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Five Coins

Age 5 to 11
Challenge Level

We had quite a variety of kinds of solutions to this problem with some sending in one solution, some sending in a variety and some considering all the possibilities.

Jonathan sent in the following idea;

The answer could be for example £2.03.
Ben could have two pound coins and three one penny coins total would be £2.03.
Or he could have £1.37 - he would have 1 x 2p coin, 1 x 5p coin, 1 x 10p coin, 1 x £1 coin and 1 x 20p coin making a total of £1.37.

Another solution was sent in by Lily;

I thought of a two pound and tried to make £2.01. So I times 50p by 4 which equaled £2.00.Then that meant I had one coin left which had to be 1p.So that only left adding them together. £2.00+1p= £2.01

Rachel sorted out what the range could be;

The most Ben can have in his pocket is £10 because he could have five £2 coins.
The least he could have is 5p i ­f he had five 1p coins in his pocket.

It is always good to hear how, when provided with a simple starting point, children can take mathematics beyond our expectations. Here is an account of such an event from a teacher ...

"Our school, Barford St Peters Primary in Warwickshire, has an after-school Maths Club for juniors where we often look at NRICH problems. I explained this problem to them, and most of them decided to work on it from a slightly different angle; they looked at the different possible total amount, from 1p up to 100p, and worked out if this was possible with five coins, and if so, which five coins would do it. Dean, Liam and Andrew worked up to about 40p in the time available, and decided that only 1p, 2p, 3p and 4p were impossible with five coins.

Then, in his own time, Andrew decided that he wanted to consider, for a given number of coins, what was the smallest total which required that number of coins. His first answer was that 38p was the smallest amount which required fivecoins, 88p the smallest to need six coins. He worked out that £5.88 was the smallest to need ninecoins ( £2, £2, £1, 50p, 20p, 10p, 5p, 2p, 1p), but that £7.88 needed ten coins (another £2 plus the ones before). He reckoned £17.88 needed fifteen coins, £27.88 needed twenty coins, £37.88 needed twenty five coins, £187.88 needed one hundred coins, and then went off into the realms of trillions of coins and googols of coins!"