Question: 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?
1. **State the problem:** We want to find the time it takes for Jane and Bill to clean the park together.
2. **Formula used:** When two people work together, their combined work rate is the sum of their individual work rates. The formula is:
$$\frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2}$$
where $t$ is the time working together, $t_1$ is Jane's time alone, and $t_2$ is Bill's time alone.
3. **Identify given values:**
- Jane's time alone: $t_1 = 6$ hours
- Bill's time alone: $t_2 = 3$ hours
4. **Calculate combined rate:**
$$\frac{1}{t} = \frac{1}{6} + \frac{1}{3}$$
5. **Find common denominator and add:**
$$\frac{1}{t} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6}$$
6. **Simplify the fraction:**
$$\frac{1}{t} = \frac{\cancel{3}}{\cancel{6}} = \frac{1}{2}$$
7. **Solve for $t$:**
$$t = 2$$
8. **Interpretation:** It will take Jane and Bill 2 hours to clean the park together.
**Final answer:** a.) 2 hours