changing averages
Find the value of $m$ from these statements about a group of numbers
Age
14 to 16
Challenge level
A list of integers has a mode of $32$, mean $22$ and median $m$. $m$ is one of the numbers on the list.
The smallest number on the list is $10$.
If $m$ were replaced with $m + 10$, the mean would be $24$.
If $m$ were instead replaced with $m-8$, the median would be $m - 4$.
What is $m$?
This problem is adapted from the World Mathematics Championships
Using each piece of information, we can construct the list, with the numbers ordered from smallest to largest.
'The smallest number in the list is $10$' allows us to begin constructing the list: $$ 10, \underline{ }, \underline{ }, ... $$
'The median is $m$. $m$ is one of the numbers on the list', means $m$ must be the middle number on the list.
The 'list has a mode of $32$', so $32$ must appear at least twice on the list. We can't be sure, but it is likely that $32$ is greater than $m$: $$10, ..., m, ..., 32, 32, ...$$
The mean is $22$, but 'If $m$ were replaced with $m + 10$, the mean of the new list would be $24$.' So increasing one of the numbers by $10$ increases the mean by $2$, which is $10\div5$. So there must be $5$ numbers on the list: $$10, \underline{ }, m, 32, 32$$
'If $m$ were instead replaced with $m - 8$, the median of the new list would be $m - 4$.' So $m-4$ must also be a number on the list: $$10, m-4, m, 32, 32$$
The mean is $22$, so we can set up an equation to find $m$:
$$\begin{align} \frac{10+(m-4)+m+32+32}5&=22\\
\Rightarrow \frac{70+2m}{5}&=22\\
\Rightarrow \frac{35}{5}+\frac{m}{5}&=11\\
\Rightarrow 7+\frac{m}5&=11\\
\Rightarrow \frac{m}5&=11-7\\
\Rightarrow m&=4\times5\\
\Rightarrow m&=20\end{align}$$