Question 1105118: In one month a store sold pads of paper, some for $9, some for $2 and some for $0.50. A total of 90 pads were sold for a total of $90. Evaluate the total number of the $0.50 pads sold minus the number of $2 pads sold.
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website! Let x, y, and z be the numbers of $9, $2, and $0.50 pads sold, respectively. Then
 a total of 90 pads were sold
 the total cost of the pads sold was $90
This is a system of Diophantine equations: 3 variables but only 2 equations; but we can find the solution(s) using the fact that the variables have to have non-negative integer values. And in this problem, since it says that all three types of pads were sold, the values in fact must be positive integers.
(1) Eliminate one of the variables to get a system of one equation in two variables;
(2) solve that equation for one variable in terms of the other; and
(3) use the fact that the variable values must be positive integers to find the solution(s).
We can double the second equation and subtract the two equations to eliminate z:
We can see by inspection that (0,30) would be a solution if x could be 0; but we know x has to be a positive integer.
With the coefficients 17 and 3 being relatively prime, all other potential solutions can be found starting with (0,30) and adding 3 to the value of x while subtracting 17 from the value of y. We get
(0,30), (3,13), (6,-4)...
But the variable values have to be positive integers.
So the only solution to this problem is x=3 and y=13, which makes z=74.
CHECK:
3+13+74 = 90 check
3(9)+13(2)+74(.5) = 27+26+37 = 90 check
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