SOLUTION: A,B, and C make widgets. A alone needs 10 days. If A and B work together, they need 6 days. B and C need 8 days. A,B, and C work together. when they finish, A made 240 widgets mo

Question 1163839: A,B, and C make widgets. A alone needs 10 days. If A and B work together, they
need 6 days. B and C need 8 days. A,B, and C work together. when they finish,
A made 240 widgets more than C did. How many widgets did C make?
Found 3 solutions by Theo, ikleyn, Edwin McCravy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
let w = the number of widgets produced.
let a = the rate that A makes widgets.
let b = the rate that B makes widgets.
let c = the rate that c makes widgets.

rate * time = quantity

you are given that A can make w widgets in 10 days.
the formula becomes a * 10 = w
solve for a to get a = w/10

you are given that A and B can make w widgets in 6 days when they work together.
therefore (a + b) * 6 = w
solve for a + b to get a + b = w/6
since a = w/10, the formula becomes w/10 + b = w/6
solve for b to get b = w/6 - w/10 = 10w/60 - 6w/60 = 4w/60 = w/15

you are given that B and C can make w widgets in 8 days when they work together.
therefore (b + c) * 8 = w
solve for b + c to get b + c = w/8
since b = w/15, the formula becomes w/15 + c = w/8
solve for c to get c = w/8 - w/15 = 15w/120 - 8w/120 = 7w/120

your rates for A, B, and C are:
a = w/10
b = w/15
c = 7w/120

when all 3 work together, the formula becomes:
(a + b + c) * T = w
this becomes (w/10 + w/15 + 7w/120) * T = w
this becomes 27w/120 * T = w
solve for T to get T = w * 120/27w = 120/27 days

the contribution of A to w is equal to w/10 * 120/27 = 120/270 * w
the contribution of B to w is equal to w/15 * 120/27 = 120/405 * w
the contribution of C to w is equal to 7w/120 * 120/27 = 7/27 * w

the total contribution is 120/270 * w + 120/405 * w + 7/27 * w = w

the contribution of A is 240 more widgets than the contribution of C.

if you let x = the number of widgets that A contributes and y = the number of widgets that C contributes, you get x = y + 240
subtract y from both sides of this equation to get x - y = 240
since A contributes 120w/270 widgets and C contributes 7w/27 widgets, then x = 120w/270 and y = 7w/27, so you get 240 = 120w/270 - 7w/27
solve for w to get w = 1296

what this says is that, when A and B and C work together, they make 1296 widgets in 120/27 days.

from above:

the contribution of A to w is equal to w/10 * 120/27 = 120/270 * w
the contribution of B to w is equal to w/15 * 120/27 = 120/405 * w
the contribution of C to w is equal to 7w/120 * 120/27 = 7/27 * w
since w = 1296, these formulas become:
the contribution of A to 1296 is equal to 1296/10 * 120/27 = 576
the contribution of B to 1296 is equal to 1296/15 * 120/27 = 384
the contribution of C to 1296 is equal to 7*1296/120 * 120/27 = 336

the contribution of A is 240 more than the contribution of C.
since the contribution of C is 336, then 336 + 240 = 576.

your solution is that C made 336 widgets, assuming i did all of the above correctly.
Answer by ikleyn(52756)   (Show Source): You can put this solution on YOUR website!
.
A,B, and C make widgets. A alone needs 10 days. If A and B work together, they need 6 days. B and C need 8 days.
A,B, and C work together. when they finish, A made 240 widgets more than C did. How many widgets did C make?
~~~~~~~~~~~~~~~

From the condition, rate of work for A is of the job per day; the combined rate of work of A and B working together is of the job per day. Hence, the individual rate of B is - = = of the job per day. Next, the combined rate of work of B and C working together is of the job per day. Hence, the individual rate of C is - = of the job per day. Then the combined rate of A, B and C, working together, is + + = = = . It means that they complete the job in days, working all the three together. During this time, A made = = of the job, while C made = of the job. If w is the total number of widgets then the contribution of A is widgets, while the contribution of C is widgets. We are given that this difference, - is 240 - = 240, or - = 240 = 240 w = = 1296. Hence, C made = 336 widgets. ANSWER

Solved.

Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!

I added that they needed to make N widgets to make the question clearer.

A,B, and C make widgets, AND THEY NEED TO MAKE N WIDGETS. A alone needs 10
days TO MAKE N WIDGETS. If A and B work together, they need 6 days TO MAKE N
WIDGETS. B and C need 8 days TO MAKE N WIDGETS. A,B, and C work together TO
MAKE N WIDGETS. When they finish MAKING N WIDGETS, A made 240 widgets more
than C did. How many widgets did C make?

Let the total number of widgets that need be made = N. Let A's rate be x widgets/day. A alone needs 10 days TO MAKE N WIDGETS. So 10x = N If A and B work together, they need 6 days TO MAKE N WIDGETS Let B's rate be y widgets/day. So 6x + 6y = N 6x + 6y = 10x -4x + 6y = 0 6y = 4x y = 4x/6 y = 2x/3 B and C need 8 days TO MAKE N WIDGETS. So 8y + 8z = N 8y + 8z = 10x 4y + 4z = 5x 4(2x/3) + 4z = 5x 8x/3 + 4z = 5x 8x + 12z = 15x -7x + 12z = 0 12z = 7x z = 7x/12 A,B, and C work together TO MAKE N WIDGETS. Let their time to make N widgets be K days. So Kx + Ky + Kz = N Kx + K(2x/3) + K(7x/12) = 10x 12Kx + 8Kx + 7Kx = 120x 27Kx = 120x 27K = 120 K = 40/9 When they finish MAKING N WIDGETS, A made 240 widgets more than C did. Kx = Kz + 240 (40/9)x = (40/9)(7x/12) + 240 Divide through by 40 (1/9)x = (1/9)(7x/12) + 6 Multiply through by 108 12x = 7x + 648 5x = 648 x = 648/5 y = 2(648/5)/3 y = 432/5 z = 7x/12 z = 7(648/5)/12 z = 378/5 How many widgets did C make? C worked for 40/9 days at the rate of 378/5 widgets/day, so he made (40/9)(378/5) = 336 widgets. That's the answer. But to answer all questions that could have been asked in order to check: A worked for 40/9 days at the rate of 648/5 widgets/day, so he made (40/9)(648/5) = 576 widgets. (which is 240 more than the 336 A made) B worked for 40/9 days at the rate of 432/5 widgets/day, so he made (40/9)(432/5) = 384 widgets. So N, the number of widgets they needed to make was 336+576+384 = 1296 widgets. Edwin

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