SOLUTION: In one month a store sold pads of paper, some for $9, some for $2, and some for 50 cents. A total of 90 pads were sold for a total of $90. evaluate the total number of the 50 cent

Question 1104904: In one month a store sold pads of paper, some for $9, some for $2, and some for 50 cents. A total of 90 pads were sold for a total of $90. evaluate the total number of the 50 cent pads sold minus the number of $2 pads sold.

Answer by greenestamps(13198) (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. Then
[the total number of pads sold was 90]
[the total cost of the pads sold was $90]

This is a Diophantine equation -- three unknowns but only two equations; but we can find a solution using the fact that the unknowns have positive integer values.

The general technique for solving a Diophantine system with three variables and two equations is to eliminate one of the variables to obtain a single equation in two variables; then analyze that equation to find integer solutions.

You can choose any one of the variables to be the one to be eliminated. The amount of work required to finish the problem can vary a great deal depending on which variable you eliminate first. Unfortunately, it takes a huge amount of experience with solving this kind of problem to know ahead of time which variable is the best one to eliminate.

So I will choose to eliminate variable z -- mostly because of the fractional coefficient on z in the second equation. So double the second equation and then subtract one equation from the other:

Quick inspection of that equation shows that one solution is (x,y) = (0,30). However, since the problem implies that some of each size pad were sold, we can reject that solution.

To find other solutions in positive integers, since the coefficients 3 and 17 are relatively prime, we can take our rejected (0,30) solution and add 3 to x and subtract 17 from y, giving us (x,y) = (3,13).

The next possible solution would be found by again adding 3 to x and subtractin 17 from y; but that would give us a negative value for y.

So our only possible solution is (x,y) = (3,13).

That gives us z = (90-(3+13)) = 74.

So the store sold 3 $9 pads, 13 $2 pads, and 74 50-cent pads.

CHECK: The total cost is
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