecutive integer is x + 1 [*]x^2 - 2(x + 1) = 22 [/LIST] Multiply through: x^2 - 2x - 2 = 22 Subtract 22 from each side: x^2 - 2x - 24 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2-2x-24%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get: x = 6 and x = -4 Since the problem states [U]positive integers[/U], we use: x = 6 and x + 1 = 7 [B](6, 7)[/B]

Write this out, let the number be x. sqrt(2x) = x - 4 since 4 less means subtract Square each side: sqrt(2x)^2 = (x - 4)^2 2x = x^2 - 8x + 16 Subtract 2x from both sides x^2 - 10x + 16 = 0 Using the [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2+-+10x+%2B+16+%3D+0&pl=Solve+Quadratic+Equation&hintnum=0']quadratic calculator[/URL], we get two potential solutions x = (2, 8) Well, 2 does not work, since sqrt(2*2) = 2 which is not 4 less than 2 However, 8 does work: sqrt(2*8) = sqrt(16) = 4, which is 4 less than the number 8.

The sum of 2 numbers is 18. 3 times the greater number exceeds 4 times the smaller number by 5. Find the numbers. Let the first number be x. The second number is y. We have: [LIST=1] [*]x + y = 18 [*]3x = 4y + 5 [/LIST] Rearrange (2), by subtracting 4y from each side: 3x - 4y = 5 So we have a system of equations: [LIST=1] [*]x + y = 18 [*]3x - 4y = 5 [/LIST] Using our [URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+y+%3D+18&term2=3x+-+4y+%3D+5&pl=Cramers+Method']simultaneous equations calculator[/URL], we get: [B]x = 11 y = 7[/B]

The sum of 2 numbers is 70. The difference of these numbers is 24. Write and solve a system of equations to determine the numbers. Let the two numbers be x and y. We have the following equations: [LIST=1] [*]x + y = 70 [*]x - y = 24 [/LIST] Add (1) to (2): 2x = 94 Divide each side by 2 [B]x = 47[/B] Plug this into (1) 47 + y = 70 Subtract 47 from each side, we have: [B]y = 23[/B]

[I]Is at least [/I]means greater than or equal to: 5x + 2x >= 70 If we combine like terms, we have: 7x >=70 We can further simplify by dividing each side of the inequality by 7 x >=10 If you want the interval notation for that, use the [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E%3D10&pl=Show+Interval+Notation']interval notation calculator[/URL].

The sum of a number and its square is 72. find the numbers? Let the number be n. We have: n^2 + n = 72 Subtract 72 from each side: n^2 + n - 72 = 0 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-72%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we have: [B]n = 8 or n = -9 [/B] Since the numbers do not state positive or negative, these are the two solutions.

A number means an arbitrary variable, let's call it x. The sum of a number and itself means adding the number to itself x + x Simplified, we have 2x The word is means equal to, so we have an algebraic expression of: [B]2x= 8 [/B] IF you need to solve this equation, divide each side by 2 [B]x = 4[/B]

Twice n means we multiply n by 2 2n The sum of n and twice n means we add n + 2n The word [I]is[/I] means equal to, so we set that expression above equal to 12 n + 2n = 12 Combine like terms: 3n = 12 Divide each side of the equation by 3 to isolate n n = 4 Check our work Twice n is 2*4 = 8 Add that to n = 4 8 + 4 12

The sum of the ages of levi and renee is 89 years. 7 years ago levi's age was 4 times renees age. How old is Levi now? Let Levi's current age be l. Let Renee's current age be r. Were given two equations: [LIST=1] [*]l + r = 89 [*]l - 7 = 4(r - 7) [/LIST] Simplify equation 2 by multiplying through: [LIST=1] [*]l + r = 89 [*]l - 7 = 4r - 28 [/LIST] Rearrange equation 1 to solve for r and isolate l by subtracting l from each side: [LIST=1] [*]r = 89 - l [*]l - 7 = 4r - 28 [/LIST] Now substitute equation (1) into equation (2): l - 7 = 4(89 - l) - 28 l - 7 = 356 - 4l - 28 l - 7 = 328 - 4l To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=l-7%3D328-4l&pl=Solve']type the equation into our search engine[/URL] and we get: l = [B]67[/B]

The sum of the digits of a 2 digit number is 10. The value of the number is four more than 15 times the unit digit. Find the number. Let the digits be (x)(y) where t is the tens digit, and o is the ones digit. We're given: [LIST=1] [*]x + y = 10 [*]10x+ y = 15y + 4 [/LIST] Simplifying Equation (2) by subtracting y from each side, we get: 10x = 14y + 4 Rearranging equation (1), we get: x = 10 - y Substitute this into simplified equation (2): 10(10 - y) = 14y + 4 100 - 10y = 14y + 4 [URL='https://www.mathcelebrity.com/1unk.php?num=100-10y%3D14y%2B4&pl=Solve']Typing this equation into our search engine[/URL], we get: y = 4 Plug this into rearranged equation (1), we get: x = 10 - 4 x = 6 So our number xy is [B]64[/B]. Let's check our work against equation (1): 6 + 4 ? 10 10 = 10 Let's check our work against equation (2): 10(6)+ 4 ? 15(4) + 4 60 + 4 ? 60 + 4 64 = 64

The sum of the digits of a certain two-digit number is 16. Reversing its digits increases the number by 18. What is the number? Let x and (16-x) represent the original ten and units digits respectively Reversing its digits increases the number by 18 Set up the relational equation [10x + (16-x)] + 18 = 10(16 - x) + x Solving for x 9x + 34 = 160 - 9x Combine like terms 18x = 126 Divide each side of the equation by 18 18x/18 = 126/18 x = 7 So 16 - x = 16 - 7 = 9 The first number is 79, the other number is 97. and 97 - 79 = 18 so we match up. The number in our answer is [B]79[/B]

The sum of the squares of two consecutive positive integers is 61. Find these two numbers. Let the 2 consecutive integers be x and x + 1. We have: x^2 + (x + 1)^2 = 61 Simplify: x^2 + x^2 + 2x + 1 = 61 2x^2 + 2x + 1 = 61 Subtract 61 from each side: 2x^2 + 2x - 60 = 0 Divide each side by 2 x^2 + x - 30 Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL], we get: x = 5 and x = -6 The question asks for [I]positive integers[/I], so we use [B]x = 5. [/B]This means the other number is [B]6[/B].

