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

Introduction / Context:
This question tests relative speed in running races and how results from different race lengths can be combined. Instead of working with time directly, we compare how far each runner covers when another runner finishes the race. This type of aptitude problem is very common in competitive exams because it checks comfort with proportional reasoning and careful handling of ratios.

Given Data / Assumptions:
- In a 1000 m race, A finishes when B has covered only 900 m. - In an 800 m race, B finishes when C has covered only 700 m. - All runners keep constant speeds in every race. - We must find how many metres A will be ahead of C in a 600 m race.

Concept / Approach:
The key concept is that speed is proportional to distance covered in the same time. From race results we can find ratios of speeds of A, B, and C. Once we know the ratio of speeds, we can calculate how far C runs while A completes 600 m, and the difference between 600 m and that distance gives the lead of A over C.

Step-by-Step Solution:
Step 1: From the first race, A runs 1000 m while B runs 900 m in the same time, so vA : vB = 1000 : 900 = 10 : 9. Step 2: From the second race, B runs 800 m while C runs 700 m in the same time, so vB : vC = 800 : 700 = 8 : 7. Step 3: Combine both results. Write vA = 10k and vB = 9k. Also write vB = 8t and vC = 7t. Step 4: Since vB is the same, 9k = 8t, so t = 9k / 8 and vC = 7t = 7 * 9k / 8 = 63k / 8. Step 5: Now vA : vC = 10k : (63k / 8) = 80 : 63. Step 6: In a 600 m race, time taken by A to finish is T = 600 / vA. Step 7: In the same time, distance covered by C = vC * T = vC * 600 / vA = 600 * (vC / vA). Step 8: Using vC / vA = 63 / 80, distance by C = 600 * 63 / 80 = 472.5 m. Step 9: Lead of A over C = 600 - 472.5 = 127.5 m.

Verification / Alternative check:
We can also work with unit speeds. Take vA = 80 units and vC = 63 units. Time for A to run 600 m is 600 / 80 = 7.5 time units. Distance of C in this time is 63 * 7.5 = 472.5 m. This again gives a lead of 127.5 m, so the calculation is consistent and confirms the answer.

Why Other Options Are Wrong:
- 254 m: This would require C to be extremely slow compared to A, which contradicts the given race data. - 184 m: This does not match any consistent ratio derived from the two races. - 212 m: This is larger than possible given that C is only slightly slower than B, not extremely slower than A.

Common Pitfalls:
Many learners subtract distances directly, for example by trying to scale 100 m and 100 m from the original races without computing speed ratios. Others average the two gaps, which is not correct. The safe method is always to convert the information into speed ratios and then apply those ratios carefully to the new race distance.

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