## 92. Math Puzzle $1 / 73$

A shipment of 50 entertainment booklets with prices of 0.50 M, 1.00 M, and 2.50 M is delivered to a summer camp. Invoice amount: $62.50 \mathrm{M}+$ expenses.

The delivery note was lost; however, it was remembered that the majority of the booklets were in the cheapest price category and that approximately the same number of booklets were delivered in the other two price categories. How many booklets of each price category were in the shipment?
The shipment included $x$ notebooks for M -,50, $y$ for M 1,- and $z$ for M 2,50. This results in: $x+y+z=50$ and $0.5 x+y+2.5 z=62.5$ or after eliminating $x: y+4 z=75$.

This system of equations is not sufficiently determined; as an additional condition, it is possible that $x, y$, and $z$ are integers and positive.

$$
z=\frac{75-y}{4}=18+\frac{3-y}{4}
$$

is only an integer if $\frac{3-y}{4}$ is also an integer. Then $z=18+u, y=3-4 u$ and $x=29+3 u$ with the following list of possible solutions:

| $u$ | $x$ | $y$ | $z$ |  |
| :---: | :---: | :---: | :---: | :---: |
| 0 | 29 | 3 | 18 |  |
| -1 | 26 | 7 | 17 |  |
| -2 | 23 | 11 | 16 | $*$ |
| -3 | 20 | 15 | 15 | $*$ |
| -4 | 17 | 19 | 14 |  |

* Solution most likely based on the additional problem