Answer
The required solution is
$34\frac{2}{7}$.
Work Step by Step
We know that the average rate on a round-trip commute having a one-way distance $d$ is given by the complex rational expression,
$\frac{2d}{\frac{d}{{{r}_{1}}}+\frac{d}{{{r}_{2}}}}$.
We know that a complex rational expression or a complex fraction is an algebraic rational expression in which either the numerator contains a rational expression or the denominator contains a rational expression or both the numerator and denominator contain a rational expression.
Now simplify the denominator of the given complex rational expression:
$\begin{align}
& \frac{d}{{{r}_{1}}}+\frac{d}{{{r}_{2}}}=\frac{d}{{{r}_{1}}}\times \frac{{{r}_{2}}}{{{r}_{2}}}+\frac{d}{{{r}_{2}}}\times \frac{{{r}_{1}}}{{{r}_{1}}} \\
& =\frac{d{{r}_{2}}}{{{r}_{1}}{{r}_{2}}}+\frac{d{{r}_{1}}}{{{r}_{2}}{{r}_{1}}} \\
& =\frac{d{{r}_{2}}+d{{r}_{1}}}{{{r}_{1}}{{r}_{2}}} \\
& =\frac{d\left( {{r}_{1}}+{{r}_{2}} \right)}{{{r}_{1}}{{r}_{2}}}
\end{align}$.
Also, simplifying the given complex rational expression:
$\begin{align}
& \frac{2d}{\frac{d}{{{r}_{1}}}+\frac{d}{{{r}_{2}}}}=\frac{2d}{\frac{d\left( {{r}_{1}}+{{r}_{2}} \right)}{{{r}_{1}}{{r}_{2}}}} \\
& =\frac{2d\times {{r}_{1}}{{r}_{2}}}{d\left( {{r}_{1}}+{{r}_{2}} \right)} \\
& =\frac{2{{r}_{1}}{{r}_{2}}}{\left( {{r}_{1}}+{{r}_{2}} \right)}
\end{align}$.
Put, ${{r}_{1}}=40$ and ${{r}_{2}}=30$. The above simplified expression becomes
$\begin{align}
& \frac{2{{r}_{1}}{{r}_{2}}}{\left( {{r}_{1}}+{{r}_{2}} \right)}=\frac{2\times 40\times 30}{\left( 40+30 \right)} \\
& =\frac{2400}{70} \\
& =34\frac{2}{7}
\end{align}$.
Thus, the average rate is $34\frac{2}{7}$ miles per hour.
