X, Y and Z are three contestants in a race of 1000 m. Assume that all run with different uniform speeds. X gives Y a start of 40 m and X gives Z a start of 64 m. If Y and Z were to compete in a race of 1000 m, how many metres start will Y give to Z ?
- (a) 20
- (b) 25 ✓ UPSC's answer
- (c) 30
- (d) 35
Why the answer is (b)
• Let the speeds of X, Y, and Z be $V_x$, $V_y$, and $V_z$ respectively.
• Since X gives Y a 40 m start in a 1000 m race, X runs 1000 m while Y runs 960 m in the same time, so $\frac{V_x}{V_y} = \frac{1000}{960} = \frac{25}{24}$.
• Since X gives Z a 64 m start, X runs 1000 m while Z runs 936 m in the same time, so $\frac{V_x}{V_z} = \frac{1000}{936} = \frac{125}{117}$.
• To find the ratio of Y's speed to Z's speed, divide the first ratio by the second: $\frac{V_y}{V_z} = \frac{V_x/V_z}{V_x/V_y} = \frac{125/117}{25/24} = \frac{125}{117} \times \frac{24}{25} = \frac{5 \times 24}{117} = \frac{120}{117} = \frac{40}{39}$.
• This means when Y runs 40 m, Z runs 39 m. In a 1000 m race, if Y runs 1000 m, Z runs $1000 \times \frac{39}{40} = 975$ m.
• Therefore, Y must give Z a start of $1000 - 975 = 25$ m.
Why the other options are wrong
- (a) 20
- Option (a) is incorrect because the calculated start distance is 25 m, not 20 m.
- (c) 30
- Option (c) is incorrect because the calculated start distance is 25 m, not 30 m.
- (d) 35
- Option (d) is incorrect because the calculated start distance is 25 m, not 35 m.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.