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Going to the Cinema

Age 11 to 16

This resource is part of "Freedom and Constraints"

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.

What other interesting mathematical questions can you think of to explore next?
We have thought of some possibilities:

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   Adults £5.00
Pensioners £1.00            or      Pensioners £2.50
Children £0.50   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?

We'd love you to share the questions you've come up with. Tell us also how you got started and any conclusions you have arrived at. 
Send us your thoughts; we'll be publishing a selection.