A runs 7/4 times as fast as B. If A gives B a start of 90 m, how far must the…

Concept

In a race where the runners' speeds are in a fixed ratio, both runners take the SAME time to finish. Since distance = speed × time, in equal time the distances they cover are in the same ratio as their speeds. A "start of d metres" means the slower runner is allowed to run d metres less than the full course. So if the winning post is at distance x, the faster runner covers x while the slower runner need cover only (x − d), and for a dead heat these two distances must be in the speed ratio.

Application

1. Fix the speed ratio.

   A runs 7/4 times as fast as B, so speed of A : speed of B = 7 : 4. In equal time, distance(A) : distance(B) = 7 : 4.

2. Express the gain per unit race.

   While A covers 7 m, B covers 4 m, so A gains 7 − 4 = 3 m on B for every 7 m that A runs.

3. Convert the required 90 m start into A's distance.

   The 90 m start is exactly the lead A must build up. A gains 3 m for every 7 m it runs, so the distance A must run to gain 90 m is x = 7 × (90 ÷ 3) = 7 × 30 = 210 m.

So the winning post must be 210 metres from the start.

Cross-check

If the post is at 210 m, A runs the full 210 m while B (given a 90 m start) runs only 210 − 90 = 120 m. Check the ratio of distances: 210 : 120 = 7 : 4, which is exactly the speed ratio — so both reach the post in the same time. The answer 210 m is confirmed.