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### Number and algebra

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### Advanced mathematics

# Going to the Cinema

*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:

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:

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?

Or search by topic

Age 11 to 16

Alison's cinema has

One day, Alison notices that her cinema is full,

and she has taken exactly

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!

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.

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.

Send us your thoughts; we'll be publishing a selection.