You may also like

problem icon

Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

problem icon

Tea Cups

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

problem icon

Counting on Letters

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

Cinema Problem

Age 11 to 14 Challenge Level:

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 extension

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

Possible support

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