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.

## Comments

### Mathematics

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.

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### Going to the Cinemas

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

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### Re: Going to the Cinemas

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?

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### Re: Mathematics

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.

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### Going to the Cinemas

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.

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