Let the 3 integers be x, y, and z. y = x + 1 z = y + 1, or x + 2. Set up our equation: x + (x + 1) + (x + 2) = 42 Group our variables and constants: (x + x + x) + (1 + 2) = 42 3x + 3 = 42 Subtract 3 from each side: 3x = 39 Divide each side of the equation by 3: [B]x = 13 So y = x + 1 = 14 z = x + 2 = 15 (x,y,z) = (13,14,15)[/B]

The sum of twice an integer and 3 times the next consecutive integer is 48 Let the first integer be n This means the next consecutive integer is n + 1 Twice an integer means we multiply n by 2: 2n 3 times the next consecutive integer means we multiply (n + 1) by 3 3(n + 1) The sum of these is: 2n + 3(n + 1) The word [I]is[/I] means equal to, so we set 2n + 3(n + 1) equal to 48: 2n + 3(n + 1) = 48 Solve for [I]n[/I] in the equation 2n + 3(n + 1) = 48 We first need to simplify the expression removing parentheses Simplify 3(n + 1): Distribute the 3 to each term in (n+1) 3 * n = (3 * 1)n = 3n 3 * 1 = (3 * 1) = 3 Our Total expanded term is 3n + 3 Our updated term to work with is 2n + 3n + 3 = 48 We first need to simplify the expression removing parentheses Our updated term to work with is 2n + 3n + 3 = 48 [SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE] (2 + 3)n = 5n [SIZE=5][B]Step 2: Form modified equation[/B][/SIZE] 5n + 3 = + 48 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 3 and 48. To do that, we subtract 3 from both sides 5n + 3 - 3 = 48 - 3 [SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE] 5n = 45 [SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE] 5n/5 = 45/5 Cancel the 5's on the left side and we get: n = [B]9[/B]

The sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifteen The sum of two y and the quantity of three plus y 2y + (3 + y) twice the quantity two y minus two 2(2y - 2) The sum of two y and the quantity of three plus y plus twice the quantity two y minus two 2y + (3 + y) + 2(2y - 2) Equals 15 to get our algebraic expression [B]2y + (3 + y) + 2(2y - 2) = 15[/B] [B][/B] If the problem asks you to solve for yL 2y + 3 + y + 4y - 4 = 15 Group like terms: 7y - 1 = 15 Add 1 each side: 7y = 16 Divide each side by 7: y = [B]16/7[/B]

The sum of two y and three 2y + 3 the difference of three y and one 3y - 1 is the same as means equal to. Set 2y + 3 equal to 3y - 1 for our final algebraic expression: [B]2y + 3 = 3y - 1[/B] [B][/B] If the problem asks you to solve for y, subtract 2y from each side: 3 = y - 1 Add 1 to each side: y = [B]4[/B]

The temperature inside the lab refrigerator is no more than 35 . Use t to represent the temperature (in ) of the refrigerator. The phrase [I]no more than[/I] means less than or equal to. We have this inequality: [B]t <= 35[/B]

The total age of three cousins is 48. Suresh is half as old as Hakima and 4 years older than Saad. How old are the cousins? Let a be Suresh's age, h be Hakima's age, and c be Saad's age. We're given: [LIST=1] [*]a + h + c = 48 [*]a = 0.5h [*]a = c + 4 [/LIST] To isolate equations in terms of Suresh's age (a), let's do the following: [LIST] [*]Rewriting (3) by subtracting 4 from each side, we get c = a - 4 [*]Rewriting (2) by multiply each side by 2, we have h = 2a [/LIST] We have a new system of equations: [LIST=1] [*]a + h + c = 48 [*]h = 2a [*]c = a - 4 [/LIST] Plug (2) and (3) into (1) a + (2a) + (a - 4) = 48 Group like terms: (1 + 2 + 1)a - 4 = 48 4a - 4 = 48 [URL='https://www.mathcelebrity.com/1unk.php?num=4a-4%3D48&pl=Solve']Typing this equation into our search engine[/URL], we get: [B]a = 13 [/B]<-- Suresh's age This means that Hakima's age, from modified equation (2) above is: h = 2(13) [B]h = 26[/B] <-- Hakima's age This means that Saad's age, from modified equation (3) above is: c = 13 - 4 [B]c = 9[/B] <-- Saad's age [B] [/B]

The volleyball team and the wrestling team at Clarksville High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $4 per car. In addition, they have already brought in $81 from past fundraisers. The wrestling team has raised $85 in the past, and they are making $2 per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take? Set up the earnings equation for the volleyball team where w is the number of cars washed: E = Price per cars washed * w + past fundraisers E = 4w + 81 Set up the earnings equation for the wrestling team where w is the number of cars washed: E = Price per cars washed * w + past fundraisers E = 2w + 85 If the raised the same amount in total, set both earnings equations equal to each other: 4w + 81 = 2w + 85 Solve for [I]w[/I] in the equation 4w + 81 = 2w + 85 [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 4w and 2w. To do that, we subtract 2w from both sides 4w + 81 - 2w = 2w + 85 - 2w [SIZE=5][B]Step 2: Cancel 2w on the right side:[/B][/SIZE] 2w + 81 = 85 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 81 and 85. To do that, we subtract 81 from both sides 2w + 81 - 81 = 85 - 81 [SIZE=5][B]Step 4: Cancel 81 on the left side:[/B][/SIZE] 2w = 4 [SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE] 2w/2 = 4/2 w = [B]2 <-- How many cars it will take [/B] To get the total earnings, we take either the volleyball or wrestling team's Earnings equation and plug in w = 2: E = 4(2) + 81 E = 8 + 81 E = [B]89 <-- Total Earnings[/B]

120/3/4 = x/6 Cross multiply: 0.75x = 720 Divide each side of the equation by 0.75 [B]x = 960[/B]

There are 13 animals in the barn. some are chickens and some are pigs. there are 40 legs in all. How many of each animal are there? Chickens have 2 legs, pigs have 4 legs. Let c be the number of chickens and p be the number of pigs. Set up our givens: (1) c + p = 13 (2) 2c + 4p = 40 [U]Rearrange (1) to solve for c by subtracting p from both sides:[/U] (3) c = 13 - p [U]Substitute (3) into (2)[/U] 2(13 - p) + 4p = 40 26 - 2p + 4p = 40 [U]Combine p terms[/U] 2p + 26 = 40 [U]Subtract 26 from each side:[/U] 2p = 14 [U]Divide each side by 2[/U] [B]p = 7[/B] [U]Substitute p = 7 into (3)[/U] c = 13 - 7 [B]c = 6[/B] For a shortcut, you could use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+p+%3D+13&term2=2c+%2B+4p+%3D+40&pl=Cramers+Method']simultaneous equations calculator[/URL]

There are 2 consecutive integers. Twice the first increased by the second yields 16. What are the numbers? Let x be the first integer. y = x + 1 is the next integer. We have the following givens: [LIST=1] [*]2x + y = 16 [*]y = x + 1 [/LIST] Substitute (2) into (1) 2x + (x + 1) = 16 Combine x terms 3x + 1 = 16 Subtract 1 from each side 3x = 15 Divide each side by 3 [B]x = 5[/B] So the other integer is 5 + 1 = [B]6[/B]

There are 32 female performers in a dance recital. The ratio of men to women is 3:8. How many men are in the dance recital? 3:8 = x:32 3/8 = x/32 Cross multiply: 8x = 96 Divide each side by 8 x = 12 Check our work: 12:32 Divide each part by 4 12/4 = 3 and 32/4 = 8 so we have 3:8 :)

