Math behind a story
In a 10000 m race between a hare and a tortoise, the hare's speed is 5 times of the tortoise's speed. They depart the starting point at the same time. The tortoise runs continuously, while the hare stops to sleep after reaching a certain point. When the hare wakes up, the tortoise is already 4000 m in front of it. The hare tries to catch up using his original speed, but by the time the tortoise reaches the finish line, the hare is still 200 m behind. What is the distance traveled by the tortoise during the time the hare is sleeping?
The answer is 8040.
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1 solution
Let x be the tortoise's speed, and t be the time that the hare sleeps. The quantity we want to compute, namely the distance travelled by the tortoise while the hare is sleeping, is given by x t .
The tortoise takes time 1 0 0 0 0 / x to reach the finish line. During this time, the hare runs at speed 5 x for some time, then sleeps for time t and then runs again at speed 5 x . When the tortoise crosses the finish line, he is 200 m short of doing so. It follows that the hare ran 9800 m in time x 1 0 0 0 0 − t at speed 5 x . Thus 5 x ( x 1 0 0 0 0 − t ) = 9 8 0 0 or in other words x t = 1 0 0 0 0 − 5 9 8 0 0 = 1 0 0 0 0 − 1 9 6 0 = 8 0 4 0
Note that the information that when the hare woke up, the tortoise was 4000 m ahead is unnecessary for solving the problem.