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Maths Classic Race Puzzle

Maths Classic Race Puzzle Solution - 19 November

Lavesh, Bolt, and Lewis race each other in a 100 meters race. All of them run at a constant speed throughout the race.

Lavesh beats Bolt by 20 meters.
Bolt beats Lewis by 20 meters.

How many meters does Lavesh beat Lewis by ?

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Solution
Lavesh beats Lewis by 36 meters.

It is tempting to guess that Lavesh beats Lewis by 40 meters, but when Lavesh finishes and is 20 meters ahead of Bolt, Bolt is NOT 20 meters ahead of Lewis (he's only 16 meters ahead), and it will take a couple more seconds before Bolt increases his lead over Lewis to 20 meters.

To figure out the answer, we let Lavesh's speed be A meter/second. So it takes him 100/A seconds to finish the race. At this point, we know that Bolt has run 80 meters (since Lavesh beats him by 20 meters). So Bolt runs 80 meters in 100/A seconds, meaning that he is running at a speed of (80/(100/A)) meters/second, or (8A/10) meters per second.

So we then know that it takes Bolt 100/(8A/10) seconds to finish the race, or 125/A seconds. At this point, we know that Lewis has run 80 meters (since Bolt beats him by 20 meters). So Lewis runs 80 meters in 125/A seconds, meaning that he is running at a speed of (80/(125/A)) meters/second, or 80A/125 meters per second.

Now that we know Lewis's speed, we just need to figure out how far he had run when Lavesh finished the race. Since Lavesh finished in 100/A seconds, we can determine that Lewis had run (100/A) * (80A/125) = 8000/125 = 64 meters when Lavesh finished the race. And so Lavesh beat him by (100 - 64) = 36 meters.