UPSC Prelims 2025 CSAT, question 37

Topic: Arithmetic

Q37

$P$ and $Q$ walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. $P$ walks at an average speed of 5 rounds per hour and $Q$ walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m. ?

(a) 12
(b) 13Correct answer
(c) 14
(d) 15

Solution.

Answer: (b) 13

Since $P$ ( $5$ rounds/hour) and $Q$ ( $3$ rounds/hour) walk in opposite directions from the same point at 5:00 a.m., their relative speed is $5 + 3 = 8$ rounds per hour, so they cross each other every $\frac{60}{8} = 7.5$ minutes.
In the first $20$ minutes (from 5:00 a.m. to 5:20 a.m.), they complete $8 \times \frac{1}{3} = 2\frac{2}{3}$ relative rounds and cross $2$ times (at 5:07:30 a.m. and 5:15:00 a.m.).
By 7:00 a.m. ( $2$ hours from start), they complete $8 \times 2 = 16$ relative rounds; strictly between 5:20 a.m. and 7:00 a.m. are crossings $3$ through $15$ , which is $15 - 2 = 13$ times.

Trap: If the boundary meeting at exactly 7:00 a.m. (crossing 16) is included in 'between 5:20 a.m. and 7:00 a.m.', the count is 14, which is option (c).