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

Feb 2, 2017

There are many possible solutions.

$\text{ "5," "color(red)(9," "11)," "15" }$ is just one of them

#### Explanation:

There are many sets of four numbers which will meet these requirements.

If the mean if the four numbers is 10, it means their total is $40$

$\frac{40}{4} = 10$

If the median is 10, then two of the numbers have to be less than 10 and two are greater than 10.

The two middle numbers have to be an equal distance from 10.

So we could have  " "?," "color(red)(9," "11)," "?
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} \uparrow$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 10$ is exactly in the middle

$9 + 11 = 20 \text{ }$ The other two numbers also have to add up to 20.

So we could have $\text{ "5," "color(red)(9," "11)," } 15$

or we could have $\text{ "1," "color(red)(9," "11)," } 19$

or$\text{ "4," "color(red)(8," "12)," } 16$

or$\text{ "3," "color(red)(7," "13)," } 17$

even $\text{ "5," "color(red)(5," "15)," } 15$ meets the requirements