UPSC Prelims 2010 Q101: Science & Technology – Quantitative Aptitude
Three men start together to travel the same way around a circular track of 11 km. Their
speeds are 4, 5.5 and 8 kmph respectively. When will they meet at the starting point for the first
time?
- (a) After 11 hours
- (b) After 21 hours
- (c) After 22 hours
- (d) After 33 hours
Answer: (c) After 22 hours
Explanation
When three runners first meet at the starting point
Three men circle an 11 km track at speeds 4, 5.5, and 8 km/h; we need when they're all simultaneously back at the start.
| Step | Calculation |
|---|---|
| Lap time, runner 1 (4 km/h) | 11/4 = 2.75 hours |
| Lap time, runner 2 (5.5 km/h) | 11/5.5 = 2 hours |
| Lap time, runner 3 (8 km/h) | 11/8 = 1.375 hours |
| Find LCM of 2.75, 2, and 1.375 hours | Testing multiples shows the smallest common time is 22 hours |
| Verify | 22/2.75 = 8 laps; 22/2 = 11 laps; 22/1.375 = 16 laps — all whole numbers |
Why LCM of lap times works:
| Point | Detail |
|---|---|
| Each runner returns to start | Only after completing a whole number of laps |
| All three together | Only at a time that is a common multiple of all three lap times |
Answer: (c) After 22 hours.
Why each option is right or wrong
- (a) Incorrect. 11 hours is not a common multiple of all three individual lap times.
- (b) Incorrect. 21 hours also fails to be an exact multiple of all three lap times.
- (c) CORRECT. 22 hours is the least common multiple of 2.75, 2, and 1.375 hours, verified by whole-lap counts of 8, 11, and 16.
- (d) Incorrect. 33 hours, while also a common multiple, is not the FIRST (smallest) time they meet — 22 hours comes first.
Key point
All three runners meet at the start only at a time that is a common multiple of their individual lap times (2.75h, 2h, 1.375h) — the smallest such time is 22 hours.
Remember
Three runners on an 11 km track at 4, 5.5, 8 km/h: first meet back at the start after 22 hours.
Common mistakes
- Picking a later common multiple (like 33 hours) instead of the FIRST (smallest) one.
- Making an arithmetic error while finding the LCM of non-integer lap times.
- Forgetting to verify that the chosen time gives whole-number lap counts for all three runners.
Exam tip
'When do they all meet at the start' = LCM of each runner's individual lap time. Always pick the SMALLEST such common multiple.
Sources
- NCERT Class X Mathematics — Real Numbers
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (H.C.F. and L.C.M. of Numbers)