[SIZE=6]There are 812 students in a school. There are 36 more girls than boys. How many girls are there? Let b be boys Let g be girls We're given two equations:[/SIZE] [LIST=1] [*][SIZE=6]b + g = 812[/SIZE] [*][SIZE=6]g = b + 36[/SIZE] [/LIST] [SIZE=6]Rearrange equation 2 to subtract b from each side: [/SIZE] [LIST=1] [SIZE=6] [LIST][*]b + g = 812[/LIST] [LIST][*]-b + g = 36[/LIST][/SIZE] [/LIST] [SIZE=6]Add equation (1) to equation (2): b - b + 2g = 812 + 36 The b's cancel: 2g = 848 Divide each side by 2: 2g/2 = 848/2 g = [B]424[/B] [B][/B] To find b, we put g= 424 into equation 1: b + 424 = 812 b = 812 - 424 b = [B]388[/B] [MEDIA=youtube]JO1b7qVwWoI[/MEDIA] [/SIZE]

There are 85 students in a class, 40 of them like math,31 of them like science, 12 of them like both, how many don't like either. We have the following equation: Total Students = Students who like math + students who like science - students who like both + students who don't like neither. Plug in our knowns, we get: 85 = 40 + 31 - 12 + Students who don't like neither 85 = 59 + Students who don't like neither Subtract 59 from each side, we get: Students who don't like neither = 85 - 59 Students who don't like neither = [B]26[/B]

There are two numbers. The sum of 4 times the first number and 3 times the second number is 24. The difference between 2 times the first number and 3 times the second number is 24. Find the two numbers. Let the first number be x and the second number be y. We have 2 equations: [LIST=1] [*]4x + 3y = 24 [*]2x - 3y = 24 [/LIST] Without doing anything else, we can add the 2 equations together to eliminate the y term: (4x + 2x) + (3y - 3y) = (24 + 24) 6x = 48 Divide each side by 6: [B]x = 8 [/B] Substitute this into equation (1) 4(8) + 3y = 24 32 + 3y = 24 [URL='https://www.mathcelebrity.com/1unk.php?num=32%2B3y%3D24&pl=Solve']Type 32 + 3y = 24 into our search engine[/URL] and we get [B]y = 2.6667[/B].

There is an escalator that is 1090.3 feet long and drops a vertical distance of 193.4 feet. What is its angle of depression? The sin of the angle A is the length of the opposite side / hypotenuse. sin(A) = Opposite / Hypotenuse sin(A) = 193.4 / 1090/3 sin(A) = 0.1774 [URL='https://www.mathcelebrity.com/anglebasic.php?entry=0.1774&pl=arcsin']We want the arcsin(0.1774)[/URL]. [B]A = 10.1284[/B]

Set up two equations: (1) b = g + 16 (2) b + g = 150 Substitute equation (1) into (2) (g + 16) + g = 150 Combine like terms 2g + 16 = 150 Subtract 16 from each side 2g = 134 Divide each side by 2 to isolate g g = 67 Substitute this into equation (1) b = 67 + 16 [B]b = 83[/B]

Twice x means we multiply x by 2: 2x five less than twice x 2x - 5 Three x 3x The word [I]is[/I] means equal to. Set 2x - 5 equal to 3x for our algebraic expression: [B]2x - 5 = 3x [/B] If the problem asks you to solve for x, subtract 2x from each side [B]x = -5[/B]

Tiffany is 59 years old. The sum of the ages of Tiffany and Maria is 91. How old is Maria? Tiffany + Maria = 91 59 + Maria = 91 Subtract 59 from each side Maria = 91 - 59 [B]Maria = 32[/B]

Let h be the number of hours that pass when Charlie starts. We have the following equations: [LIST] [*]Charlie: D = 40h + 9 [*]Danny: D = 55h [/LIST] Set them equal to each other: 40h + 9 = 55h Subtract 40h from both sides: 15h = 9 h = 3/5 [B]3/5 of an hour = 3(60)/5 = 36 minutes[/B]

Thank you so much [QUOTE="math_celebrity, post: 1003, member: 1"]Let h be the number of hours that pass when Charlie starts. We have the following equations: [LIST] [*]Charlie: D = 40h + 9 [*]Danny: D = 55h [/LIST] Set them equal to each other: 40h + 9 = 55h Subtract 40h from both sides: 15h = 9 h = 3/5 [B]3/5 of an hour = 3(60)/5 = 36 minutes[/B][/QUOTE]

Tina's mom made brownies. When tinas friend came over they ate 1/3 of the brownies. Her sister ate 2 and her dad ate 4. If there are 26 brownies left. How many did her mom make Let b denote the number of brownies Tina's mom made. We're given: b - 1/3b - 2 - 4 = 26 Combining like terms, we have: 2b/3 - 6 = 26 Add 6 to each side, we get: 2b/3 = 32 To solve this equation for b, we [URL='https://www.mathcelebrity.com/prop.php?num1=2b&num2=32&den1=3&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']type it in our math engine[/URL] and we get: b = [B]48[/B]

Today is my birthday! Four-fifths of my current age is greater than three-quarters of my age one year from now. Given that my age is an integer number of years, what is the smallest my age could be? Let my current age be a. We're given: 4/5a > 3/4(a + 1) Multiply through on the right side: 4a/5 > 3a/4 + 3/4 Let's remove fractions by multiply through by 5: 5(4a/5) > 5(3a/4) + 5(3/4) 4a > 15a/4 + 15/4 Now let's remove the other fractions by multiply through by 4: 4(4a) > 4(15a/4) + 4(15/4) 16a > 15a + 15 [URL='https://www.mathcelebrity.com/1unk.php?num=16a%3E15a%2B15&pl=Solve']Typing this inequality into our search engine[/URL], we get: a > 15 This means the smallest [I]integer age[/I] which the problem asks for is: 15 + 1 = [B]16[/B]

Tom has 6 fewer pencils than ari. Tom has 7 pencils. How many pencils does Ari have? 6 fewer means less. Let a = Ari's pencils. We have: a - 6 = 7 Add 6 to each side [B]a = 13[/B]

Free Triangle Inequality Calculator - This calculator displays 2 scenarios
1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle
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Triangle KLM has vertices at . k(-2,-2), l(10,-2), m(4,4) What type of triangle is KLM? [URL='https://www.mathcelebrity.com/slope.php?xone=-2&yone=-2&slope=+2%2F5&xtwo=10&ytwo=-2&pl=You+entered+2+points']Side 1: KL[/URL] = 12 [URL='https://www.mathcelebrity.com/slope.php?xone=10&yone=-2&slope=+2%2F5&xtwo=4&ytwo=4&pl=You+entered+2+points']Side 2: LM[/URL] = 8.4853 [URL='https://www.mathcelebrity.com/slope.php?xone=-2&yone=2&slope=+2%2F5&xtwo=4&ytwo=4&pl=You+entered+2+points']Side 3: KM[/URL] = 6.3246 Then, we want to find the type of triangle. Using our [URL='https://www.mathcelebrity.com/tribasic.php?side1input=12&side2input=8.4853&side3input=6.3246&angle1input=&angle2input=&angle3input=&pl=Solve+Triangle']triangle solver with our 3 sides[/URL], we get: [B]Obtuse, Scalene[/B]

Also classifies triangles based on sides and angles entered.

