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## 'Buying a Balloon' printed from http://nrich.maths.org/

Mrs Fother's class sent us in their answers
to this problem:

First we made a list of all the possible coins Lolla might have
used to pay the clown. These were 1p, 2p, 5p, 10p, 20p, 50p,
£1 and £2.

If Lolla paid using only 1p pieces, she would have paid 6p, which
we thought was too little for a balloon.

If she paid using only £2 coins, she would have paid
£12, which we thought was far too much for a balloon.

Then we looked at how many different prices Lola could have paid
using exactly two different types of coin.

With the 1p and the 2p she could have paid 7p, 8p, 9p, 10p or 11p.

With the 1p and the 5p she could have paid 10p, 14p, 18p, 22p or
26p.

Here we thought we saw a pattern. We started off with five 1p coins
and one of the other type of coin, and then to get the next largest
amount we took away one 1p coin and added another of the other type
of coin. So, as we did above, to go from 10p (five 1p coins and one
5p coin) to 14p (four 1p coins and two 5p coins) we took away 1p
and added 5p. This is the same as adding 4p.

With the 1p and the 10p the smallest amount she could have paid was
five 1p coins and one 10p coin, which makes 15p. 10p - 1p = 9p so
we need to add 9p to 15p to get the next smallest amount of money -
this is 24p, which is four 1p coins and two 10p coins.

Having found this pattern, we then split into groups to look at how
many different prices Lola could have paid using exactly three
different types of coin.

Here are some of our results:

With the 1p, the 2p and the 5p, she could have paid 11p, 12p, 13p,
14p, 15p, 16p, 17p, 19p, 20p or 23p.

With the 5p, the 10p and the 50p, she could have paid 80p, 85p,
90p, 95p, £1.25, £1.30, £1.35,
£1.70, £1.75 or £2.15.

Thank you very much, Mrs Fother's
class!