Going to the Cinema

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!


How many adults, pensioners and children were present?


You may want to pick some combinations to start off with, and then tweak them to get closer to the answer.

This spreadsheet might help.

Once you've explored this, you might like to take a look at Cinema Surprises.



Assuming that there x Adults, y Pensioners and z Children, we get two equations:
(1) x + y + z = 100, and
(2) 350x + 100y + 85z = 10000 on converting pounds to pence.

Now divide second equation by 5 to get:
70x + 20y + 17z = 2000
and multiply first equation with 20 to get:
20x + 20y + 20z = 2000

On subtracting, we get
50x - 3z = 0

As Adults, Pensioners and Children can be whole number and all audience are not pensioners. So, x or z need to be greater than zero.

Therefore x=3 and z =50 and y = 47.


So starting with the facts, $100 was paid in total, all 100 seats taken. Adults pay $3.50, Children; $0.85 and Pensioners pay $1.00.
I started off trying it so all adults, children and pensioners are the same number.
33 adult tickets would cost $115.50. Crossing that suggestion out; I tried again with 3 adults, 50 children and the 47 that were left as pensioners.
There now is 100 people and this sort of working out:
3 adults= $10.50, 50 children= $42.50, 47 pensioners= $47.00
Total $100

You've got an answer, which is good, but can you explain why you decided to try with three adults? How did you arrive at these numbers?

Well done! This is a really nice solution. Can you use a similar method to have a go at Cinema Surprises?

I've hidden the answer at the bottom of your solution so that others don't see it while they're trying to solve the problem.

Assuming A adults, P pensioners, and C children.
Then A + P + C = 100,
and 3.5A + P + 0.85C = 100.

If we multiply the first equations by -1,
then -A - P - C = -100.

Added to the second equation...

2.5A - 0.15C = 0
2.5A = 0.15C

This equality occurs when A=3, C = 50 (at 7.5).

If these solutions are used, then 3 + P + 50 = 100
and (3.5 x 3) + P + (0.85 x 50) = 100.

For both, P = 47.

Thus, there are 3 adults, 47 pensioners, and 50 children.