A triangle with a perimeter of 120. What degree are the three sides?

What kind of triangle? Do you have side lengths? I need more information.

Free True False Equations Calculator - Determines if a set of addition and subtraction of numbers on each side of an equation are equivalent. Also known as true or false equations

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e. Let's choose x. Twice a number: 2x decreased by eight 2x - 8 [I]is [/I]means equal to. Set 2x - 8 equal to zero for our algebraic expression: [B]2x - 8 = 0 [/B] If the problem asks you to solve for x, add 8 to each side: 2x = 8 Divide each side by 2: x= [B]4[/B]

twice the difference of a number and 3 is equal to 3 times the sum of a number and 2. We've got 2 algebraic expressions here. Let's take them in parts. Left side algebraic expression: twice the difference of a number and 3 [LIST] [*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x. [*]The word [I]difference[/I] means we subtract 3 from the variable x [*]x - 3 [*]Twice this difference means we multiply (x - 3) by 2 [*]2(x - 3) [/LIST] Right side algebraic expression: 3 times the sum of a number and 2 [LIST] [*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x. [*]The word [I]sum[/I] means we add 2 to the variable x [*]x + 2 [*]3 times the sum means we multiply (x + 2) by 3 [*]3(x + 2) [/LIST] Now, we have both algebraic expressions, the problem says [I]is equal to[/I] This means we have an equation, where we set the left side algebraic expression equal to the right side algebraic expression using the equal sign (=) to get our answer [B]2(x - 3) = 3(x + 2)[/B]

twice the difference of a number and 55 is equal to 3 times the sum of a number and 8 Take this algebraic expression in pieces. Left side: The phrase [I]a number[/I] means an arbitrary variable, let's call it x. x The difference of this number and 55 means we subtract 55 from x x - 55 Twice the difference means we multiply x - 55 by 2 2(x - 55) Right side: The phrase [I]a number[/I] means an arbitrary variable, let's call it x. x The sum of a number and 8 means we add 8 to x x + 8 3 times the sum means we multiply x + 8 by 3 3(x + 8) Now that we have the left and right side of the expressions, we see the phrase [I]is equal to[/I]. This means an equation, so we set the left side equal to the right side: [B]2(x - 55) = 3(x + 8)[/B]

seven plus x 7 + x Twice the quantity of seven plus x 2(7 + x) Difference of x and seven x - 7 The phrase [I]is the same as[/I] means equal to. This is our algebraic expression: [B]2(7 + x) = x - 7 [/B] If the problem asks you to solve for x, distribute 2 on the left side: 14 + 2x = x - 7 Subtract x from the right side 14 + x = -7 Subtract 14 from each side [B]x = -21[/B]

[SIZE=6]Twice the sum of a number and 6 is equal to three times the difference of the number and 3. Find the number. The phrase [/SIZE][I][SIZE=7]a number[/SIZE][/I][SIZE=6] means an arbitrary variable, let's call it x. The sum of a number and 6 means we add 6 to x: x + 6 Twice the sum of a number and 6 means we multiply x + 6 by 2: 2(x + 6) the difference of the number and 3 means we subtract 3 from x x - 3 three times the difference of the number and 3 means we multiply x - 3 by 3: 3(x- 3) The word [I]is[/I] means we set 2(x + 6) equal to 3(x - 3) 2(x + 6) = 3(x - 3) Use the distributive property to multiply through: 2x + 12 = 3x - 9 Subtract 2x from each side: 2x - 2x + 12 = 3x - 2x - 9 x - 9 = 12 Add 9 to each side: x - 9 + 9 = 12 + 9 x = [B]21[/B] [B][/B] [B][MEDIA=youtube]CeZl_oZnSiw[/MEDIA][/B][/SIZE]

two mechanics worked on a car. the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. together they charged a total of 1225. what was the rate charged per hour by each mechanic if the sum of the two rates was 170 per hour? Set up two equations: (1) 10x + 5y = 1225 (2) x + y = 170 Rearrange (2) x = 170 - y Substitute that into (1) 10(170 - y) + 5y = 1225 1700 - 10y + 5y = 1225 1700 - 5y = 1225 Move 5y to the other side 5y + 1225 = 1700 Subtract 1225 from each side 5y =475 Divide each side by 5 [B]y = 95[/B] Which means x = 170 - 95, [B]x = 75[/B]

Two numbers have a sum of 20. If one number is p, express the other in terms of p. If the sum is 20 and one number is p, then let the other number be q. We have: p + q = 20 We want q, so we subtract p from each side: [B]q = 20 - p[/B]

Two numbers total 50 and have a difference of 28. Find the two numbers. Using x and y as our two numbers, we write the following 2 equations: [LIST=1] [*]x + y = 50 [*]x - y = 28 [/LIST] Add the 2 rows: 2x = 78 Divide each side by 2: [B]x = 39[/B] If x = 39, then from (1), we have y = 50 - 39 [B]y = 11[/B]

Let the numbers be x and y. Set up our givens: [LIST=1] [*]x + y = 83 [*]x - y = 17 [/LIST] [U]Add equation (1) to equation (2)[/U] x + x + y - y = 83 + 17 [U]The y-terms cancel out:[/U] 2x = 100 [U]Divide each side by 2:[/U] 2x/2= 100/2 x = [B]50[/B] [U] Plug x = 50 into equation (1)[/U] 50 + y = 83 [U]Subtract 50 from each side:[/U] 50 - 50 + y = 83 - 50 [U]Cancel the 50 on the left side:[/U] y = [B]33 [/B] So our two numbers (x, y) = (33, 50) [MEDIA=youtube]jajO043ChUM[/MEDIA]

Cross multiply: ub = ak Divide each side of the equation by k to isolate a: a = ub/k [MEDIA=youtube]A3NW3Y68iNY[/MEDIA]

U=ak/b, for a [U]Cross multiply:[/U] Ub = ak [U]Divide each side by k[/U] [B]a = Ub/k[/B]

V ? E + F = 2 for e To solve this literal equation, we want to isolate e. Add E to both sides: V ? E + F + E = 2 + E The E's cancel on the left side, so we have: V + F = 2 + E Subtract 2 from each side: V + F - 2 = 2 + E + 2 The 2's cancel on the right side, so we have: E = [B]V + F - 2[/B]

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimited rentals. What number of movies rentals is plan B less than plan A? Let x equal the number of movies rented and C the cost for rentals Plan A: C = 1.25x + 25 Plan B: C = 40 Set up the inequality: 1.25x + 25 > 40 Subtract 25 from each side: 1.25x > 15 Divide each side of the inequality by 1.25 x > 12 So [B]13[/B] rentals or more make Plan B less than Plan A.

