Speed–time inverse relationship: The speeds of three cars are in the ratio 3 : 4 : 5. For the same distance, what is the ratio of the times taken by them? Provide the simplified integer ratio.

Introduction / Context:
For a fixed distance, time is inversely proportional to speed. Thus, if speeds are in one ratio, times will be in the reciprocal ratio. Converting these reciprocals into integers makes the answer clear and comparable.

Given Data / Assumptions:

• Speed ratio = 3 : 4 : 5.
• All travel the same distance.

Concept / Approach:
If v ∝ 3 : 4 : 5, then time t ∝ 1/3 : 1/4 : 1/5. Multiply by the least common multiple of denominators (here 60) to eliminate fractions and obtain an integer ratio.

Step-by-Step Solution:

Verification / Alternative check:
Check consistency: times multiplied by speeds should be equal (distance). 20*3 = 60, 15*4 = 60, 12*5 = 60—valid.

Why Other Options Are Wrong:
3 : 4 : 5 is the speed ratio, not time; 5 : 4 : 3 is not the correct inverse; 12 : 15 : 20 reverses from fastest to slowest.

Common Pitfalls:
Forgetting that time and speed are inversely related; copying the speed ratio directly for time.

Final Answer:
20 : 15 : 12