Question

Mia can clean her apartment in 6 hours while her roommate can clean the apartment in 5 hours. If they work together, how long would it take them to clean the apartment?

Answer 1

To find out how long it would take Mia and her roommate to clean the apartment together, we can calculate their rates of work and then combine them.

1. **Mia's work rate**: Mia can clean the apartment in 6 hours, so her rate is:
\[
\text{Rate of Mia} = \frac{1 \text{ apartment}}{6 \text{ hours}} = \frac{1}{6} \text{ apartments per hour}
\]

2. **Roommate's work rate**: Her roommate can clean the apartment in 5 hours, so her rate is:
\[
\text{Rate of Roommate} = \frac{1 \text{ apartment}}{5 \text{ hours}} = \frac{1}{5} \text{ apartments per hour}
\]

3. **Combined work rate**: When they work together, their rates add up:
\[
\text{Combined Rate} = \frac{1}{6} + \frac{1}{5}
\]

To add these fractions, we need a common denominator. The least common multiple of 6 and 5 is 30.

4. **Convert to a common denominator**:
\[
\frac{1}{6} = \frac{5}{30}
\]
\[
\frac{1}{5} = \frac{6}{30}
\]

Now, we can add the two rates:
\[
\text{Combined Rate} = \frac{5}{30} + \frac{6}{30} = \frac{11}{30} \text{ apartments per hour}
\]

5. **Find the time taken to clean one apartment together**: To find the time \( T \) (in hours) to clean one apartment together, we take the reciprocal of their combined rate:
\[
T = \frac{1 \text{ apartment}}{\frac{11}{30} \text{ apartments per hour}} = \frac{30}{11} \text{ hours}
\]

6. **Convert this into hours and minutes**:
\[
\frac{30}{11} \approx 2.727 \text{ hours}
\]

To convert the decimal into minutes, we multiply the decimal part by 60:
\[
0.727 \times 60 \approx 43.64 \text{ minutes}
\]

Thus, approximately, it takes Mia and her roommate **2 hours and 44 minutes** to clean the apartment together.

So, the final answer is:
\[
\frac{30}{11} \text{ hours} \text{ or approximately } 2 \text{ hours and } 44 \text{ minutes}.
\]