Problem 10.5. Hooligan Vasya loves to run on the escalator in the subway, and he runs down twice as fast as he runs up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving up? (The escalator always moves at a constant speed.)
Answer: 324.

![](https://cdn.mathpix.com/cropped/2024_05_06_6da73bfd3e09e8b55e3fg-41.jpg?height=702&width=711&top_left_y=89&top_left_x=371)

Fig. 6: to the solution of problem 10.4

Solution. Let the length of the escalator be 1, the speed of Vasya going down the escalator be $x$, and the speed of the escalator be $y$ (speeds are expressed in the length of the escalator divided by a minute). According to the problem, $6=\frac{1}{x}+\frac{1}{x / 2}=\frac{3}{x}$, from which $x=\frac{1}{2}$.

From the condition, it also follows that $\frac{1}{x+y}+\frac{1}{x / 2-y}=13.5$. Substituting $x=\frac{1}{2}$ into this equation and multiplying it by the product of the denominators, we get

$$
\frac{1}{4}-y+\frac{1}{2}+y=\frac{27}{2}\left(\frac{1}{2}+y\right)\left(\frac{1}{4}-y\right)
$$

Transferring everything to the left, we obtain the quadratic equation $\frac{27}{2} y^{2}+\frac{27}{8} y-\frac{15}{16}=0$, whose roots are $y_{1}=\frac{1}{6}$ and $y_{2}=-\frac{5}{12}$ (the second root does not fit, as the speed is positive).

It remains to calculate the value of the expression $\frac{1}{x-y}+\frac{1}{x / 2+y}$. Substituting $x=\frac{1}{2}$ and $y=\frac{1}{6}$ into it, we get 5.4 minutes. Converting this to seconds, we get $5.4 \cdot 60=324$ seconds.