Answer
$\displaystyle \frac{15}{4}$
Work Step by Step
The harmonic mean of $a$ and $b$ is:
$\displaystyle \frac{1}{\frac{\frac{1}{a}+\frac{1}{b}}{2}}$
We plug in $a=3$ and $b=5$:
$\displaystyle \frac{1}{\frac{\frac{1}{3}+\frac{1}{5}}{2}}$
$\displaystyle \frac{1}{\frac{\frac{5}{15}+\frac{3}{15}}{2}}$
$\displaystyle \frac{1}{\frac{\frac{8}{15}}{2}}$
$\displaystyle \frac{1}{\frac{8}{15*2}}$
$\displaystyle \frac{1}{\frac{8}{30}}$
$\displaystyle \frac{1}{\frac{4}{15}}$
$\displaystyle \frac{15}{4}$
