Nancy can complete a typing job in 8 hours. When Michael helps her, they can do the job together in 5 hours. How many hours would it take Michael to do the job alone?
The correct answer and explanation is:
Let’s solve the problem step-by-step.
Problem restatement:
- Nancy can finish the typing job alone in 8 hours.
- Together, Nancy and Michael can finish the job in 5 hours.
- Question: How long will it take Michael alone to complete the job?
Step 1: Understand the rates
- Nancy’s work rate is the fraction of the job she completes per hour.
- Since Nancy finishes the job in 8 hours, her rate is:
Nancy’s rate=1 job8 hours=18 jobs/hour\text{Nancy’s rate} = \frac{1 \text{ job}}{8 \text{ hours}} = \frac{1}{8} \text{ jobs/hour}
- Let Michael’s time to finish the job alone be xx hours.
- Then Michael’s rate is:
Michael’s rate=1 jobx hours=1x jobs/hour\text{Michael’s rate} = \frac{1 \text{ job}}{x \text{ hours}} = \frac{1}{x} \text{ jobs/hour}
Step 2: Use combined work rate
- When Nancy and Michael work together, their rates add up.
- Together, they finish the job in 5 hours, so their combined rate is:
Combined rate=15 jobs/hour\text{Combined rate} = \frac{1}{5} \text{ jobs/hour}
- The combined rate is the sum of their individual rates:
18+1x=15\frac{1}{8} + \frac{1}{x} = \frac{1}{5}
Step 3: Solve the equation for xx
18+1x=15\frac{1}{8} + \frac{1}{x} = \frac{1}{5}
Subtract 18\frac{1}{8} from both sides: 1x=15−18\frac{1}{x} = \frac{1}{5} – \frac{1}{8}
Find a common denominator (40): 15=840,18=540\frac{1}{5} = \frac{8}{40}, \quad \frac{1}{8} = \frac{5}{40}
So, 1x=840−540=340\frac{1}{x} = \frac{8}{40} – \frac{5}{40} = \frac{3}{40}
Now invert both sides to find xx: x=403≈13.33 hoursx = \frac{40}{3} \approx 13.33 \text{ hours}
Final answer:
Michael alone will take approximately 13.33 hours (or 13 hours and 20 minutes) to complete the job.
Explanation in 300 words:
This problem is a classic example of combined work rates, often used to find the time it takes for two or more people working together to complete a job, and then to find the individual time of the second person.
First, we recognize that the rate at which Nancy works is the portion of the job she completes per hour. Since she can finish the whole job in 8 hours, her rate is 18\frac{1}{8} of the job per hour. Similarly, we assume Michael’s work rate is 1x\frac{1}{x} jobs per hour, where xx is the unknown time Michael would take alone.
When two people work together, their rates add up. That means the combined rate of Nancy and Michael working simultaneously is the sum of their individual rates. We know that together they complete the job in 5 hours, so their combined rate is 15\frac{1}{5} jobs per hour.
By setting the sum of their rates equal to the combined rate, we form the equation 18+1x=15\frac{1}{8} + \frac{1}{x} = \frac{1}{5}. Solving this equation involves isolating 1x\frac{1}{x}, which represents Michael’s rate, and subtracting Nancy’s rate from the combined rate.
After finding 1x=340\frac{1}{x} = \frac{3}{40}, we invert to find xx, yielding 403\frac{40}{3} hours, or approximately 13 hours and 20 minutes. This means Michael is slower than Nancy since he takes longer to finish the job alone.
This problem nicely illustrates how work rates combine and how to solve for an unknown individual rate when given the combined rate and one known individual rate. It’s an important concept in time and work problems frequently encountered in math and real-world scenarios such as teamwork, project planning, and task delegation.