## 147. Math Puzzle $8 / 77$

A student buys drawing pens for $0.45 \mathrm{M}$ each and pencils for $0.20 \mathrm{M}$ each. He pays a total of 6,- M. The receipt is no longer legible. However, more detailed information is needed for a later review.

What quantities of each item could the student possibly have bought?
The student bought $x$ felt-tip pens and $y$ pencils, where it is clear from the outset that $x$ and $y$ are whole positive numbers. For the total price, we have

$$
0.45 x + 0.20 y = 6.00 \quad \text{or} \quad 9 x + 4 y = 120
$$

from which $y = 30 - \frac{9}{4} x$

Since $y$ is an integer, $x$ must be divisible by 4. This leads to the following possibilities: $x=0, y=30$ (i.e., only pencils were bought, which is unlikely), $x=4, y=21$; $x=8, y=12$; $x=12, y=3$. For other combinations, $x$ or $y$ would be negative!