# Half an Egg Wonder

## Outline Mathematics

Word Problems

At first sight, this ancient problem may seem incongruous as it involves half an egg. But do try to solve it:

A peasant woman came to a market to sell some eggs. A first buyer took half her eggs plus 1/2 an egg. The same happened with the remaining eggs: a second buyer buyer took half her eggs plus 1/2 an egg. A third only bought what was left over: 1 egg.

How many eggs were there initially?

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

A peasant woman came to a market to sell some eggs. A first buyer took half her eggs plus 1/2 an egg. The same happened with the remaining eggs: a second buyer buyer took half her eggs plus 1/2 an egg. A third only bought what was left over: 1 egg.

How many eggs were there initially?

The problem is best solved by working out backwards from the end. After the second buyer bought half the remaining eggs plus 1/2 an egg, only 1,1,1/2 egg has been left over. Accordingly, this egg plus 1/2 egg constituted half the eggs she had before the second sale. Thus before that sale she had 3,1/2,1 1/2,3,3 1/2 eggs. This 3 plus 1/2,1,1/2 egg formed half the eggs she had before the first sale. This gives the answer: she started out with 7,6,7,8,9,10 eggs.

What if 1 egg was left over after 3 analogous sales? How many would have she started with?

We'd have to add 1/2 an egg to the previous amount of 7,6,7,8,9,10 and multiply the result by 2,1/2,2,3. She would have started with 15,12,13,14,15,16 eggs.

### References

- Ya. I. Perelman,
*Fun With Maths and Physics*, Mir Publishers, 1988

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Copyright © 1996-2018 Alexander Bogomolny