Runner A can cover 22.5 metres in the same time that runner B covers 25 metres. In a 1 kilometre race, by how many metres will B beat A?

Introduction / Context:
This question is a straightforward application of speed ratios in a race. You are told that in the same time, A runs 22.5 m while B runs 25 m. From this, you can find the ratio of their speeds. The race distance is 1 kilometre, and you must determine by how much B beats A at the finish line. Such questions are important for building intuition about relative speed and proportional reasoning.

Given Data / Assumptions:

• In equal time intervals, A covers 22.5 m and B covers 25 m.
• Race distance = 1 km = 1000 m.
• Both runners maintain constant speeds.
• We need the distance by which B beats A at the end of the 1000 m race.

Concept / Approach:
Speeds are proportional to distances covered in the same time:

• v_A : v_B = 22.5 : 25.
• Simplify this to a simple integer ratio.
• When B runs the full 1000 m, compute how much distance A covers in the same time using the speed ratio.
• The difference between 1000 m and A's distance is the winning margin of B.

Step-by-Step Solution:
Step 1: v_A : v_B = 22.5 : 25. Step 2: Write 22.5 as 45/2 and 25 as 25. Step 3: Ratio v_A : v_B = (45 / 2) : 25 = 45 : 50. Step 4: Simplify 45 : 50 by dividing both by 5 to get 9 : 10. Step 5: So v_A : v_B = 9 : 10. Step 6: Let B cover 10 speed units while A covers 9 units in the same time. Step 7: In the actual race, B runs 1000 m. This corresponds to 10 units of distance. Step 8: So 1 unit = 1000 / 10 = 100 m. Step 9: A's distance in that time = 9 units = 9 * 100 = 900 m. Step 10: Therefore, B beats A by 1000 - 900 = 100 m.

Verification / Alternative check:
We can verify with explicit speeds:

• Assume a unit time t where A covers 22.5 m and B covers 25 m.
• Speed v_A = 22.5 / t, v_B = 25 / t.
• To run 1000 m, B's time = 1000 / v_B = 1000t / 25 = 40t.
• In 40t, A covers distance = v_A * 40t = (22.5 / t) * 40t = 900 m.
• Difference = 1000 - 900 = 100 m.

This consistency check confirms that B is indeed 100 metres ahead when B finishes the 1 km race.

Why Other Options Are Wrong:
75 m, 25 m, and 50 m occur if one mis-scales the ratio 9 : 10 or forgets to map 10 units to 1000 m correctly. Any smaller difference would contradict the ratio v_B / v_A = 10 / 9, which predicts that A covers only 900 m when B completes 1000 m. Thus, only 100 m matches the correct calculation.

Common Pitfalls:
The most common error is failing to simplify the ratio correctly and making arithmetic mistakes when scaling it up to the actual distance. Some students also mistakenly average the distances 22.5 and 25 or think the difference 2.5 m directly translates to 1000 m without using ratios. Always express the relationship in terms of speed ratio and then scale it carefully to the given race distance.

Final Answer:
In the 1 kilometre race, runner B will beat runner A by 100 metres.