This problem is in two parts. The
first part provides some building blocks which will help you to
solve the final challenge. These can be attempted in any order. Of
course, you are welcome to go straight to the Final Challenge!

Click a question from below to get started.

Question A
A cinema has $100$ seats.
The ticket prices are:

$£10$ for adults

$50$p for pensioners

What is the minimum that the cinema's takings could be, if all
$100$ tickets are sold?

What is the next smallest amount that the takings could be? And the
next? And the next...?

The cinema took $£449$ one day. Work out how many adults
and how many pensioners bought tickets, if all the seats were
filled.

Question B
A cinema has $100$ seats.
The ticket prices are:

$£10$ for adults

$10$p for children

What is the minimum that the cinema's takings could be, if all
$100$ tickets are sold?

What is the next smallest amount that the takings could be? And the
next? And the next...?

The cinema took $£208$ one day. Work out how many adults
and how many children bought tickets, if all the seats were
filled.

Question C
A cinema has $100$
seats.

The ticket prices are:

$50$p for pensioners

$10$p for children

What is the minimum that the cinema's takings could be, if all
$100$ tickets are sold?

What is the next smallest amount that the takings could be? And the
next? And the next...?

The cinema took $£20$ one day. Work out how many
pensioners and how many children bought tickets, if all the seats
were filled.

Question D
The ticket prices at a
cinema are:

$£10$ for adults

$50$p for pensioners

How many different ways can you find of selling tickets so that the
takings are exactly $£60$?

The cinema introduces a child ticket for $10$p. The cinema manager
starts to list the different combinations of adults, children and
pensioners that will earn him exactly $£60$. Part of his
table is shown below:

Without listing all the
combinations work out how many different ways there are to
earn exactly $£60$.

What is special about the total number of tickets sold when exactly
4 adult tickets are sold (and the takings are exactly
$£60$)? Explain why.

Final Challenge
A cinema has $100$
seats.

Show how it is possible to sell exactly $100$ tickets and take
exactly $£100$ if the prices are:

$£10$ for adults

$50\text{p}$ for pensioners

$10\text{p}$ for children

Is there only one solution?

This problem is based on
Cinema Problem from SIGMA 1 by David Kent and Keith
Hedger