A sandwich and a meal plate cost on average $R \$ 5.00$ and $R \$ 7.00$, respectively. In how many ways can one buy sandwiches and meal plates with $R \$ 90.00$, without leaving any change?
If $S$ corresponds to the number of sandwiches and $P$ the number of meal plates, then $5 S+7 P=90$. We need to find integer solutions $P$ and $Q$ for this equation. We have:

$$
5 S+7 P=90 \Longrightarrow P=\frac{90-5 S}{7}=5 \times \frac{18-S}{7}
$$

Since $P$ is a natural number, 7 must divide $18-S$, so $S=4$, 11, or 18, and in each of these cases, $P$ is equal to 10, 5, and 0, respectively. Therefore, we have only three ways to make the purchase.