Mike can paint a certain room in \(x\) hours and Michelle can paint the same room in half the time. How much time, in hours, would it take for Mike and Michelle to paint the room together?

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We can write a simple expression for the rates.

$$ \text{Mike's Rate}= \frac{1 \text{ room}}{x \text{ hours}} $$ $$ \text{Michelle's Rate}= \frac{1 \text{ room}}{\frac{x}{2}\text{ hours}} \tag*{ \scriptsize Michelle takes half the time} $$

We can find their combined rate by adding:

$$ \frac{1 \text{ room}}{x \text{ hours}} + \frac{1 \text{ room}}{\frac{x}{2}\text{ hours}} $$ $$ = \frac{1}{x}+\frac{2}{x} $$ $$ = \frac{3}{x} $$

Since we added two terms with the same units, we can interpret the result with the same units:

$$ =\frac{3 \text{ rooms}}{x \text{ hours}} $$

Which is the same as:

$$ =\frac{1 \text{ room}}{\frac{x}{3} \text{ hours}} $$