vw^2+y=x for w This is an algebraic expression. Subtract y from each side: vw^2 + y - y = x - y The y's cancel on the left side, so we're left with: vw^2 = x - y Divide each side by v w^2 = (x - y)/v Take the square root of each side: w = [B]Sqrt((x - y)/v)[/B]

[QUOTE="Jahn, post: 78, member: 5"]Water flows from tank A to tank B at the rate of 2 litres per minute. Initially tank A has 200 litres in it and tank B has 100 Litres in it. Water drains from tank B at 0.5 litres per minute. After how many minutes are there equal volumes of water in the 2 tanks? Write an equation and solve it.[/QUOTE] Tank A: V = 200 - 2x Tank B: V = 100 - 0.5x Where x is the number of minutes passed. Set them equal to each other 200 - 2x = 100 - 0.5x Subtract 100 from each side: 100 - 2x = -0.5x Add 2x to each side: 1.5x = 100 Divide each side of the equation by x: x = 66.66666667

Wendy is paid $7.50 per hour plus a bonus of $80 each week. Last week Wendy earned $312.50. How many hours did Wendy work last week? Setup the earnings equation with h hours: 7.5h + 80 = 312.50 Solve for [I]h[/I] in the equation 7.5h + 80 = 312.50 [SIZE=5][B]Step 1: Group constants:[/B][/SIZE] We need to group our constants 80 and 312.50. To do that, we subtract 80 from both sides 7.5h + 80 - 80 = 312.50 - 80 [SIZE=5][B]Step 2: Cancel 80 on the left side:[/B][/SIZE] 7.5h = 232.5 [SIZE=5][B]Step 3: Divide each side of the equation by 7.5[/B][/SIZE] 7.5h/7.5 = 232.5/7.5 h = [B]31 [URL='https://www.mathcelebrity.com/1unk.php?num=7.5h%2B80%3D312.50&pl=Solve']Source[/URL][/B]

what integer is tripled when 9 is added to 3 fourths of it? Let the integer be n. Tripling an integer means multiplying it by 3. We're given: 3n = 3n/4 + 9 Since 3 = 12/4, we have: 12n/4 = 3n/4 + 9 Subtract 3n/4 from each side: 9n/4 = 9 [URL='https://www.mathcelebrity.com/prop.php?num1=9n&num2=9&den1=4&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Typing this equation into the search engine[/URL], we get: [B]n = 4[/B]

What is the annual nominal rate compounded daily for a bond that has an annual yield of 5.4%? Round to three decimal places. Use a 365 day year. [U]Set up the accumulation equation:[/U] (1+i)^365 = 1.054 [U]Take the natural log of each side[/U] 365 * Ln(1 + i) = 1.054 Ln(1 + i) = 0.000144089 [U]Use each side as a exponent to eulers constant e[/U] (1 + i) = e^0.000144089 1 + i = 1.000144099 i = 0.000144099 or [B].0144099%[/B]

What is the sample space for a 10 sided die? Sample space means the set of all possible outcomes. For a 10-sided die, we have: [B]{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}[/B]

Answer Choices: A. 6 B. 1/2 C. 1/3 D. 1/4 [U]Multiply through by 6:[/U] 2 * 6x/3 + 6/6 = 6/3 4x + 1 = 2 [U]Subtract 1 from each side:[/U] 4x + 1 - 1 = 2 - 1 4x = 1 [U]Divide each side by 4:[/U] 4x/4 = 1/4 x = [B]1/4[/B] [MEDIA=youtube]jywMlPs3c2w[/MEDIA]

What number when multiplied by four exceeds itself by 42? Let the number be n. We have: 4n = n + 42 Subtract n from each side: 3n = 42 Divide each side by 3 [B]n = 14[/B]

What pair of consecutive integers gives the following: 7 times the smaller is less than 6 times the larger? Let x and y be consecutive integers, where y = x + 1 We have 7x < 6y as our inequality. Substituting x, y = x + 1, we have: 7x < 6(x + 1) 7x < 6x + 6 Subtracting x from each side, we have: x < 6, so y = 6 + 1 = 7 (x, y) = (6, 7)

whats the probability of rolling a 5 and then rolling a number less then 2 [U]Roll a 5:[/U] There's only one 5 on a six sided die P(X = 5) = 1/6 A number less than 2 is only 1: P(X < 2) = P(X = 1) P(X = 1) = 1/6 Since each event is independent, we multiply: P(X = 5) * P(X = 1) = 1/6 * 1/6 P(X = 5) * P(X = 1) = [B]1/36[/B]

When 28 is subtracted from the square of a number, the result is 3 times the number. Find the negative solution. Let the number be n. Square of a number: n^2 28 is subtracted from the square of a number: n^2 - 28 3 times the number: 3n [I]The result is[/I] mean an equation, so we set n^2 - 28 = 3n n^2 - 28 = 3n Subtract 3n from each side: n^2 - 3n - 28 = 3n - 3n The right side cancels to 0, so we have: n^2 - 3n - 28 = 0 This is a quadratic equation in standard form, so we [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-28%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']use our quadratic calculator[/URL] to solve: We get two solutions for n: n = (-4, 7) The question asks for the negative solution, so our answer is: [B]n = -4[/B]

When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 integers. What is the difference of the product minus the sum? Let the 3 consecutive positive integers be: [LIST=1] [*]x [*]x + 1 [*]x + 2 [/LIST] The product is: x(x + 1)(x + 2) The sum is: x + x + 1 + x + 2 = 3x + 3 We're told the product is equivalent to: x(x + 1)(x + 2) = 16(3x + 3) x(x + 1)(x + 2) = 16 * 3(x + 1) Divide each side by (x + 1) x(x + 2) = 48 x^2 + 2x = 48 x^2 + 2x - 48 = 0 Now subtract the sum from the product: x^2 + 2x - 48 - (3x + 3) [B]x^2 - x - 51[/B]

When 4 is subtracted from the square of a number, the result is 3 times the number. Find the positive solution. Let the number be n. We have: n^2 - 4 = 3n Subtract 3n from each side: n^2 - 3n - 4 = 0 [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-3n-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Typing this quadratic equation into the search engine[/URL], we get: n = (-1, 4) The problem asks for the positive solution, so we get [B]n = 4[/B].

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the number. Find the number The phrase [I]a number, [/I]means an arbitrary variable, let's call it "x". 4 times a number, increased by 40, means we multiply 4 times x, and then add 40 4x + 40 100 decreased by the number means we subtract x from 100 100 - x The problem tells us both of these expressions are the same, so we set them equal to each other: 4x + 40 = 100 - x Add x to each side: 4x + x + 40 = 100 - x + x The x's cancel on the right side, so we have: 5x + 40 = 100 [URL='https://www.mathcelebrity.com/1unk.php?num=5x%2B40%3D100&pl=Solve']Typing this equation into the search engine[/URL], we get [B]x = 12[/B].

