Average of ten positive numbers is $\bar{x}$. If each number is increased by $10\%$, then $\bar{x}$

Let the ten positive numbers be $a_1,a_2,\dots ,a_{10}$. Their initial average is:
$$\bar{x}=\frac{a_1+a_2+\cdots +a_{10}}{10}$$

If each number is increased by $10\%$, the new values become $1.1a_1,1.1a_2,\dots ,1.1a_{10}$.
The new average ${\bar{x}}^{\prime }$ will be:

$${\bar{x}}^{\prime }=\frac{1.1a_1+1.1a_2+\cdots +1.1a_{10}}{10}=1.1\left(\frac{a_1+a_2+\cdots +a_{10}}{10}\right)=1.1\bar{x}$$

An increase to $1.1\bar{x}$ represents exactly a $10\%$ increase in the average value.