In a 100 metre race, runner A beats runner B by 10 metres and beats runner C by 13 metres. If they now run a 180 metre race at the same constant speeds, by how many metres will B beat C?

Introduction / Context:
This is a classic races and games question involving relative speeds of different runners. You know how much A is ahead of B and C in a 100 metre race. From this, you can find the ratio of their speeds. The question then asks you to use these speed ratios to determine by how much B will beat C in a longer race of 180 metres. It tests your understanding of proportional reasoning and constant speed motion.

Given Data / Assumptions:

• In a 100 m race, A finishes first.
• When A has completed 100 m, B has completed 90 m (A beats B by 10 m).
• When A has completed 100 m, C has completed 87 m (A beats C by 13 m).
• All runners maintain constant speeds across races.
• We consider a new race of 180 m and must find by how many metres B beats C.

Concept / Approach:
From the first race:

• Speeds are proportional to distances covered in the same time.
• So v_A : v_B : v_C = 100 : 90 : 87.

To find how much B beats C in a 180 m race, we:

• Consider the time when B has just finished 180 m.
• Compute how far C runs in that same time, using the speed ratio between B and C.
• The difference between 180 m and C's distance is the margin by which B beats C.

Step-by-Step Solution:
Step 1: From the first race, v_A : v_B : v_C = 100 : 90 : 87. Step 2: Focus on B and C. Their speed ratio v_B : v_C = 90 : 87 = 30 : 29. Step 3: In the new 180 m race, we consider the moment B finishes 180 m. Step 4: Let the time taken by B to run 180 m be T seconds. Step 5: Since distance = speed * time, we have 180 = v_B * T. Step 6: In time T, C covers distance = v_C * T. Step 7: Using ratio v_B / v_C = 30 / 29, we get v_C = (29 / 30) * v_B. Step 8: So distance run by C in time T = v_C * T = (29 / 30) * v_B * T = (29 / 30) * 180 = 174 m. Step 9: Therefore, when B has finished 180 m, C has run 174 m. Step 10: B beats C by 180 - 174 = 6 m.

Verification / Alternative check:
You can also think directly in terms of scaling:

• When A runs 100 m, B runs 90 m and C runs 87 m.
• Ratio of B's speed to C's speed = 90 : 87 = 30 : 29.
• If B runs 30 units of distance, C runs 29 units in the same time.
• Scale this up so that B runs 180 m: 30 units correspond to 180 m, so 1 unit = 6 m, and 29 units = 174 m.
• Again, B beats C by 6 m.

Both methods agree perfectly and show the margin is 6 metres.

Why Other Options Are Wrong:
5.4m, 4.5m, and 5m all come from incorrect ratios or from mixing up who beats whom. None of them arises when you correctly use the speed ratio 30 : 29 and scale distances to 180 m. Only 6m exactly matches the difference between 180 m and 174 m in the new race.

Common Pitfalls:
Learners often try to directly proportion the given 10 m and 13 m advantages to the new distance without first finding speed ratios, which leads to incorrect results. Another common error is to think the ratio is 100 : 90 : 87 and then scale everything incorrectly without isolating B and C. Always convert the scenario to speed ratios and then compute distances for the new race carefully.

Final Answer:
In the 180 metre race, runner B will beat runner C by 6 metres.