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the number The phrase [I]a number[/I] means an arbitrary variable, let's call it x. x 4 times a number means we multiply x by 4: 4x Increased by 40 means we add 40 to 4x: 4x + 40 100 decreased by the number means we subtract x from 100: 100 - x The phrase [I]is the same as[/I] means equal to, so we set 4x + 40 equal to 100 - x 4x + 40 = 100 - x Solve for [I]x[/I] in the equation 4x + 40 = 100 - x [SIZE=5][B]Step 1: Group variables:[/B][/SIZE] We need to group our variables 4x and -x. To do that, we add x to both sides 4x + 40 + x = -x + 100 + x [SIZE=5][B]Step 2: Cancel -x on the right side:[/B][/SIZE] 5x + 40 = 100 [SIZE=5][B]Step 3: Group constants:[/B][/SIZE] We need to group our constants 40 and 100. To do that, we subtract 40 from both sides 5x + 40 - 40 = 100 - 40 [SIZE=5][B]Step 4: Cancel 40 on the left side:[/B][/SIZE] 5x = 60 [SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE] 5x/5 = 60/5 x = [B]12[/B] Check our work for x = 12: 4(12) + 40 ? 100 - 12 48 + 40 ? 100 - 12 88 = 88

When Ms. Thelma turned on her oven, the temperature inside was 70 degrees F. The temperature began to rise at a rate of 20 degrees per minute. How Long did it take for the oven to reach 350 degrees F? Figure out how many degrees we have left: 350 - 70 = 280 Let m = minutes 20m = 280 Divide each side by m [B]m = 14[/B]

When the circumference of a circle is increased from 10 pi inches to 15 pi inches, by how many inches is the radius increased? C = 2pir Smaller circle: 2pir = 10pi Divide each side by 2pi: r = 5 Larger circle: 2pir = 15pi Divide each side by 2pi: r = 7.5 Difference = 7.5 - 5 = [B]2.5 or 2&1/2 [MEDIA=youtube]HvMNNffcv78[/MEDIA][/B]

When the side of a square is doubled in length, its area increases by 432 square inches. What is the size of the original square? Original square side length is s Area = s^2 Double the side lengths to 2s New area = (2s)^2 = 4s^2 Setup the difference relation: 4s^2 - s^2 = 432 3s^2 = 432 Divide each side by 3: 3s^2/3 = 432/3 s^2 = 144 s = [B]12[/B]

Which of the following equations represents a line that is parallel to the line with equation y = -3x + 4? A) 6x + 2y = 15 B) 3x - y = 7 C) 2x - 3y = 6 D) x + 3y = 1 Parallel lines have the same slope, so we're looking for a line with a slope of 3, in the form y = mx + b. For this case, we want a line with a slope of -3, like our given line. If we rearrange A) by subtracting 6x from each side, we get: 2y = -6x + 15 Divide each side by 2, we get: y = -3x + 15/2 This line is in the form y = mx + b, where m = -3. So our answer is [B]A[/B].

Write a fraction with a denominator of 9. The fraction should be less than 1/2 Let n be the numerator. We have: n/9 < 1/2 multiply each side by 9: 9n/9 < 9/2 n < 9/2 Examples are 8/2, 7/2, 6/2, 5/2, 4/2, 3/2,

Your answer is correct. Here is how I set up the profit equation where h is the hours worked and x is the supply cost: P(h) = 15.35h + x We know P(4) = 141.73 P(4) = 15.35(4) + x 141.73 = 15.35(4) + x Simplify 141.73 = 61.4 + x Subtract 61.4 from each side: [B]x = 80.33[/B]

wy - ma = ay/n for y Subtract ay/n from each side: wy - ma - ay/n = ay/n - ay/n wy - ma - ay/n = 0 Now add ma to each side: wy - ay/n = ma Factor out y: y(w - a/n) = ma Divide each side by (w - a/n) y = [B]ma/(w - a/n)[/B]

x + 8y/4 = 9y for x Step 1: Isolate x by subtracting 8y/4 from each side: x + 8y/4 - 8y/4 = 9y - 8y/4 Cancel 8y/4 on the left side: x =[B] 9y - 8y/4 [MEDIA=youtube]5NLDNw_T8GU[/MEDIA][/B]

x - 5y = 6 for x Add 5y to each side to solve this literal equation for x. x - 5y + 5y = 6 + 5y Cancel the 5y on each side, we get: x = [B]6 + 5y[/B]

x is a multiple of 6 and 1 ? x ? 16. We want multiples of 6 between 1 and 16. We start with 6. Another multiple of 6 is 12 The next multiple of 6 is 18, which is out side the range of 1 ? x ? 16. So our number set is [B]x = {6, 12}[/B]

x is a woman who served as US president before 2000 No woman US presidents before 2000, so we have the empty set. [B]x = {}[/B]

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4 Varies directly means there is a constant k such that: x = ky^(1/3) When x = 1 and y = 27, we have: 27^1/3(k) = 1 3k = 1 To solve for k, we[URL='https://www.mathcelebrity.com/1unk.php?num=3k%3D1&pl=Solve'] type in our equation into our search engine[/URL] and we get: k = 1/3 Now, the problem asks for y when x = 4. We use our variation equation above with k = 1/3 and x = 4: 4 = y^(1/3)/3 Cross multiply: y^(1/3) = 4 * 3 y^(1/3) =12 Cube each side: y^(1/3)^3 = 12^3 y = [B]1728[/B]

(X+y)/3=5 for x Cross multiply: x + y = 15 Subtract y from each side: [B]x = 15 - y[/B]

x-m=n+p, for x Add m to each side: x - m + m = n + p + m Cancelling the m's on each side: x = [B]n + p + m[/B]

x/3 - g = a for x Add g to each side so we can isolate the x term: x/3 - g + g = a + g Cancel the g terms on the left side and we get: x/3 = a + g Multiply each side by 3 to isolate x: 3(x/3) = 3(a + g) Cancelling the 3's on the left side, we get: x = [B]3(a + g)[/B]

x/5-7=2q for x Add 7 to each side: x/5 -7 + 7 = 2q + 7 Cancel the 7's on the left side, we get: x/5 = 2q + 7 Cross multiply the 5: x = 5(2q + 7) x = [B]10q + 35[/B]

x/r - h = 4 for x Add h to each side: x/r - h + h = h + 4 Cancel the h's on the left side, we get: x/r = h + 4 Multiply each side by r to isolate x: xr/r = r(h + 4) Cancel the r's on the left side, we get: x = [B]r(h + 4)[/B]

x/y + 9 = n for x Subtract 9 from each side to isolate the x term: x/y + 9 - 9 = n - 9 Cancel the 9's on the left side and we get: x/y = n - 9 Because we have a fraction on the left side, we can cross multiply the denominator y by n - 9 [B]x =[/B] [B]y(n - 9)[/B]

x/y + 9 = n for y First, subtract 9 from each side to isolate the y term: x/y + 9 - 9 = n - 9 Cancel the 9's on the left side, and we get: x/y = n - 9 Cross multiply: x = y(n - 9) Divide each side by (n - 9): x/(n - 9) = y(n - 9)/(n - 9) Cancel the (n - 9) on the right side, and we get: y = [B]x/(n - 9)[/B]

x/y = z - 8 for x Multiply each side by y to isolate x: y*(x/y) = y(z - 8) The y's cancel out on the left side, so we have: x = [B]y(z - 8)[/B]

x/y = z - 8 for x We start by seeing that x is isolated. To remove y from the left side, we multiply each side of the equation by y: xy/y = y(z - 8) Cancelling y on the left side, we get our answer of: x = [B]y(z - 8) [MEDIA=youtube]_HNyGlnnQdQ[/MEDIA][/B]

