Inverse-time concept for equal distances: The speeds of three cars are in the ratio 2 : 3 : 4. Find the ratio of the times taken by these cars to cover the same distance.

Introduction / Context:
For a fixed distance, time is inversely proportional to speed. This problem tests whether you can correctly invert a speed ratio to obtain a time ratio and express it in integers.

Given Data / Assumptions:

• Speed ratio = 2 : 3 : 4.
• Distance is the same for all cars.
• Time ∝ 1 / Speed.

Concept / Approach:
If v1 : v2 : v3 = 2 : 3 : 4, then t1 : t2 : t3 = 1/2 : 1/3 : 1/4. To avoid fractions, multiply by the least common multiple of denominators (LCM = 12).

Step-by-Step Solution:
Time ratio (fractions) = 1/2 : 1/3 : 1/4.Multiply each term by 12 ⇒ 6 : 4 : 3.

Verification / Alternative check:
Pick a sample distance, say 12 units. Times would be 12/2 = 6, 12/3 = 4, 12/4 = 3, confirming 6 : 4 : 3.

Why Other Options Are Wrong:

• 2 : 3 : 4 repeats the speed ratio instead of inverting.
• 4 : 3 : 2 and 4 : 3 : 6 do not match 1/2 : 1/3 : 1/4 scaled.

Common Pitfalls:

• Confusing direct and inverse proportion for time vs speed.
• Forgetting to scale fractional ratios to whole numbers.

Final Answer:
6 : 4 : 3