9.5. At the exchange office, only the following operations can be performed:

1) exchange 2 gold coins for three silver coins and one copper coin;
2) exchange 5 silver coins for three gold coins and one copper coin.

Nikolai had only silver coins. After several visits to the exchange office, he had fewer silver coins, no gold coins appeared, but he got 50 copper coins. By how much did the number of silver coins decrease for Nikolai? Justify your answer. Note that exchanging money at the bank is not an equivalent operation, meaning that with each exchange, the purchasing power of the coins Nikolai had slightly decreased.
Solution: As a result of each operation, Nikolai acquires exactly 1 copper coin, which means there were exactly 50 operations in total. Of these, some (let's say \(a\)) were of the first type, and the rest \(50-a\) were of the second type. On operations of the first type, Nikolai spent \(2a\) gold coins, and on operations of the second type, he earned \(3(50-a)\) gold coins. Since he has no gold coins left, \(2a = 3(50-a)\), from which \(a = 30\). This means that on operations of the first type, Nikolai received \(30 \cdot 3 = 90\) silver coins, and on operations of the second type, he lost \((50-30) \cdot 5 = 100\) such coins. Thus, the number of silver coins decreased by \(100 - 90 = 10\) coins.

Answer: By 10 coins.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| Correct solution path, but the answer is incorrect due to  arithmetic errors | 6 points |
| Correctly and justifiedly found the number of operations of one  type (either the first or the second) | 5 points |
| Proven that the ratio of the number of operations of the 1st and 2nd  types is 3:2 AND/OR the total number of operations of both types is justified | 3 points |
| The problem is correctly solved under the assumption of equal value of all  exchanges (the general case is not investigated) | 2 points |
| A specific example of exchanges is provided, showing that  the answer 10 is possible, but its uniqueness is not justified | 1 point |
| Correct answer without justification (with incorrect justification) |  |