Question

If the median and range of four numbers $\left\{\mathrm{x},\mathrm{y},2\mathrm{x}+\mathrm{y},\mathrm{x}-\mathrm{y}\right\}$, where $0<\mathrm{y}<\mathrm{x}<2\mathrm{y}$, are $10$ and $28$ respectively, then the mean of the numbers is:

Answer 1

Step 1: Find the value of $\mathrm{x}$ and $\mathrm{y}$:

Given numbers are $\left\{\mathrm{x},\mathrm{y},2\mathrm{x}+\mathrm{y},\mathrm{x}-\mathrm{y}\right\}$, the median is $10$ and range is $28$.

$$\begin{gathered} \because 0<\mathrm{y}<\mathrm{x}<2\mathrm{y} \\ \therefore \mathrm{y}>\frac{\mathrm{x}}{2},\mathrm{y}<\mathrm{x} \\ \Rightarrow \mathrm{x}-\mathrm{y}<\frac{\mathrm{x}}{2} \\ \Rightarrow \mathrm{x}-\mathrm{y}<\mathrm{y}<\mathrm{x}<2\mathrm{x}+\mathrm{y} \\ \mathrm{Median}=\frac{\mathrm{y}+\mathrm{x}}{2} \\ 10=\frac{\mathrm{y}+\mathrm{x}}{2} \\ 20=\mathrm{y}+\mathrm{x}-\left[1\right] \end{gathered}$$

The range of the numbers is

$$\begin{gathered} \mathrm{Range}=2\mathrm{x}+\mathrm{y}-(\mathrm{x}-\mathrm{y}) \\ 28=\mathrm{x}+2\mathrm{y}-\left[2\right] \end{gathered}$$

From equations $\left(1\right)$ and $\left(2\right)$ we get

$x=12,y=8$

Step 2: Compute the mean:

$$\begin{gathered} \mathrm{Mean}=\frac{\mathrm{x}+\mathrm{y}+2\mathrm{x}+\mathrm{y}+(\mathrm{x}-\mathrm{y})}{4} \\ \mathrm{Mean}=\frac{4\mathrm{x}+\mathrm{y}}{4} \\ \mathrm{Mean}=\frac{4\times 12+8}{4} \\ \mathrm{Mean}=\frac{56}{4}=14 \end{gathered}$$

Therefore mean of the number is $14$.