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Three girls, Ann, Betty and Cynthia, each have a younger brother, Dylan,
Ernie and Frank, respectively. All six children do some fruit picking for
the local farmer. The farmer agrees to pay each child as many dollars per
basket as the number of baskets of fruit collected by that child. Each of
the girls earned $45 more than her brother, and all six children collected
a different number of baskets. How much did the farmer pay them all in total ?
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Let x be the number of baskets collected by some girl, and y be the number of basket collected by her brother.
Then for each such a pair, we have this equation
x^2 - y^2 = 45.
Factor left side to get
(x-y)*(x+y) = 45.
45 has these decompositions in the product of factors 1*45, 3*15 and 9*5.
These decompositions produce these systems of equations
a) x - y = 1
x + y = 45
with the solution x = 23, y = 22;
b) x - y = 3
x + y = 15
with the solution x = 9, y = 6;
c) x - y = 5
x + y = 9
with the solution x = 7, y = 2.
These are all possible solutions to the numbers of baskets collected by the children.
There is NO other solutions.
Hence, the total pay was = 1183 dollars. ANSWER
Solved.