Problem 4. If a passenger travels from Moscow to St. Petersburg by a regular train, they will arrive in 10 hours. If they take an express train, which they have to wait more than 2.5 hours for, they will arrive 3 hours earlier than the regular train. Find the ratio of the speeds of the express train and the regular train, given that 2 hours after departure, the express train will be at the same distance from Moscow as the regular train.
Answer. The express is 2.5 times faster.

Solution. Let $x$ be the time spent waiting for the express. Then the express traveled for $10-3-x=7-x$ hours, covering in 2 hours the same distance that the train covers in $x+2$ hours. Therefore, $\frac{10}{7-x}=\frac{x+2}{2}$. We get $x^{2}-5 x+6=0$. But by the condition $x>2.5$, so $x=3$, and the required ratio is $\frac{x+2}{2}=2.5$.