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Going to the Cinema

Stage: 3 and 4

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.