# Problem of the Week Problem D and Solution Stacks and Stacks

## Problem

Virat has a large collection of $2 bills and$5 bills. He makes stacks that have a value of $100. Each stack has a least one$2 bill, at least one $5 bill, and no other types of bills. If each stack has a different number of$2 bills than any other stack, what is the maximum number of stacks that Virat can create?

## Solution

Consider a stack of bills with a total value of $100 that includes $$x$$$2 bills and $$y$$ $5 bills. The$2 bills are worth $2$$x$$ and the$5 bills are worth $5$$y$$, and so $$2x + 5y = 100$$. Determining the number of possible stacks that the teller could have is equivalent to determining the numbers of pairs $$(x, y)$$ of integers with $$x \geq 1$$ and $$y \geq 1$$ and $$2x + 5y = 100$$ or $$5y=100-2x$$. (We must have $$x \geq 1$$ and $$y \geq 1$$ because each stack includes at least one$2 bill and at least one \$5 bill.)

Since $$x \geq 1$$, then:

\begin{aligned} 2x &\geq 2\\ 2x + 98 &\geq 100\\ 98 &\geq 100 - 2x \\\end{aligned}

This can be rewritten as $$100 - 2x\leq 98$$.

Also, since $$5y = 100-2x$$, this becomes $$5y \leq 98$$.

This means that $$y \leq \frac{98}{5} = 19.6$$. Since $$y$$ is an integer, then $$y \leq 19$$.

Notice that since $$5y = 100 - 2x$$, then the right side is the difference between two even integers and is therefore even. This means that $$5y$$ (the left side) is itself even, which means that $$y$$ must be even.

Since $$y$$ is even, $$y \geq 1$$, and $$y \leq 19$$, then the possible values of $$y$$ are 2, 4, 6, 8, 10, 12, 14, 16, 18.

Each of these values gives a pair $$(x, y)$$ that satisfies the equation $$2x + 5y = 100$$. These ordered pairs are:

$$(x, y) = (45, 2),(40, 4),(35, 6),(30, 8),(25, 10),(20, 12),(15, 14),(10, 16),(5, 18)$$.

Therefore, we see that the maximum number of stacks that Virat could have is 9.