For how many values of $x$ are the median and the mean are equal?

Question

I know how to calculate the mean and the median, but I do not know how to solve this. Could anyone help me please?

Answer 1

Hints: Mean = $5+x/5$, and median depends on the value of $x$. Possible arrangements are $x4579,4x579,45x79,457x9,4579x$. Calculate the median and check out if the mean can equal the median or not.

Answer 2

Hint:

$\mu(x)=\frac{4+9+7+5+x}{5}=\frac{25+x}{5}$.

The median must be either $5, x$, or $7$.

Answer 3

The mean of the numbers will be $$\frac{4+9+7+5+x}{5}=5+\frac{1}{5}x$$

Furthermore, if you put the first four numbers in order from least to greatest, $$4,5,7,9$$ and then observe all possible placements of $x$ in this ordering: $$x,4,5,7,9$$ $$4,x,5,7,9$$ $$4,5,x,7,9$$ $$4,5,7,x,9$$ $$4,5,7,9,x$$ You will see that the median cannot be $4$ or $9$, thus it must be $5$, $7$, or $x$. So you can set up and solve the following equations to find the value(s) of $x$: $$5+\frac{1}{5}x=5$$ $$5+\frac{1}{5}x=7$$ $$5+\frac{1}{5}x=x$$ Can you do that?