Going to the Cinema
The 100 seats 100 pounds problem

Alison's cinema has 100 seats. One day, Alison notices that her cinema is full, and she has taken exactly £100.

The prices were as follows:

Adults £3.50
Pensioners £1.00
Children £0.85

She knows that not everyone in the audience was a pensioner!

Is it possible to work out how many adults, pensioners and children were present?

You may want to start by trying different ways of filling all 100 seats. e.g. 5 adults, 20 pensioners and 75 children

Does this earn you £100?

Too much? Too little?

Can you tweak the numbers to get closer to £100?

You may find this spreadsheet useful

Slightly changing the problem

What other interesting mathematical questions can you think of to explore next?

We have thought of some possibilities of changing slightly the question

Is there only one possible combination of adults, pensioners and children that add to 100 with takings of exactly £100?

Can there be 100 people and takings of exactly £100 if the prices are:

Adults £4.00
Pensioners £1.00
Children £0.50
OR
Adults £5.00
Pensioners £2.50
Children £0.50

Can you find alternative sets of prices that also offer many solutions? What about exactly one solution? If I can find one solution, can I use it to help me find other solutions?

If a children's film has an audience of 3 children for every adult (no pensioners), how could the prices be set to take exactly £100 when all the seats are sold?

What about a family film where adults, children and pensioners come along in the ratio 2:2:1?

Combinging Coins
Combining Coins

Charlie has 3 coins in his pocket that add up to 32p: 20p 10p 2p. Alison has 7 coins in her pocket that also add up to 32p: 10p, 5p, 5p, 5p, 5p, 1p, 1p.

Matt has 32 coins in his pocket that also add up to 32p. I'm not going to list them, you should be able to work out what he has!

Is there a way of making 32p that uses 4 coins? or 5 coins? or ANY number of coins between 3 and 32?

What about other totals? Is it always possible to make a total using every number of coins between the minimum and the maximum?

"I'm thinking of a value that I can make in 5 different ways using just 1p and 2p coins."

"Well it could be 8p, because I can make that using: 4x2 3x2, 2x1 2x2, 4x1 1x2, 6x1 0x2, 8x1"

"Oh yes, that does work, doesn't it! I was actually thinking of 9p though..."

How many ways are there of making different amounts using just 1p and 2p coins?

Which totals have the same number of ways?

Is there a quick way of working out the number of ways of making an amount with just 1p and 2p coins?

What if we just had 1p and 3p coins instead?

Or just 1p and 4p coins?

Emmy has been working out different ways to make 12p using 1p, 2p, 5p and 10p coins.

So far, this is what she has done: With a 10p I need 2p more so there's 2 ways of doing that. With two 5ps, I also need 2p more so there's 2 ways of doing that...

10p 5p 2p 1p Number of ways
1 of these none of these     2
none of these 2 of these     2
none of these 1 of these      

Can you help Emmy to finish off? You should find there are 15 ways altogether!

Can you describe a method for working out how many ways there are to make any amound using 10p, 5p, 2p and 1p coins?

What if you also had 20p, 50p coins?

It might get very time-consuming even if you have a good method!

Could you write a computer program to help?