Add w/m to each side: p = X + w/m

Xavier has $132 to buy a video game. Each game costs $12. Write an equation to find the number of games Xavier can purchase. Let g be the number of games, we have a cost function C(g) C(g) = 12g We want to find g such that C(g) = 132 12g = 132 Divide each side by 12 [B]g = 11[/B]

Multiply each side by 2 to isolate y. y +2c = 2d Subtract 2c from each side of the equation: y = 2d - 2c This can also be written y = 2(d - c)

Yesterday a car rental agency rented 4 convertibles and 30 other vehicles. Considering this data, how many of the first 68 vehicles rented today should you expect to be convertibles? 30 other vehicles + 4 convertibles = 34 cars 34 * 2 = 68 30 * 2 other vehicles + 4 * 2 convertibles = 68 cars 60 other vehicles and [B]8 convertibles[/B]

Yolanda runs each lap in 7 minutes. She will run less than 35 minutes today. What are the possible numbers of laps she will run today? 7 minutes per lap must be less than 35 minutes. Let l be the number of laps 7l < 35 Divide each side by 7 [B]l < 5[/B]

You and a friend want to start a business and design t-shirts. You decide to sell your shirts for $15 each and you paid $6.50 a piece plus a $50 set-up fee and $25 for shipping. How many shirts do you have to sell to break even? Round to the nearest whole number. [U]Step 1: Calculate Your Cost Function C(s) where s is the number of t-shirts[/U] C(s) = Cost per Shirt * (s) Shirts + Set-up Fee + Shipping C(s) = $6.50s + $50 + $25 C(s) = $6.50s + 75 [U]Step 2: Calculate Your Revenue Function R(s) where s is the number of t-shirts[/U] R(s) = Price Per Shirt * (s) Shirts R(s) = $15s [U]Step 3: Calculate Break-Even Point[/U] Break Even is where Cost = Revenue. Set C(s) = R(s) $6.50s + 75 = $15s [U]Step 4: Subtract 6.5s from each side[/U] 8.50s = 75 [U]Step 5: Solve for s[/U] [URL='https://www.mathcelebrity.com/1unk.php?num=8.50s%3D75&pl=Solve']Run this through our equation calculator[/URL] to get s = 8.824. We round up to the next integer to get [B]s = 9[/B]. [B][URL='https://www.facebook.com/MathCelebrity/videos/10156751976078291/']FB Live Session[/URL][/B]

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen? Let n be our original number. Square the number means we raise n to the power of 2: n^2 Multiply the result by 2: 2n^2 And then add three: 2n^2 + 3 If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53: 2n^2 + 3 = 53 To solve for n, we subtract 3 from each side, to isolate the n term: 2n^2 + 3 - 3 = 53 - 3 Cancel the 3's on the left side, and we get: 2n^2 = 50 Divide each side of the equation by 2: 2n^2/2 = 50/2 Cancel the 2's, we get: n^2 = 25 Take the square root of 25 n = +-sqrt(25) n = +-5 We are told the number is positive, so we discard the negative square root and get: n = [B]5[/B]

You and your friend are saving for a vacation. You start with the same amount and save for the same number of weeks. You save 75 per week, and your friend saves 50 per week. When vacation time comes, you have 950, and your friend has 800. How much did you start with, and for how many weeks did you save? [U]Let w be the number of weeks. Set up two equations where s is the starting amount:[/U] (1) s + 75w =950 (2) s + 50w = 800 [U]Rearrange (1) into (3) to solve for s by subtracting 75w[/U] (3) s = 950 - 75w [U]Rearrange (2) into (4) to solve for s by subtracting 50w[/U] (4) s = 800 - 50w [U]Set (3) and (4) equal to each other so solve for w[/U] 950 - 75w = 800 - 50w [U]Add 75w to each side, and subtract 950 from each side:[/U] 25w = 150 [U]Divide each side by w[/U] [B]w = 6[/B] Now plug w = 6 into (3) s = 950 - 75(6) s = 950 - 450 [B]s = 500[/B]

You are baking muffins for your class. There are 17 total students in your class and you have baked 5 muffins. Write and solve an equation to find the additional number x of muffins you need to bake in order to have 2 muffins for each student. Write your equation so that the units on each side of the equation are muffins per student. 2 muffins per student = 17*2 = 34 muffins. We have an equation with a given 5 muffins, how much do we need (x) to get to 34 muffins (2 per student): x + 5 = 34 To solve for x, we type this equation into our search engine and we get: x = [B]29[/B]

You are baking muffins for your class. There are 17 total students in your class and you have baked 5 muffins. Write and solve an equation to find the additional number x of muffins you need to bake in order to have 2 muffins for each student. Write your equation so that the units on each side of the equation are muffins per student. [U]Calculate total muffins:[/U] Total muffins = 2 muffins per student * 17 students Total muffins = 34 [U]Set up the equation using x for muffins:[/U] [B]x + 5 = 34 [/B] [U]To Solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B5%3D34&pl=Solve']type it in our search engine[/URL] and we get:[/U] x = [B]29 [/B]

You are comparing the costs of producing shoes at two different manufacturers. Company 1 charges $5 per pair of shoes plus a $650 flat fee. Company 2 charges $4 per pair of shoes plus a $700 flat fee. How many pairs of shoes are produced when the total costs for both companies are equal? Let s be the number of shoes. We have two equations: (1) C = 5s + 650 (2) C = 4s + 700 Set the costs equal to each other 5s + 650 = 4s + 700 Subtract 4s from each side s + 650 = 700 Subtract 650 from each side [B]s =50[/B]

You are offered two different sales jobs. The first company offers a straight commission of 6% of the sales. The second company offers a salary of $330 per week plus 2% of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good? Let s be the sales and C be the weekly commission for each sales job. We have the following equations: [LIST=1] [*]C = 0.06s [*]C = 330 + 0.02s [/LIST] Set them equal to each other: 0.06s = 330 + 0.02s Subtract 0.02s from each side: 0.04s = 330 Using our [URL='http://www.mathcelebrity.com/1unk.php?num=0.04s%3D330&pl=Solve']equation solver[/URL], we get [B]s = 8,250[/B]

You borrowed $25 from your friend. You paid him back in full after 6 months. He charged $2 for interest. What was the annual simple interest rate that he charged you? Use the formula: I = Prt. We have I = 2, P = 25, t = 0.5 2 = 25(r)0.5 Divide each side by 0.5 4 = 25r Divide each side by 25 r = 4/25 [B]r = 0.16[/B] As a percentage, this is [B]16%[/B]

