Cakes and Buns
Helen buys some cakes and some buns for her party. Can you work out how many of each she buys?
Age
11 to 14
Challenge level
Problem
The Pythagoras Patisserie sells triangular cakes at 39p each and square buns at 23p each.
Helen spent exactly £5.12 on an assortment of these cakes and buns.
How many cakes and how many buns did she buy?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 9 cakes and 7 buns
Spotting patterns in the numbers
One cake + one bun = 23p + 39p = 62p
512 $\div$ 62 = 8 remainder 16, so buying 8 cakes and 8 buns costs 16p less
16 = 39 - 23 is the difference between them
So buy one more bun and one less cake
7 cakes and 9 buns
Using algebra
$23a$ $+$ $39b$ $=$ $512$
Those numbers are not very easy to work with so try writing it differently...
$23(a+b)$ $+\underbrace{16b}_{\text{multiple
of 16}}=\underbrace{512}_{\text{multiple
of 16}}$
So $(a+b)$ must be a multiple of $16$ as well.
Try $a+b=16$
$23\times16$ $+$ $16b$ $=$ $512$
$ 368$ $ +$ $ 16b$ $ =$ $ 512$
$16b = 512 - 386 = 144$
$144=9\times16$ so $b=9$ and so $a=7$