In a 100 m race, A beats B by 10 m. If B is given a start of 10 m and they run…

Concept: If A beats B by x m in a race of d m, then in the same time A covers d m while B covers only (d - x) m, giving speed ratio A : B = d : (d - x). If, in another race of the same distance d, B is given a start of exactly x m (so B only has to cover d - x m while A covers the full d m), the two always finish together - this is the standard 'dead heat' result in races-and-games problems, not a coincidence.

1. From 'A beats B by 10 m in a 100 m race': when A covers 100 m, B covers only 90 m in the same time, so the speed ratio is A : B = 100 : 90 = 10 : 9.

2. In the second 100 m race, B is given a start of 10 m, so B only needs to cover 100 minus 10 = 90 m while A covers the full 100 m.

3. Time for A to finish = 100 / vA. Time for B to finish = 90 / vB.

4. Substituting vA = (10/9) vB: time for A = 100 / ((10/9) vB) = 90 / vB, which is exactly B's time.

Cross-check with concrete speeds: let vB = 9 m/s so vA = 10 m/s (ratio 10 : 9). A covers 100 m in 100/10 = 10 s. B covers the remaining 90 m in 90/9 = 10 s - the two times match exactly.

Result: A and B cross the finish line at the same instant, so the race ends in a TIE (dead heat).