Head-start with speed ratio: Two men A and B run a 500 m race. A has a 140 m head-start, and their speeds are in the ratio 3 : 4 (A : B). By how many metres does A win?

Introduction / Context:
In head-start problems with speed ratios, determine where the opponent is when the leading runner finishes. That distance shortfall is the winning margin.

Given Data / Assumptions:

• Race length = 500 m.
• A starts 140 m ahead, so A must run only 360 m to finish.
• Speeds A : B = 3 : 4 (constant).

Concept / Approach:
Equal times to the finish for A and B when A wins: if A runs 360 m, B runs (4/3)*360 = 480 m. The deficit from 500 m is the margin by which A wins.

Step-by-Step Solution:

Verification / Alternative check:
At the instant B reaches 500 m, A would have covered (3/4)*500 = 375 m from his own start; since A only needed 360 m, he would have finished 15 m earlier. Both checks are consistent with a 20 m win margin definition (distance B is short when A finishes).

Why Other Options Are Wrong:
10/40/60 do not match the direct proportional calculation.

Common Pitfalls:
Subtracting the head-start from the race length without using the speed ratio.

Final Answer:
20 m