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