In a 1 km race, runner A gives runner B a start of 100 metres, and in another 1 km race, runner B gives runner C a start of 80 metres. In a 1 km race among A, B and C, who will win and by what distance will the winner beat the slowest runner?

Introduction / Context: This is a classic race and relative speed problem involving three runners and different head starts. The head start tells you about the ratio of speeds between runners. Once you know those ratios, you can find how far each runner covers in the time taken by the winner to finish a race and hence calculate the difference in distances between them. Given Data / Assumptions:

• In a 1 km race between A and B, A gives B a start of 100 metres and they finish together.
• In a 1 km race between B and C, B gives C a start of 80 metres and they finish together.
• In a 1 km race among A, B and C, we must find how much the winner beats the slowest runner by in terms of distance.
• All runners maintain constant speeds in all races.

Concept / Approach: From the first condition, when A covers 1000 metres, B covers 900 metres in the same time. Thus, speed ratio v_B / v_A = 900 / 1000 = 9 / 10. From the second condition, when B covers 1000 metres, C covers 920 metres, so v_C / v_B = 920 / 1000 = 23 / 25. By multiplying these ratios, we can find v_C relative to v_A. Then, in a common 1 km race, we measure how far B and C run in the time it takes A to cover 1000 metres. Step-by-Step Solution: Step 1: From A versus B race: v_B / v_A = 900 / 1000 = 9 / 10. Step 2: From B versus C race: v_C / v_B = 920 / 1000 = 23 / 25. Step 3: Combine ratios to express v_C in terms of v_A: v_C / v_A = (v_C / v_B) * (v_B / v_A) = (23 / 25) * (9 / 10) = 207 / 250. Step 4: Now in a common race, let v_A = 1 unit speed. Then v_B = 9 / 10, v_C = 207 / 250. Step 5: Time taken by A to complete 1000 m = 1000 / v_A = 1000 units of time. Step 6: Distance covered by B in this time = v_B * time = (9 / 10) * 1000 = 900 metres. Step 7: Distance covered by C in this time = v_C * time = (207 / 250) * 1000 = 828 metres. Step 8: C is the slowest runner, covering 828 m when A finishes 1000 m. Difference = 1000 - 828 = 172 metres. Verification / Alternative check: We know that A is the fastest because A beats B with a head start, and B beats C with another head start. It is consistent that C has the smallest speed. The computed distances 1000, 900 and 828 clearly reflect that A is first, B is next and C is last. The gap between A and C is 172 metres, which matches our calculation. Why Other Options Are Wrong:

• 182 m: Slightly larger than the correct value, likely due to arithmetic errors when multiplying fractions.
• 152 m: Underestimates the gap between A and C.
• 162 m: Also underestimates the difference and does not align with the precise ratios.

Common Pitfalls:

• Confusing which runner is faster when interpreting the given head starts.
• Using 1000 - 920 or 1000 - 900 directly without combining the two races through speed ratios.
• Making mistakes in fraction multiplication when computing v_C relative to v_A.

Final Answer: In the 1 km race, the fastest runner A beats the slowest runner C by 172 m.