Two different numbers are given. If we subtract a quarter of the smaller number from each of them, we get numbers, one of which will be five times larger than the other.

How many times larger is the given larger number than the smaller one?

(L. Hozová)

Hint. Does the sign of the numbers under consideration play any role?
Given numbers will be denoted by $m$ and $v$, where $m<v$. This relation will not change if we subtract (any) the same number from both numbers. In particular, it holds that

$$
\frac{3}{4} m<v-\frac{1}{4} m
$$

For one of the numbers in the previous inequality to be five times the other, both must have the same sign:

- If both numbers are positive, then

$$
5 \cdot \frac{3}{4} m=v-\frac{1}{4} m
$$

From this, we get $v=\frac{16}{4} m=4 m$.

- If both numbers are negative, then

$$
\frac{3}{4} m=5 \cdot\left(v-\frac{1}{4} m\right)
$$

From this, we get $5 v=\frac{8}{4} m$, thus $v=\frac{2}{5} m$.

The relationship between the given numbers is either $v=4 m$ or $v=\frac{2}{5} m$.

Note. From the above, it follows, among other things, that the given numbers have the same sign.