Races & Games

"A beats B by x metres" is shorthand for: in the time A ran the whole course d, B ran only d − x. Because the time is the same, that immediately gives the speed ratio:

A's speed : B's speed = d : (d − x)

Four types:

1. A beats B by x m — B has covered d − x when A finishes.
2. Beats by t seconds — convert using B's speed: margin in metres = B's speed × t.
3. "A gives B a start of x m" — B starts x m ahead, so A runs d while B runs d − x; the speed ratio is the same as type 1.
4. A beats B by a, B beats C by b — by how much does A beat C? Chain the ratios: when A runs d, B runs d − a; when B runs d − a, C runs (d − a)(d − b)/d.

The single most common mistake is type 4: adding the two margins. They do not add, because the second margin applies to a shorter run. In a 100 m race, 10 m plus 5 m gives 14.5 m, not 15 m — always slightly less than the sum.

A "dead heat" means both finish together, so the start given exactly cancels the speed difference.

A beats B by x in a d race → speeds d : (d − x) · beats by t s → margin = B's speed × t · A beats C: d − (d−a)(d−b)/d

In a 100 m race A beats B by 10 m and B beats C by 5 m. By how much does A beat C? → When A runs 100, B runs 90. When B runs 100, C runs 95, so when B runs 90, C runs 90 × 95/100 = 85.5 m. A beats C by 100 − 85.5 = 14.5 m. Sanity check: it must be less than 10 + 5 = 15, because C loses only 5% of the 90 m B actually ran, not of the full 100 m — and 5% of 90 is 4.5, giving 10 + 4.5 = 14.5.