## Problem 4

A group of boys and girls have eaten at a restaurant where only pizzas cut into 12 slices are served. Each boy ate 6 or 7 slices and each girl 2 or 3 slices.

It is known that 4 pizzas were not enough and that with 5 pizzas there was more than enough.

Calculate the number of boys and girls in the group.
Solution:

Let $x$ and $y$ be the number of boys and girls, respectively.

We have

$$
7 x+3 y \leq 59
$$

and

$$
6 x+2 y \geq 49
$$

Subtracting member to member, we get

$$
x+y \leq 10
$$

and from (2),

$$
6 x+2(10-x) \geq 49
$$

from which $x \geq 8$.

But (1) implies that $x \leq 8$.

Therefore, $x=8$.

Substituting this value into the previous inequalities, it must be that $y=\mathbf{1}$.