24. Alex drives from point $P$ to point $Q$, then immediately returns to $P$. Bob drives from point $Q$ to point $P$, then immediately returns to $Q$. They travel on the same road, start at the same time and each travels at a constant speed. Alex's speed is three times Bob's speed. They meet each other for the first time 15 minutes after the start. How long after the start will they meet each other for the second time?
A $20 \mathrm{~min}$
B $25 \mathrm{~min}$
C $30 \mathrm{~min}$
D $35 \mathrm{~min}$
E $45 \mathrm{~min}$
Solution
C

Let Bob's speed be $V \mathrm{~km} / \mathrm{h}$. Therefore Alex's speed is $3 V \mathrm{~km} / \mathrm{h}$. Since they meet each other for the first time after 15 minutes, or $\frac{1}{4}$ of an hour, the total distance between $P$ and $Q$, in km, is $\frac{1}{4} \times V+\frac{1}{4} \times 3 V=V$. Let the distance Bob has travelled when Alex passes him for the second time be $Y \mathrm{~km}$. Since time $=\frac{\text { distance }}{\text { speed }}$, we have $\frac{Y}{V}=\frac{V+Y}{3 V}$ and hence $3 Y=V+Y$, which has solution $Y=\frac{V}{2}$. Hence the time in hours that Bob has travelled when Alex passes him for the second time is $\frac{\frac{V}{2}}{V}=\frac{1}{2}$. Therefore they will meet for a second time after 30 minutes. Note: When Alex passes Bob for the second time, they are both travelling in the same direction towards $P$, whereas when he passed Bob for the first time they were travelling in opposite directions.