Question

table of contents 〉 module 7: rational expressions and equations 〉 practice: work
practice: work
score: 0/5 answered: 0/5
question 1
mari and liam can each wash a car and vacuum its interior in 3 hours. zach needs 5 hours to do this same job alone. if zach, liam, and mari work together, how long will it take them to clean a car?
150 minutes
120 minutes
69.2 minutes
60 minutes

Explanation:

Step1: Define work rates

Let the total work (cleaning a car) be \( W = 1 \) job.
Mari's rate: \( \frac{1}{3} \) job per hour (since she takes 3 hours alone).
Liam's rate: \( \frac{1}{3} \) job per hour (same as Mari).
Zach's rate: \( \frac{1}{5} \) job per hour (takes 5 hours alone).

Step2: Combined rate

When working together, their combined rate is the sum of individual rates:
\( \text{Combined rate} = \frac{1}{3} + \frac{1}{3} + \frac{1}{5} \)
Simplify: \( \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \), then \( \frac{2}{3} + \frac{1}{5} = \frac{10 + 3}{15} = \frac{13}{15} \) job per hour.

Step3: Time to complete 1 job

Time \( t \) (in hours) is \( t = \frac{W}{\text{Combined rate}} = \frac{1}{\frac{13}{15}} = \frac{15}{13} \) hours.

Step4: Convert hours to minutes

\( \frac{15}{13} \) hours \( \times 60 \) minutes/hour \( = \frac{900}{13} \approx 69.2 \) minutes.

Answer:

69.2 minutes