## 17. Paris - Deauville

Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They leave together and arrive in Deauville simultaneously. However, Monsieur Dubois spent one-third of the time his wife continued driving on stops, while Madame Dubois spent one-quarter of the time her husband was driving on stops.

What is the ratio of the average speeds of each of the spouses' cars?
17. Let $a$ and $a^{\prime}$ be the stopping times of Madame and Monsieur Dubois, respectively, and $r$ and $r^{\prime}$ be the times they spent driving. Since they both left and arrived in Deauville at the same time,

$$
a+r=a^{\prime}+r^{\prime}
$$

On the other hand,

$$
a^{\prime}=r / 3 \quad \text { and } \quad a=r^{\prime} / 4 .
$$

Therefore,

$$
\frac{r^{\prime}}{4}+r=\frac{r}{3}+r^{\prime} \text {, or } \frac{2}{3} r=\frac{3}{4} r^{\prime} \text {, i.e., } r=\frac{9}{8} r^{\prime} \text {. }
$$

Thus, the average speed of Madame Dubois is nine eighths of her husband's average speed.