Why do this problem?

This problem requires flexibility of thought and can be solved in many different ways. Once students have solved the initial problem, there are alternative pricings so they can adapt their solution method to other situations. For each part of the problem, the question "How many solutions are there? How do you know you have found them all" can be considered.

Possible approach

Display the initial problem to fill the cinema with 100 people for £100 with the prices:
Adults £3.50
Pensioners £1.00
Children £0.85
The problem is available on slides: CinemaProblem

Give students some time to try to come up with a solution in pairs, and then encourage them to work in small groups to share strategies.

If strategies are not forthcoming, the following questions might help:

What's the maximum number of adults I could include?
What must be true about the number of children?
If I swap an adult for a child, how does the total change?
If I swap an adult for a pensioner, how does the total change?
If I swap a pensioner for a child, how does the total change?

Once students have had a chance to tackle the original problem, the second part of the task invites them to consider varying the prices:

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

Adults £4.00
Pensioners £1.00
Children £0.50

What if the prices are:

Adults £5.00
Pensioners £2.50
Children £0.50

Students could create a poster or presentation showing all the possible solutions and how they know they have found them all.

Possible support

Students could start by exploring the different possible totals if the cinema contains just adults and pensioners, or just pensioners and children.

Possible extension

Invite students to come up with their own pricing schemes where there is exactly one solution, exactly two solutions, exactly three solutions...