Questions: Jane and Bill are cleaning up a park. If she had to clean up the park herself, it would take Jane 6 hours to complete the job. If Bill alone were to clean the park, it would take him 3 hours. Working together, how long will it take Jane and Bill to clean the park? a.) 3.5 hours b.) 2 hours c.) 5.5 hours d.) 4 hours

Solution Steps

First, we need to determine the rate at which Jane and Bill can clean the park individually.

• Jane can clean the park in 6 hours, so her rate is \( \frac{1}{6} \) of the park per hour.
• Bill can clean the park in 3 hours, so his rate is \( \frac{1}{3} \) of the park per hour.

To find the combined rate of work when Jane and Bill work together, we add their individual rates:

\[ \text{Combined rate} = \frac{1}{6} + \frac{1}{3} \]

Convert \(\frac{1}{3}\) to have a common denominator with \(\frac{1}{6}\):

\[ \frac{1}{3} = \frac{2}{6} \]

Now, add the rates:

\[ \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \]

So, together, they can clean \(\frac{1}{2}\) of the park per hour.

Since their combined rate is \(\frac{1}{2}\) of the park per hour, it will take them:

\[ \text{Time} = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{1}{2}} = 2 \text{ hours} \]

Final Answer