In a 100 m race, runner A beats B by 10 m and beats C by 13 m. In a 180 m race, by how many metres will B beat C?

Introduction / Context:
Here we compare three runners using A as a reference. We know how much A beats B and C by in a 100 m race and must then predict the result of a longer race between B and C only. This is another application of relative speeds, where we first find B and C in terms of A and then obtain a direct comparison of B and C.

Given Data / Assumptions:
- A beats B by 10 m in a 100 m race, so at A's finish, B has run 90 m. - A beats C by 13 m in a 100 m race, so at A's finish, C has run 87 m. - Speeds of A, B, and C are constant and do not change across races. - We must find B's lead over C in a 180 m race.

Concept / Approach:
From the 100 m race, we find the speed ratios: vB / vA and vC / vA. Dividing these gives vB / vC directly. Knowing the ratio vB : vC lets us compute how far C runs in the time that B covers 180 m. The difference between 180 m and that distance is B's lead over C.

Step-by-Step Solution:
Step 1: From A vs B, vB / vA = 90 / 100 = 9 / 10. Step 2: From A vs C, vC / vA = 87 / 100. Step 3: Therefore vB / vC = (vB / vA) / (vC / vA) = (9 / 10) / (87 / 100) = (9 / 10) * (100 / 87) = 90 / 87 = 30 / 29. Step 4: So vC / vB = 29 / 30. Step 5: Let B run 180 m in time TB = 180 / vB. Step 6: In time TB, C covers distance = vC * TB = 180 * (vC / vB) = 180 * 29 / 30 = 174 m. Step 7: Lead of B over C = 180 - 174 = 6 m.

Verification / Alternative check:
Assume vB = 30 units and vC = 29 units. Time for B to cover 180 m is 180 / 30 = 6 units. In that time C covers 29 * 6 = 174 m. The difference is again 6 m, matching our earlier computation. Also, if we scale these speeds to a 100 m race with A, the original leads can be reproduced, which supports the internal consistency of the ratios used.

Why Other Options Are Wrong:
- 5.4 m and 4.5 m: These numbers usually arise from approximate or incorrect use of decimal ratios instead of exact fractions. - 5 m: A rounded guess close to the correct answer but not supported by the exact ratio 30 : 29.

Common Pitfalls:
The main trap is to treat the 10 m and 13 m as independent gaps and try to scale them directly to 180 m, instead of working through speed ratios. Another mistake is to subtract 10 and 13 in the wrong direction, leading to negative or meaningless values. Always convert the leads into ratios before scaling to a different race distance.

Final Answer:
In a 180 m race, B will beat C by 6 m.