Question

Four numbers have a mean and median of $10$, none of the numbers are $10$. What are the four numbers?

Answer 1

Hint: We will first use the given conditions that the median and mean of the four numbers is $10$, keeping in mind the other conditions. Then we will find some other facts and conditions about the four numbers like their sum and the property of the median numbers among the four numbers. Then, from the conditions we can deduce the four numbers. So, let us see how to solve the problem.

Complete step by step answer:
There are many sets of four numbers which will meet these requirements.
Given, the mean and median of the four numbers is $10$.
If the mean of the four numbers is $10$, it means their total is $40$, as.
$Mean = \dfrac{{{\text{Sum of numbers}}}}{{{\text{Total number of observations}}}}$
$ \Rightarrow 10 = \dfrac{{Sum}}{4}$
Now, multiplying both sides by $4$, we get,
$ \Rightarrow Sum = 40$
If the median is $10$, then two of the numbers have to be less than $10$ and two are greater than $10$.
Moreover, the two middle numbers have to be an equal distance from $10$.
That is, the average of the two numbers has to be $10$.
But there can be many sets of such four numbers that satisfy the given conditions.
So we could have, $?,9,11,?$, as,
$\dfrac{{9 + 11}}{2} = \dfrac{{20}}{2} = 10$
The other two numbers also have to add up to $20$, as the sum of the numbers is $40$.
So we could have, $5,9,11,15$
or we could have, $1,9,11,19$
or $4,8,12,16$
or, $3,7,13,17$
or, $5,5,15,15$
As all these numbers meet the requirements that the sum of the numbers have to be $40$ and the average of middle two numbers have to be $10$.

Note:
From the above problem we are able to conclude that there can be many numbers with similar statistical conditions. If in this question, there would be one condition that one of the numbers is $10$, then we could have concluded that the four numbers are, $10,10,10,10$.