Counting kangaroos
Can you work out how many kangaroos Rachel draws in one colour?
Age
14 to 16
Challenge level
Problem
Rachel draws 36 kangaroos using three different colours.
25 of the kangaroos are drawn using some grey, 28 are drawn using some pink and 20 are drawn using some brown.
Five of the kangaroos are drawn using all three colours.
How many kangaroos did she draw that use only one colour?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
The Venn diagrams below show the three colours, and the possible combinations of colours. In the version on the right, the kangaroos drawn using one colour are in the red parts of the diagram, the kangaroos drawn using two colours are in the yellow parts of the diagram, and the $5$ kangaroos drawn using three colours are in the blue part of the diagram.
Image
Adding up the kangaroos in each circle (or using each colour) will count the kangaroos in the red area once, the kangaroos in the yellow area twice (since they are counted for each of their two colours) and the kangaroos in the blue area three times (since they are counted for each of their three colours).
So, if there are $r$ kangaroos in the red area and $y$ kangaroos in the yellow area, $$\begin{align}25 + 28 + 20 &= r + 2\times y + 3\times5\\
\Rightarrow 15+10+28+20&=r+2 y+15\\
\Rightarrow58&=r+2y\end{align}$$
There are $36$ kangaroos in total, so $r+y+5=36\Rightarrow r+y=31$.
The difference between $r+y$ and $r+2y$ is $y$, so the difference between $58$ and $31$ is $y$, so $y=27$.
Since $r+y=31$, if $y=27$, then $r$ must be $4$.
So there are $4$ kangaroos drawn using only one colour.