The speeds of three racers are in the ratio 3 : 4 : 6. What is the ratio of the times taken by them to cover the same distance?

Introduction / Context:
This question explores the inverse relationship between speed and time for a fixed distance. When distance is constant, time is inversely proportional to speed: higher speed means less time, and lower speed means more time. The problem gives the ratio of speeds of three racers and asks for the ratio of times taken to cover the same distance. Such inverse proportion questions are very common in time, speed and distance topics.

Given Data / Assumptions:

• Ratio of speeds of three racers = 3 : 4 : 6.
• They all cover the same distance.
• We need the ratio of time taken by each racer.
• All speeds are positive and non zero.

Concept / Approach:
For a fixed distance, Time = Distance / Speed. This means time is inversely proportional to speed. If speeds are in the ratio v1 : v2 : v3, then times are in the ratio 1 / v1 : 1 / v2 : 1 / v3. Therefore, if speeds are 3 : 4 : 6, then times are 1 / 3 : 1 / 4 : 1 / 6. We then find an equivalent integer ratio by taking a common multiple of the denominators.

Step-by-Step Solution:
Speeds are in ratio 3 : 4 : 6.Let the common distance be D.Time taken by each racer: T1 = D / (3k), T2 = D / (4k), T3 = D / (6k) for some constant k.So T1 : T2 : T3 = 1 / 3 : 1 / 4 : 1 / 6.To remove denominators, multiply all terms by 12 (the least common multiple of 3, 4 and 6).(1 / 3) * 12 = 4, (1 / 4) * 12 = 3, (1 / 6) * 12 = 2.Therefore, T1 : T2 : T3 = 4 : 3 : 2.

Verification / Alternative check:
Assume the distance D = 12 units and speeds are proportional to actual speeds 3, 4 and 6 units per hour.Time for first racer = 12 / 3 = 4 hours.Time for second racer = 12 / 4 = 3 hours.Time for third racer = 12 / 6 = 2 hours.This confirms the time ratio 4 : 3 : 2.

Why Other Options Are Wrong:
The ratio 3 : 4 : 6 is the speed ratio, not the time ratio. Ratios 6 : 4 : 3 or 1 : 2 : 3 do not correspond to the inverse of 3 : 4 : 6 when calculated properly. Option 2 : 3 : 5 is also unrelated. Only 4 : 3 : 2 matches the correct inverse proportional relation.

Common Pitfalls:
A typical mistake is to think time and speed are directly proportional and to answer 3 : 4 : 6. Another error is to invert the ratio partially or incorrectly. Always remember that for a fixed distance, time is inversely proportional to speed, so you must take reciprocals and then simplify.

Final Answer:
The ratio of times taken by the three racers is 4 : 3 : 2.