Cinema Problem
Cinema Problem printable sheet
Alison's cinema has 100 seats.
One day, Alison notices that her cinema is full, and she has taken exactly £100.
The prices were:
Adults £3.50
Pensioners £1.00
Children £0.85
She knows that not everyone in the audience was a pensioner.
How many adults, pensioners and children were present?
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
Here are some questions you might like to consider:
- How many solutions are there for each set of prices?
- If I can find one solution, can I use it to help me find all the other solutions?
- Can you find alternative sets of prices that offer many solutions?
- What about exactly one solution?
- 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?
This problem is based on Cinema Problem from SIGMA 1 by David Kent and Keith Hedger
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...