Help Solving Fraction Math Question

Question

Im stumped by this question on a practice ACT math test:

If $\frac 1x + \frac 1y = \frac 1z$ then $z =$?

The correct answer is $\frac {xy}{x + y}$.

How do you arrive at this answer? I don't know how to even begin with this problem.

Answer 1

Multiply both sides of the expression $\frac {1}{x} + \frac {1}{y} = \frac {1}{z}$ by the number $xyz$.

So you get $\frac {xyz}{x} + \frac {xyz}{y} = \frac {xyz}{z}$, which is, after cancelling, equal to $yz+xz=xy$.

Now, $yz+xz=z(x+y)=xy$.

Divide $z(x+y)=xy$ with $(x+y)$ to obtain $z= \frac {xy}{x+y}$.

Answer 2

Adding up the two fractions in the usual way leads to $$\frac1x+\frac1y=\frac{y}{xy}+\frac{x}{xy}=\frac{x+y}{xy}.$$ Now solving $\tfrac{x+y}{xy}=\tfrac1z$ gives us $$z=\frac{xy}{x+y}.$$

Answer 3

1. Multiply the equation by $xyz$
2. Move all values with $z$ on one side of the equation.
3. Factor out $z$. You get an expression like $A\cdot z = B$ where $A$ and $B$ are some expression.
4. Divide the equation by $A$.