## 8. At the Restaurant

Monsieur Dupont remembered that today was their wedding anniversary and invited his wife to have lunch at a good restaurant. Leaving the restaurant, he found that he had only one-fifth of the money he had brought with him left, and the number of centimes left was the same as the number of francs he initially had (1 franc $=100$ centimes), while the number of francs left was five times less than the number of centimes he initially had.

How much did Monsieur Dupont's visit to the restaurant cost?
8. Let $F$ be the initial number of francs, $C$ be the initial number of centimes, $F^{\prime}$ be the remaining number of francs, and $C^{\prime}$ be the remaining number of centimes.

Since after dining at the restaurant, Monsieur Dupont had only one-fifth of his money left, we have

$$
100 F + C = 5 \left(100 F^{\prime} + C^{\prime}\right)
$$

On the other hand, we know that

$$
C^{\prime} = F \quad \text{and} \quad F^{\prime} = C / 5.
$$

From this system of three equations with four unknowns, we can easily obtain that

$$
C = (95 / 99) F
$$

However, the number of centimes is an integer and less than 100. Therefore, the initial number of francs $F$ can only be 99, which means,

$$
C = 95, \quad F^{\prime} = C / 5 = 19, \quad C^{\prime} = F = 99
$$

Thus, initially, Monsieur Dupont had 99 francs and 95 centimes, and he was left with 19 francs and 99 centimes. Therefore, the dinner at the restaurant cost him 79 francs and 96 centimes.