Apr 12, 2018

$f = \frac{a b}{a + b}$

#### Explanation:

When we say "solve for $f$", we mean you should isolate $f$ on one side of the equation, so you have something of the form $f = \ldots$.

We wish to solve $\frac{1}{f} = \frac{1}{a} + \frac{1}{b}$ for $f$. For reasons that will become clear, we need to make the right-hand side (RHS) of the equation a single fraction. We do this by finding a common denominator.

$\frac{1}{a} + \frac{1}{b}$
$= \frac{b}{a b} + \frac{a}{a b}$
$= \frac{a + b}{a b}$

So we have $\frac{1}{f} = \frac{a + b}{a b}$. Multiply both sides by $f$ to give $1 = f \left(\frac{a + b}{a b}\right)$. Now multiply both sides by $a b$ to give $a b = f \left(a + b\right)$. Finally, divide both sides by $a + b$ to give $\frac{a b}{a + b} = f$.

Thus, our final answer is $f = \frac{a b}{a + b}$.