Answer
Average rate $\approx34.2857$ miles per hour
The answer is not $35$ miles per hour because this would be the result of finding the arithmetic mean and the expression obtained does not represent this kind of mean or average.
The expression actually represents the harmonic mean.
Work Step by Step
$\dfrac{2d}{\dfrac{d}{r_{1}}+\dfrac{d}{r_{2}}}$
Simplify the expression. Start by evaluating the sum of fractions in the denominator:
$\dfrac{2d}{\dfrac{d}{r_{1}}+\dfrac{d}{r_{2}}}=\dfrac{2d}{\dfrac{dr_{2}+dr_{1}}{r_{1}r_{2}}}=...$
Evaluate the division:
$...=\dfrac{2dr_{1}r_{2}}{dr_{2}+dr_{1}}=...$
Take out common factor $d$ from the denominator and simplify:
$...=\dfrac{2dr_{1}r_{2}}{d(r_{2}+r_{1})}=\dfrac{2r_{1}r_{2}}{r_{1}+r_{2}}$
The average rates of the outgoing and return trips are $40$ and $30$ miles per hour, respectively. Substitute $r_{1}$ by $40$ and $r_{2}$ by $30$ in the expression obtained and evaluate:
Average rate $=\dfrac{2(40)(30)}{40+30}\approx34.2857$ miles per hour
The answer is not $35$ miles per hour because this would be the result of finding the arithmetic mean and the expression obtained does not represent this kind of mean or average.
