In a 1 km race, A beats B by 100 m and B beats C by 100 m. By how many metres does A beat C?
SHORT ANSWER
190 m. When A runs 1,000 m, B runs 900 m. When B runs 1,000 m, C runs 900 m, so when B runs 900 m, C runs 810 m. Hence A beats C by 1,000 - 810 = 190 m.
Explain it simply
"Beats by 100 m" sets a ratio of speeds, not a fixed gap. B's lead over C is 100 m when B finishes the full kilometre, but in the A-B race B has only run 900 m when A finishes, so C's deficit at that moment is a bit less than 100 m. Chain the ratios instead of adding the gaps.
Example
Question: 1 km race. A beats B by 100 m, B beats C by 100 m. A beats C by?
- A : B distance ratio = 1000 : 900.
- B : C distance ratio = 1000 : 900.
- When B has run 900 m, C has run 900 x 900/1000 = 810 m.
- So when A finishes 1,000 m, C is at 810 m.
- A beats C by 190 m.
Speed ratio view: A : C = (10/9) x (10/9) = 100/81. In the time A runs 1,000 m, C runs 810 m.
js
const c = 1000 * 0.9 * 0.9;
console.log(1000 - c); // Output: 190What to say in the interview
"I convert each 'beats by' into a ratio. B runs 0.9 of A's distance and C runs 0.9 of B's, so C runs 0.81 of A's, 810 m. A beats C by 190 m, not 200."
Common mistakes
- Adding the margins and answering 200 m.
- Treating 'beats by 10 seconds' and 'beats by 100 m' as the same kind of statement; time margins need the speed first.