While writing his essay, Sanyi glanced at his watch and realized that five times as much time had already passed as the time that was still left. $M$ minutes later, this ratio had already increased to 8. What will the ratio be after another $M$ minutes?
Let the total time for writing the essay be $x$ minutes. When Sanyi glanced at his watch, $\frac{5}{6} x$ minutes had passed, and $\frac{1}{6} x$ minutes remained. $M$ minutes later - at this point, the ratio of the times is 8 - we can write that

$$
\frac{5}{6} x+M=8\left(\frac{1}{6} x-M\right)
$$

from which $M=\frac{1}{18} x$

After another $M$ minutes, denoting the ratio of the times by $L$:

$$
\frac{5}{6} x+2 \frac{1}{18} x=L\left(\frac{1}{6} x-2 \cdot \frac{1}{18} x\right)
$$

from which $L=17$.

Thus, after another $M$ minutes, the ratio will be 17.

By Sophia (Fazekas M. Főv. Gyak. Gimn., 9th grade)