In a race of 1000 m, runner A can beat B by 100 m, and in a race of 800 m, B can beat C by 100 m. By how many metres will A beat C in a 600 m race?

Introduction / Context:
This is another version of the earlier relative speed question involving three runners A, B, and C. We are given A versus B over 1000 m and B versus C over 800 m and asked to find A's lead over C over 600 m. The structure is the same, but the options are expressed with the word metres written out. It still tests the same ratio and proportional reasoning skills.

Given Data / Assumptions:
- In a 1000 m race, A finishes when B has covered 900 m. - In an 800 m race, B finishes when C has covered 700 m. - Speeds of all runners remain constant. - We must find A's lead over C in a 600 m race.

Concept / Approach:
We again convert the information into speed ratios. From each race we get vA relative to vB and vB relative to vC. Combining these gives vA relative to vC. Once we know vC / vA, we can compute how far C runs while A completes 600 m, and then subtract to get the lead. This is a direct proportional application.

Step-by-Step Solution:
Step 1: From the first race, vA : vB = 1000 : 900 = 10 : 9. Step 2: From the second race, vB : vC = 800 : 700 = 8 : 7. Step 3: Let vB = 9k and vA = 10k. Also write vB = 8t and vC = 7t. Step 4: Since vB is same, 9k = 8t, so t = 9k / 8 and vC = 7 * 9k / 8 = 63k / 8. Step 5: Ratio vC / vA = (63k / 8) / (10k) = 63 / 80. Step 6: Time taken by A to run 600 m is T = 600 / vA. Step 7: In time T, C runs distance = vC * T = 600 * (vC / vA) = 600 * 63 / 80 = 472.5 m. Step 8: Lead of A over C = 600 - 472.5 = 127.5 metres.

Verification / Alternative check:
Choose vA = 80 and vC = 63 units. Time for A to run 600 m is 600 / 80 = 7.5. In this time C runs 63 * 7.5 = 472.5 m, which produces the same difference of 127.5 m. This confirms the consistency of the computed lead with the combined ratio method.

Why Other Options Are Wrong:
- 125.5 m and 126.5 m: These are close but incorrect, often caused by premature rounding during calculations. - 128.5 m: Slight overestimate that does not match the exact fractional result 600 * 63 / 80.

Common Pitfalls:
A common mistake is to average or directly add the differences 100 m and 100 m without translating them into speed ratios. Others treat the new race distance as 1000 m or 800 m instead of 600 m. Always re-build the ratio chain carefully, then apply it only to the exact race length mentioned in the question.

Final Answer:
A will beat C by 127.5 metres in the 600 m race.