Beaten by twenty metres, twice

Question

Three runners each run at their own constant speed. In a 100 metre race, Andrew beats Britney by 20 metres: when Andrew finishes, Britney has run 80 metres. In a separate 100 metre race, Britney beats Carol by 20 metres.

If Andrew and Carol race over 100 metres, by how many metres does Andrew win?

The answer is not 40.

Accepted Answer

36 metres. Speed ratios. In the same time that Andrew runs 100 metres, Britney runs 80. So Britney's speed is 80/100 = 0.8 of Andrew's. Likewise Carol's speed is 0.8 of Britney's. Chain them. Carol's speed is 0.8 × 0.8 = 0.64 of Andrew's. When Andrew has run 100 metres, Carol has run 64 metres. Andrew wins by 100 minus 64 = 36 metres. Why not 40. The two 20-metre margins are measured against different runners. Britney's 20-metre lead over Carol is earned while Britney runs 100 metres, but in Andrew's race Britney only gets to run 80 metres before it ends, and in that shorter time her lead ove…

Beaten by twenty metres, twice

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