The average (mean) of $\frac{1}{5}$ and $\frac{1}{10}$ is $\frac{1}{x}$. The value of $x$ is
(A) $\frac{20}{3}$
(B) $\frac{3}{20}$
(C) 30
(D) $\frac{10}{3}$
(E) $\frac{2}{15}$
Solution 1

The average of two numbers is half-way between the two numbers.

So what number is half-way between $\frac{1}{5}$ and $\frac{1}{10}$ ?

We can write $\frac{1}{5}=\frac{4}{20}$ and $\frac{1}{10}=\frac{2}{20}$, so the number half-way in between is $\frac{3}{20}$ or $\frac{1}{\left(\frac{20}{3}\right)}$

Therefore, $x=\frac{20}{3}$.

## Solution 2

The average of two numbers $X$ and $Y$ is $\frac{1}{2}(X+Y)$, so

$$
\begin{aligned}
\frac{\frac{1}{5}+\frac{1}{10}}{2} & =\frac{1}{x} \\
\frac{\frac{3}{10}}{2} & =\frac{1}{x} \\
x & =\frac{2}{\left(\frac{3}{10}\right)} \\
x & =\frac{20}{3}
\end{aligned}
$$

ANSWER: (A)