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Fair Shares?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Last weekend Mrs Jenkins won £25 and she gave her winnings to her five children.
She gave her first child £1 plus $1 \over 6$ of the money remaining.
She gave her second child £2 plus $1 \over 6$ of the money remaining.
She gave her third child £3 plus $1 \over 6$ of the money remaining, and so on...

Without doing any calculations, which child do you think ended up with the most money?

Work out how much each child received. Are you surprised?  

Mrs Hobson also had some money to share with her family.
She gave her first child £1 plus $\frac{1}{5}$ of the money remaining.
She gave her second child £2 plus $\frac{1}{5}$ of the money remaining.
She gave her third child £3 plus $\frac{1}{5}$ of the money remaining, and so on...

How much money did she have to share out if all the children received the same amount?
How many children were there in the family?

In a family with $8$ children, the mother wants to give each child a lump sum plus a fraction of the remainder, in the same way that Mrs Jenkins and Mrs Hobson did. How much money will she share out, and what fraction will she use each time, in order to share the money equally?

Can money be shared out in this way for any number of children?
Explain how this could be done.
By considering what will happen when there are $n$ children, can you show that this will always work?