Track / Stage 5 / 69 of 400 #669 of 2000

Problem 669

AIME late

Algebra Difficulty 5.2 Find the answer

13.232. From the railway station to the beach is 4.5 km. A boy and a scheduled bus left the station for the beach simultaneously. After 15 minutes, the boy met the bus returning from the beach, and he managed to walk another $9 / 28$ km from the place of the first meeting with the bus, when he was caught up by the same bus, which had reached the station and set off to the beach again. Find the speeds of the boy and the bus, assuming they are constant and neither the boy nor the bus stopped en route, but the bus made stops of 4 minutes each at the beach and at the station.

Your answer

Official solution

## Solution.

Let $v_{\text {m }}, v_{\text{a}}$ be the speeds of the boy and the bus, respectively.

According to the problem, we have the system
$\left\{\begin{array}{l}\frac{15}{60} v_{M}+\frac{15-4}{60} v_{\text{a}}=2 \cdot 4.5, \\ \frac{9}{28 v_{M}}=\frac{4}{60}+\frac{2 \frac{15}{60} v_{M}+\frac{9}{28}}{v_{\text{a}}},\end{array}\right.$

from which $v_{\text{M}}=3 \text{ km/h}, v_{\text{a}}=45 \text{ km/h}$ .

Answer: 3 and 45 km/h.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.