In a 100 metre race, A finishes the race in 36 seconds while B finishes it in 45 seconds. By how many metres does A beat B?

Introduction / Context:
This is a basic races and games problem where the finishing times of two runners for the same distance are known. Because they run at constant speeds, the distances they cover in the same time are proportional to their speeds. The question asks how many metres A is ahead of B when A finishes the race. This tests your ability to use proportional reasoning with speed, distance, and time.

Given Data / Assumptions:

• Race distance = 100 m.
• A's finishing time = 36 seconds.
• B's finishing time = 45 seconds.
• Both run at constant speeds.
• We need the distance between A and B when A finishes the race.

Concept / Approach:
We know:

• B takes 45 seconds to cover 100 m.
• Therefore, B's speed = 100 / 45 m/s.
• When A finishes in 36 seconds, we compute how far B has run in those 36 seconds using B's constant speed.
• The difference between 100 m and B's distance at that time is the lead by which A beats B.

Step-by-Step Solution:
Step 1: Compute B's speed. Step 2: v_B = 100 m / 45 s = 100 / 45 m/s. Step 3: In 36 seconds, distance covered by B = v_B * 36. Step 4: Distance_B_36 = (100 / 45) * 36 = 100 * 36 / 45. Step 5: Simplify 36 / 45 = 4 / 5, so Distance_B_36 = 100 * 4 / 5 = 80 m. Step 6: When A finishes 100 m, B has covered only 80 m. Step 7: Therefore, A beats B by 100 - 80 = 20 m.

Verification / Alternative check:
You can also compare speeds:

• A's speed v_A = 100 / 36 m/s.
• B's speed v_B = 100 / 45 m/s.
• The ratio of their speeds v_A : v_B = (100 / 36) : (100 / 45) = 45 : 36 = 5 : 4.
• This means when A runs 5 units of distance, B runs 4 units in the same time.
• At the moment A has completed 100 m (5 units), B has completed (4 / 5) * 100 = 80 m, so the gap is 20 m.

Both methods give consistent results.

Why Other Options Are Wrong:
25m, 22.5m, and 9m arise from incorrect handling of ratios or incorrect simplification. None of these values match the derived relation that B covers 80 m in the time A covers 100 m. Therefore, only 20m correctly captures the distance by which A is ahead.

Common Pitfalls:
A common mistake is to confuse the time difference (45 - 36 = 9 seconds) with a distance difference, and therefore answer 9m, which is incorrect. Another error is to try to average times or distances without using proper proportional reasoning. Always work either with speeds or with ratios of speeds for a precise solution.

Final Answer:
Runner A beats runner B by 20 metres in the 100 metre race.