Question 1189984
<br>
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.<br>
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<br>
<pre>
            # 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</pre>
The three expressions for the total number of problems are equal to the same number.<br>
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<br>
(1):
{{{xy=xy+3x-2y-6}}}
{{{3x-2y-6=0}}}
{{{2y=3x-6}}}
{{{y=(3/2)x-3}}}<br>
(2):
{{{xy=x((3/2)x-3)=(3/2)x^2-3x}}}<br>
(3):
{{{6x+24=(3/2)x^2-3x}}}
{{{12x+48=3x^2-6x}}}
{{{3x^2-18x-48=0}}}
{{{x^2-6x-16)=0}}}
{{{(x-8)(x+2)=0}}}<br>
x=8 or x=-2<br>
Clearly the negative solution makes no sense; so
{{{x=8}}}
{{{y=(3/2)x-3=12-3=9}}}
{{{xy=8*9=72}}}<br>
ANSWER: The number of problems in the packet was 72<br>
CHECK:
Annie: 6(x+4) = 6(12) = 72
Ben: xy = 8(9) = 72
Claire: (x-2)(y+3) = 6(12) = 72<br>