You can afford monthly deposits of $270 into an account that pays 3.0% compounded monthly. How long will it be until you have $11,100 to buy a boat. Round to the next higher month. [U]Set up our accumulation expression:[/U] 270(1.03)^n = 11100 1.03^n = 41.1111111 [U]Take the natural log of both sides[/U] n * Ln(1.03) = 41.1111111 n = 3.71627843/0.0295588 n = 125.72 so round up to [B]126[/B]

You can pay a daily entrance fee of $3 or purchase a membership for the 12 week summer season for $82 and pay only $1 per day to swim. How many days would you have to swim to make the membership worthwhile? Set up cost equations: Daily entrance fee: 3d where d is the number of days of membership Membership fee 82 + 1d Set them equal to each other 82 + 1d = 3d Subtract d from each side: 2d = 82 Divide each side by 2 [B]d = 41[/B]

You deposit $2000 in an account that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest? The simple interest formula for the accumulated balance is: Prt = I We are given P = 2,000, r = 0.04, and I = 500. 2000(0.04)t = 500 80t = 500 Divide each side by 80 t = [B]6.25 years [MEDIA=youtube]Myz0FZgwZpk[/MEDIA][/B]

If you have $6 left over, then 8 notebooks cost $22 - $6 = $16. 8 notebooks = $16 Divide each side of the equation by 8 Each notebook is $2

You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explains how many tacos you can buy. Let's start with t as the number of tacos. We know that cost = price * quantity, so we have the following inequality for our taco spend: [B]0.5t <= 10 [/B] Divide each side of the inequality by 0.5 to isolate t: 0.5t/0.5 <= 10/0.5 Cancel the 0.5 on the left side and we get: t <= [B]20 [MEDIA=youtube]yy51EsGi1nM[/MEDIA][/B]

You have $37 to plant garden. If you spend $12.25 on seeds, how many packs of vegetable plants can you buy for 2.75 each? [U]How much do we have to spend on plants?[/U] $37 - 12.25 = $24.75 [U]Calculate how many vegetable plants we can buy. Set up an equation where x = vegetable plants[/U] 2.75x = 24.75 Divide each side by 2.75 [B]x = 9[/B]

You have a total of 42 math and science problems for homework. You have 10 more math problems than science problems. How many problems do you have in each subject? Let m be the math problems and s be the science problems. We have two equations: (1) m + s = 42 (2) m = s + 10 Substitute (2) into (1) (s + 10) + s = 42 Combine like terms 2s + 10 = 42 Subtract 10 from each side 2s = 32 Divide each side by 2 [B]s = 16[/B] So that means m = 16 + 10 --> [B]m = 26 (m, s) = (26, 16)[/B]

You roll a standard, fair, 5-sided die and see what number you get. Find the sample space of this experiment. Write your answer using { } symbols, and write your values in order with a comma but no spaces between Sample Space: [B]{1,2,3,4,5}[/B]

You roll two six-sided dice. What is the probability that the sum is less than 13? The probability is [B]1, or 100%[/B], since the maximum sum of two six-sided dice is 12.

Your friends in class want you to make a run to the vending machine for the whole group. Everyone pitched in to make a total of $12.50 to buy snacks. The fruit drinks are $1.50 and the chips are $1.00. Your friends want you to buy a total of 10 items. How many drinks and how many chips were you able to purchase? Let c be the number of chips. Let f be the number of fruit drinks. We're given two equations: [LIST=1] [*]c + f = 10 [*]c + 1.5f = 12.50 [/LIST] Rearrange equation 1 by subtracting f from both sides: [LIST=1] [*]c = 10 - f [*]c + 1.5f = 12.50 [/LIST] Substitute equation (1) into equation (2): 10 - f + 1.5f = 12.50 To solve for f, we [URL='https://www.mathcelebrity.com/1unk.php?num=10-f%2B1.5f%3D12.50&pl=Solve']type this equation into our search engine[/URL] and we get: [B]f = 5[/B] Now, substitute this f = 5 value back into modified equation (1) above: c = 10 - 5 [B]c = 5[/B]

your starting salary at a new company is 45000. Each year you receive a 2% raise. How long will it take you to make $80000? Let y be the number of years of compounding the 2% raise. Since 2% as a decimal is 0.02, we have the following equation for compounding the salary: 45000 * (1.02)^y = 80000 Divide each side by 45000: (1.02)^y = 1.77777777778 To solve this equation for y, we [URL='https://www.mathcelebrity.com/natlog.php?num=1.02%5Ey%3D1.77777777778&pl=Calculate']type it in our search engine[/URL] and we get: y = [B]29.05[/B] [B]Or just over 29 years[/B]

z = (x + y)/mx; Solve for x Cross multiply: zmx = x + y Subtract x from each side zmx - x = y Factor out x x(zm - 1) = y Divide each side by zm - 1 x = y/(zm - 1) [MEDIA=youtube]ksxCS3YlCwY[/MEDIA]

z/w=x+z/x+y for z This is a literal equation. Let's isolate z on one side. Subtract z/x from each side. z/w - z/x = x + y Factor our z on the left side: z(1/w - 1/x) = x + y Divide each side by (1/w - 1/x) z = x + y/(1/w - 1/x) To remove reciprocals in the denominator, we rewrite 1/w - 1/x with a common denominator xw (x - w)/xw Then multiply x + y by the reciprocal z = [B](x + y)xw/(x - w)[/B]

z=m-x+y, for x This is a literal equation. Let's add subtract (m + y) from each side: z - (m + y) = m - x + y - (m + y) The m + y terms cancel on the right side, so we have: z - m - y = -x Multiply each side by -1 to isolate x: -1(z - m - y) = -(-x) x = [B]m + y - z[/B]

Zombies are doubling every 2 days. If two people are turned into zombies today, how long will it take for the population of about 600,000 to turn into zombies? Let d = every 2 days. We set up the exponential equation 2 * 2^d = 600,000 Divide each side by 2: 2^d = 300000 To solve this equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=2%5Ed%3D300000&pl=Calculate']type it in our math engine[/URL] and we get d = 18.19 (2 day periods) 18.19 * days per period = 36.38 total days Most problems like this ask you to round to full days, so we round up to [B]37 days[/B].

zy-dm=ky/t for y Isolate terms with y to solve this literal equation. Subtract zy from each side: zy - dm - zy = ky/t - zy Cancel the zy terms on the left side, we get: -dm = ky/t - zy Factor out y: y(k/t - z) = -dm Divide each side by (k/t - z) y = -dm/(k/t - z) (k/t - z) can be rewritten as (k - tz)/t We multiply -dm by the reciprocal of this quotient to get our simplified literal equation: y = [B]-dmt/(k - tz)[/B]

|(x-7)/5|<=4 Set up two equations: [LIST=1] [*](x-7)/5 <= 4 [*](x-7)/5 > -4 [/LIST] Cross Multiply (1): x - 7 <= 20 Add 7 to each side: x <= 27 Cross Multiply (2): x - 7 > -20 Add 7 to each side: x > -13

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