## 18. Essay in Latin

An essay in Latin is scored on a scale from 0 to $20^{1}$. Michel's score is above the average, while Claude's score is below the average. What score did each of them receive, if it is known that when one third of the smaller of these two scores is subtracted from each of them, one of the resulting remainders will be three times the other remainder?
18. Let $g$ be the larger of the two estimates, and $p$ be the smaller of them. Then

$$
g-p / 3=3(p-p / 3)
$$

from which it follows that

$$
g=7 p / 3
$$

If $p=3$ and $g=7$, Michelle would have received a score below average, which contradicts the problem's condition. Therefore, the only possible solution is:

$$
g=14 \text{ and } p=6
$$

In other words, Michelle scored 14 points, and Claude scored 6.