Question: Question 2
To collect data on the signal strengths in a neighborhood, Tracy must drive from
house to house and take readings. She has a graduate student, Dave, to assist
her. Tracy figures it would take her eight hours to complete the task working
alone, and that it would take Dave 12 hours if he completed the task by himself.
How long will it take Tracy and Dave to complete the task together?
7.2 hours
3.6 hours
6.0 hours
4.8 hours
1. **State the problem:** Tracy takes 8 hours to complete the task alone, Dave takes 12 hours alone. We want to find how long they take working together.
2. **Formula used:** When two people work together, their combined work rate is the sum of their individual rates. If $t$ is the time to complete the task together, then:
$$\frac{1}{8} + \frac{1}{12} = \frac{1}{t}$$
3. **Calculate the combined rate:**
Find a common denominator for $\frac{1}{8}$ and $\frac{1}{12}$, which is 24.
$$\frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24}$$
4. **Sum the rates:**
$$\frac{3}{24} + \frac{2}{24} = \frac{5}{24}$$
5. **Set equal to combined rate:**
$$\frac{5}{24} = \frac{1}{t}$$
6. **Solve for $t$:**
$$t = \frac{24}{5}$$
7. **Simplify:**
$$t = 4.8$$
8. **Interpretation:** Tracy and Dave working together will complete the task in $4.8$ hours.
**Final answer:** $4.8$ hours