SOLUTION: Annie, Ben and Claire are training for MATHCOUNTS. On January 1st, they each started to solve a large packet of problems that their coach gave them. Each of them worked at a consta

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Question 1189984: Annie, Ben and Claire are training for MATHCOUNTS. On January 1st, they each started to solve a large packet of problems that their coach gave them. Each of them worked at a constant rate. Annie solved six problems each day and finished four days after Ben did. Claire solved three more problems each day than Ben did and finished two days before he did. How many problems were in the packet their coach gave them?
Answer by greenestamps(13203) About Me  (Show Source):
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Let x and y, respectively, be the number of days Ben takes to complete the packet and the number of problems he solves each day.

Then use the given information to get expressions in terms of x and y for the total number of problems for each of the three students.
Annie: 6 problems a day, taking 4 days longer than Ben
Claire: 2 fewer days to finish than Ben, working 3 more problems each day than Ben

            # of days  # solved each day  # of problems
  ------------------------------------------------------------------
    Annie      x+4            6         6(x+4)=6x+24
    Ben         x             y         xy
    Claire     x-2           y+3        (x-2)(y+3)=xy+3x-2y-6

The three expressions for the total number of problems are equal to the same number.

Strategy...
(1) Eliminate "xy" from the expressions, by using the expressions for the number of problems for Ben and Claire to get an equation for y in terms of x
(2) Substitute that in the equation for the number of problems for Ben to get an expression for the number of problems in terms of x only
(3) We now have two expressions in terms of x only for the total number of problems; set them equal to each other and solve for x

(1):
xy=xy%2B3x-2y-6
3x-2y-6=0
2y=3x-6
y=%283%2F2%29x-3

(2):
xy=x%28%283%2F2%29x-3%29=%283%2F2%29x%5E2-3x

(3):
6x%2B24=%283%2F2%29x%5E2-3x
12x%2B48=3x%5E2-6x
3x%5E2-18x-48=0
x%5E2-6x-16%29=0
%28x-8%29%28x%2B2%29=0

x=8 or x=-2

Clearly the negative solution makes no sense; so
x=8
y=%283%2F2%29x-3=12-3=9
xy=8%2A9=72

ANSWER: The number of problems in the packet was 72

CHECK:
Annie: 6(x+4) = 6(12) = 72
Ben: xy = 8(9) = 72
Claire: (x-2)(y+3) = 6(12) = 72