The average of twenty positive numbers is $x$. If each number increases by $20\%$, then $x$ will:

A fundamental property of averages states that if each observation in a data set is multiplied or changed by a certain percentage factor, the average also changes by that exact same factor.Let the numbers be $a_1,a_2,\dots ,a_{20}$.The initial average is:
$$x=\frac{a_1+a_2+\cdots +a_{20}}{20}$$
If each number increases by $20\%$, each new number becomes $1.2a_i$.The new average becomes:
$$\text{New Average}=\frac{1.2a_1+1.2a_2+\cdots +1.2a_{20}}{20}$$
$$\text{New Average}=1.2\times \left(\frac{a_1+a_2+\cdots +a_{20}}{20}\right)=1.2x$$
An average of $1.2x$ represents a $20\%$ increase over the original average $x$.