# What combination of seven numbers has a mean of 45, median of 40, and a mode 50?

Oct 21, 2015

There are many possible solutions:
$\left\{10 , 20 , 30 , 40 , 50 , 50 , 115\right\}$ and $\left\{32 , 33 , 35 , 40 , 50 , 50 , 75\right\}$
are two possibilities

#### Explanation:

Arranged in ascending order we know that the middle term must be $40$ (since it is the median).

The sum of the terms must be $7 \times 45 = 315$

Two of the terms must be $50$, the mode (we can't have Three terms $= 50$ since this would require the the three smallest terms to have an average value greater than $40$.

So we have a requirement:
$\textcolor{w h i t e}{\text{XXX}} \left\{a , b , c , 40 , 50 , 50 , z\right\}$
where $a < b < c < 40 < 50 < z$
and $a + b + c + z = 315 - \left(40 + 50 + 50\right) = 175$

There are lots of solutions which meet this requirement.