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:

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