Problem 5. A 100 denar banknote needs to be split into coins of 2 and 5 denars, with the total number of coins being 32. If such a split is possible, how many 2 denar coins and how many 5 denar coins are used?
Solution. First method. If we choose all 32 coins to be 2 denars, then we would have 64 denars, and the 100 denar banknote would not be covered. If one of the 32 coins, which are 2 denars, is replaced with a 5 denar coin, the amount increases by $5-2=3$ denars. The amount of 64 denars needs to be increased by $100-64=36$ denars. According to this, the process of replacing a 2 denar coin with a 5 denar coin should be repeated 36:3=12 times. This way, we would get 12 coins of 5 denars and 20 coins of 2 denars each. This is the required distribution.

Second method. Let $x$ be the number of 5 denar coins. Then $32-x$ is the number of 2 denar coins. We get the equation: $5x + 2(32-x) = 100$, from which $x=12$.

## V section