Anička has saved 290 coins in her piggy bank, consisting of one-crown and two-crown coins. When she uses a quarter of all the two-crown coins, she collects the same amount as when she uses a third of all the one-crown coins.

What is the total amount Anička has saved?

(L. Růžičková)
Let's start from the requirement that a quarter of the two-crown coins gives the same amount as a third of the one-crown coins. From this, it follows that the number of two-crown coins is a multiple of four and the number of one-crown coins is a multiple of three, and that these numbers are in the ratio $4: 6$. Thus, the number of two-crown coins is $4k$ and the number of one-crown coins is $6k$, where $k$ is some positive integer $\left(2 \cdot \frac{4k}{4} = \frac{6k}{3}\right)$.

With this notation, the total number of coins is $10k = 290$. From this, it follows that $k = 29$, so Anička has $4 \cdot 29 = 116$ two-crown coins and $6 \cdot 29 = 174$ one-crown coins. The total value of her savings is therefore

$$
2 \cdot 116 + 174 = 406 \text{ crowns.}
$$

Evaluation. 4 points for determining the relationships between the numbers of one-crown and two-crown coins (here using the variable $k$); 2 points for determining $k$ and the result.

Note. Using the variable $k$, the total value of Anička's savings is expressed as $(2 \cdot 4 + 6) \cdot k = 14 \cdot k$.