# The Hare and the Tortoise



## tonyt (May 25, 2005)

I vaguely remember this conundrum from my childhood but can't remember how to explain that it can't happen.

A Hare and a Tortoise decide to have a race.

As the hare can run ten times faster than the tortoise he gives the tortoise a head start.

Off the tortoise goes and when he reaches a point 100 miles/kilometres or whatever, from the start the hare begins to run. 

As he can run ten times faster than the tortoise he arrives at the “100” point in one tenth of the time the tortoise took to get there. 

The tortoise however didn’t stop when he got there but continued to race on. 

By the time the hare has reached point “100” the tortoise has reached point “110”.

The hare continues to race and reaches point 110 but of course, the tortoise has moved on to point “111” and so it goes on……

Does the hare ever catch the tortoise?


Any maths teachers out there?


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## Wizzo (Dec 3, 2007)

tonyt said:


> The hare continues to race and reaches point 110 but of course, the tortoise has moved on to point "111" and so it goes on……


Well no it doesn't because by this time the tortoise has already been overtaken! Unless of course the hare has stopped for his obligatory nap which is why in the fable he loses the race.

JohnW


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## tonyt (May 25, 2005)

Wizzo said:


> Well no it doesn't because by this time the tortoise has already been overtaken.................
> JohnW


Yep - I'd worked that bit out - I just can't get my brain to explain how to work out at what point the hare overtook the tortoise.


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## safariboy (May 1, 2005)

You have discovered the joys of the infinite series with a finite sum. There are plenty of examples.

I cannot write it in this type of space but try any A level maths book.


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## Kelcat (Apr 20, 2008)

I feel this could best be explained with a graph...

....unfortunatly as I'm spending this evening in the company of my oldest friend (Irish with a smooth dark body a creamy white top) - I am unable to help you further :lol: :lol: :lol:


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## Wizzo (Dec 3, 2007)

OK Tony, here goes......

Assuming the Tortoise covers 10 miles a day and therefore the Hare will cover 100 miles a day, and for simplicity we will say that the day is 100 minutes long, then:

The Tortoise covers 1 mile in 10 minutes; the Hare covers 10 miles in 10 minutes.

At the end of 10 days the Tortoise has covered 100 miles.

Day 11 the Tortoise covers another 10 miles and the Hare covers his first 100.

Day 12. After ten minutes the Tortoise has covered another mile so now he has done 111 miles. Meanwhile the Hare has covered 10miles in that 10 minutes so he has covered 110 miles. After 11 minutes the Tortoise has covered 111.1 miles and the Hare has done 111 miles. During the following minute the Hare will overtake the Tortoise - the Tortoise will have only covered 111.2 miles at the end of the 12th minute whereas the Hare will have covered 112 miles.

I hope that has explained it simply enough.

JohnW


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