Answer to Riddle #70: Three Runners Each Beat the Other by 20m

70. Andrew, Britney and Carol race each other in a 100 meters race. All of them run at a constant speed throughout the race.

Andrew beats Britney by 20 meters.
Britney beats Carol by 20 meters.

By how many meters does Andrew beat Carol?

Another quickie, a bit harder than the last but we should be able to dispatch it quite quickly:

A woman running in silhouetteBefore reading the answer can I interest you in a clue?

So experienced puzzlers like us will detect instantly that the answer 40m doesn't smell right. But why? The two distances of 20m are not measured at the same time, one is measured when Andrew crosses the line, the other when Britney crosses the line. They don't all stay 20m apart for the whole race they get further apart. Britney and Carol will be less than 20m apart at the time Andrew crosses the line.

We are going to need a little maths. Not much, just speed = distance over time. We will find out Carol's distance when Andrew crosses the line. So time TA is the time when Andrew crosses the line, using t = d / v we can find an expression that accounts for the fact that at this time Andrew has travelled 100m and Carol 80m:
1) TA = 100 / VA = 80 / VB
Similarly looking at the time when Britney crosses the line:
2) 100 / VB = 80 / VC

We really now have what we need, the rest is just algebra:
1) → VB = (4/5)VA
2) → VC = (4/5)VB
→ VC = (4/5)(4/5)VA = (16/25)VA

Remember we are looking for the DC the distance Carol has travelled at the time Andrew crosses the line. We already have an expression for that time in terms of VA in equation 1. 1) TA = 100 / VA. So we have a speed and we have a time. distance it trivial.
DC = VCT = (16/25)VA • 100 / VA
DC = 64m

So at the time that Adam crosses the line Carol is at 64m. Meaning Adam wins by 36m

Calculating in Your Head

The above treatment is quite formal. Since the speeds are constant we know that at any point Britney has travelled 80% as far as Andrew and Carol 80% as far as Britney....

It seems reasonable to calculate that the distance Carol has travelled is 0.8 • 0.8 DA = 0.64 DA

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