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# Sweet Shop

##### Age 11 to 14 Challenge Level:

We received many correct solutions to this problem, especially from students at Moorfield Junior School. Thank you all.

Tiffany Lau from Island School showed clearly how she arrived at her solution:

Liam bought $3$ lollipops for $84$p, so each lollipop costs $28$p.

Kenny bought a choco, egg and lollipop for $54$p, so choco + egg = $54$p - $28$p = $26$p.

Now a choco + egg + chew = $61$p, so a chew costs $61$p - $26$p = $35$p.

Mandy bought an egg, lollipop and chew for $80$p, so an egg costs $80p - 35p - 28p$ which equals $17$p.

Since the choco + egg = $26$p and an egg = $17$p, a choco bar = $9$p.

So lollipop = $28$p, chew = $35$p, egg = $17$p, choco = $9$p.
Nathan bought one of each, which costs $89$p altogether, so he had $11$p change.

1 way in which Nathan could spend all his money (£$1$) is:
$2$ lollipops, $1$ chew and $1$ choco bar ($56$p + $35$p + $9$p).

Jessica from Kowloon Junior School, Hong Kong used a slightly different method:

First, you find out that three lollypops cost $84$p, so you divide $84$ by $3$ to find the price of one lollypop, which is $28$p.

Next, you go to Kenny. He had $46$p change from his 1 pound, so you take $46$p away from $1$ pound, which means that he spent $54$p.

Next, you go to Judy. She spent her money buying the same things as Kenny - the choco bar and the mini egg - except for the lollypop - she bought a chew instead. Judy spent $61$p.

Next, you take the amount Kenny spent from the amount Judy spent - $7$p. This is the difference between the price of Kenny's lollypop and Judy's chew. The lollypop costs $28$p, so you add $7$p to it to find out the price of the chew - $35$p.

After that, we go to Mandy. She bought a mini egg, a lollypop, and a chew. You add the prices of the lollypop and the chew together, which equals $63$p. You then take $63$p away from one pound, which equals $37$p. Mandy had $20$p change from her sweets, so you take away $20$p away from $37$p, which equals $17$p - the price of the mini egg!

To find the price of the choco bar, we have to go back to Kenny. He spent $45$p on the mini egg and the lollypop. We then take away $45$p from one pound, which is $55$p - the money he had left. He had $46$p change, which you take away from $55$p - $9$p, the price of the choco bar!

So now you know that: Choco bars equal $9$p each, chews equal $35$p each, mini eggs equal $17$p each, and lollypops equal $28$p each!

Add these together and subtract it from one pound to find out the change which Nathan gets. The last question uses a bit of trial and error- my answer was $8$ choco bars and $1$ lollypop.

Jimmy Ye from Sir John A. Macdonald C.I. School used algebraic notation to arrive at the solution and suggested that there are two other ways Nathan could spend all his money:

Let $b$ represent the choco bar, $c$ the chews, $m$ the mini eggs and $l$ the lollypops.

A set of equations can be written from the known:
$b + m + c = 61$
$b + m + l = 54$
$3l = 84$
$m + l + c = 80$

we know $l = 28$ easily

substitute $l = 28$ into $m + l + c = 80$ and we get $m + c = 52$

substitute $m + c = 52$ into $b + m + c = 61$ and we get $b = 9$

substitute $l = 28$ and $b = 9$ into $b + m + l = 54$ and we get $m = 17$

substitute $m = 17$ and $l = 28$ in to $m + l + c = 80$, and we get $c = 35$

In summary:
$b = 9$
$m = 17$
$l = 28$
$c = 35$

When Nathan bought one of each of the $4$ products, he had to spend $89$ pence, so he got $11$ pence change.

There are $3$ ways Nathan could spend all his money:
he can buy $3$ choco bars, $1$ mini egg and $2$ lollypops,
or $1$ choco bar, $1$ chew and $1$ lollypops,
or $8$ choco bars, and $1$ lollypop.