Adam and Eve received a basket containing 31 apples. On the first day, Eve ate three quarters of what Adam ate. On the second day, Eve ate two thirds of what Adam ate on the same day. By the evening of the second day, the basket was empty. How many apples did Eve eat from the basket? (Adam and Eve eat apples whole and do not share them.)

(L. Hozová)
According to the problem, on the first day, Eva ate three quarters of what Adam ate. Therefore, the number of apples Adam ate on the first day must be a multiple of four. We denote it as $4a$, where $a$ is an unknown natural number. The number of apples Eva ate on the first day is then $3a$. The numbers of apples eaten on the second day are denoted similarly: Adam ate $3b$ and Eva $2b$ apples, where $b$ is an unknown natural number. We set up an equation with two unknowns and look for its solution in the set of natural numbers:

$$
4a + 3a + 3b + 2b = 31
$$

after simplification

$$
7a + 5b = 31
$$

We substitute natural numbers 1 to 4 for $a$ and each time determine whether $b$ also comes out as a natural number. (We do not try substituting larger numbers for $a$ because $b$ would come out negative.) This way, we find the only solution to the equation in the set of natural numbers:

$$
a=3, b=2 \text{.}
$$

On the first day, Eva ate $3a = 3 \cdot 3 = 9$ apples, and on the second day, $2b = 2 \cdot 2 = 4$ apples. In total, she ate 13 apples.