Question

Bill and Bob are working to build toys. Bill can build k toys in 6 hours. Bob can build k toys in 3 hours. How long would it take Bob and Bill to build 4k toys working together?

• A. 8 hours
• B. 12 hours
• C. 9 hours
• D. 2 hours
• E. 4 hours

Hint:

In work rate problems, the variable for the amount of work (like 'k' here) often cancels out in the final calculation. Don't be intimidated by it; focus on correctly setting up the rates.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a combined work rate problem. The key principle is that when people work together, their individual rates of work add up to a combined rate. The formula relating work, rate, and time is Work = Rate \(\times\) Time.
Step 2: Key Formula or Approach:
1. Calculate the individual work rate for Bill and Bob. Rate is defined as the amount of work done per unit of time. 2. Add their individual rates to find their combined work rate. 3. Use the combined rate and the total amount of work to be done (4k toys) to calculate the total time required. Time = Work / Rate.
Step 3: Detailed Explanation:
Individual Rates:
Bill's rate: \( \text{Rate}_{\text{Bill}} = \frac{\text{Work}}{\text{Time}} = \frac{k \text{ toys}}{6 \text{ hours}} = \frac{k}{6} \) toys/hour.
Bob's rate: \( \text{Rate}_{\text{Bob}} = \frac{\text{Work}}{\text{Time}} = \frac{k \text{ toys}}{3 \text{ hours}} = \frac{k}{3} \) toys/hour.

Combined Rate:
When they work together, their rates add up: \[ \text{Rate}_{\text{Combined}} = \text{Rate}_{\text{Bill}} + \text{Rate}_{\text{Bob}} = \frac{k}{6} + \frac{k}{3} \] To add these fractions, find a common denominator (6): \[ \text{Rate}_{\text{Combined}} = \frac{k}{6} + \frac{2k}{6} = \frac{3k}{6} = \frac{k}{2} \text{ toys/hour} \]
Time to Complete the Job:
The total work to be done is building 4k toys. \[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{4k \text{ toys}}{\frac{k}{2} \text{ toys/hour}} \] \[ \text{Time} = 4k \times \frac{2}{k} = 8 \text{ hours} \] Step 4: Final Answer:
It would take them 8 hours to build 4k toys together.