In a 500 metre race, the speeds of contestants A and B are in the ratio 3 : 4. A is given a head start of 140 metres. By how many metres does A win the race?

Introduction / Context:
This problem combines speed ratios with a head start in a race. Two contestants, A and B, run a 500 metre race. Their speed ratio is known, and A is given a 140 metre head start. You need to find by how many metres A wins the race. Such questions assess understanding of relative speeds, distances, and head starts in competitive scenarios.

Given Data / Assumptions:

• Total race distance = 500 m.
• Speed ratio v_A : v_B = 3 : 4.
• A is given a start of 140 m.
• Hence, A needs to run 500 - 140 = 360 m.
• B runs the full 500 m.
• Both run at constant speeds.

Concept / Approach:
We work with speed ratio and times:

• Let v_A = 3k and v_B = 4k for some k.
• Time taken by A to complete 360 m: T_A = 360 / (3k).
• Time taken by B to complete 500 m: T_B = 500 / (4k).
• Compare T_A and T_B to see who finishes first.
• Then determine how far the trailing runner has gone when the winner finishes.

Step-by-Step Solution:
Step 1: Let v_A = 3k and v_B = 4k. Step 2: Distance A runs = 360 m, so T_A = 360 / (3k) = 120 / k seconds. Step 3: Distance B runs = 500 m, so T_B = 500 / (4k) = 125 / k seconds. Step 4: Since 120 / k < 125 / k, A finishes earlier and wins the race. Step 5: At time T_A = 120 / k, distance covered by B = v_B * T_A = 4k * (120 / k) = 480 m. Step 6: At that moment, B is at 480 m while the race distance is 500 m. Step 7: So A wins by 500 - 480 = 20 m.

Verification / Alternative check:
You can use a scaling method:

• Let k = 1 for simplicity, so v_A = 3 m/s and v_B = 4 m/s.
• A's time: T_A = 360 / 3 = 120 s.
• B's time if he runs 500 m: T_B = 500 / 4 = 125 s.
• In 120 s, B travels 4 * 120 = 480 m.
• Difference between race finish line and B's position = 500 - 480 = 20 m.

This numerical check yields the same margin of victory and confirms our solution.

Why Other Options Are Wrong:
60m and 40m represent larger margins than what the speed ratio and distances allow. If A were this far ahead, the times would not match the given speeds. A margin of 10m is too small; recalculations show B would not be that close when A finishes. Only 20m is consistent with the ratio v_A : v_B = 3 : 4 and the 140 m head start.

Common Pitfalls:
Some students incorrectly adjust the starting distance rather than the distance A actually runs, or they assume both runners still cover 500 m. Others may average the speeds or distances erroneously. The correct method is to compute time for each runner based on the true distance they cover, then compare how far the slower runner has gone when the faster one finishes.

Final Answer:
Runner A wins the 500 metre race by 20 metres.