Question:
Working at a constant rate, Bob can produce \(\frac{x}{3}\) widgets in 8 minutes. Working at a constant rate, Jack can produce \(2x\) widgets in 40 minutes, where \(x \gt 0\).
Quantity A: The number of minutes it will take Bob to produce \(5x\) widgets.
Quantity B: The number of minutes it will take Jack to produce \(6x\) widgets.
Working at a constant rate, Bob can produce \(\frac{x}{3}\) widgets in 8 minutes. Working at a constant rate, Jack can produce \(2x\) widgets in 40 minutes, where \(x \gt 0\).
Quantity A: The number of minutes it will take Bob to produce \(5x\) widgets.
Quantity B: The number of minutes it will take Jack to produce \(6x\) widgets.
Quantity A: The number of minutes it will take Bob to produce \(5x\) widgets.
Quantity B: The number of minutes it will take Jack to produce \(6x\) widgets.
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For work-rate problems, setting up a proportion is often a very fast method. For Bob: \(\frac{\text{widgets}_1}{\text{time}_1} = \frac{\text{widgets}_2}{\text{time}_2} \implies \frac{x/3}{8} = \frac{5x}{\text{Time}_A}\). Solving for \(\text{Time}_A\) gives 120. For Jack: \(\frac{2x}{40} = \frac{6x}{\text{Time}_B}\). Solving for \(\text{Time}_B\) gives 120.
Updated On: Oct 3, 2025
- if Quantity A is greater;
- if Quantity B is greater;
- if the two quantities are equal;
- if the relationship cannot be determined from the information given.
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
This is a work-rate problem. The key is to first determine the rate of production for both Bob and Jack, and then use that rate to find the time required to produce a different quantity of widgets.
Step 2: Key Formula or Approach:
The fundamental relationship is: \(\text{Rate} = \frac{\text{Work}}{\text{Time}}\). Consequently, \(\text{Time} = \frac{\text{Work}}{\text{Rate}}\). We can also solve this using proportions.
Step 3: Detailed Explanation:
First, let's calculate the production rate for Bob.
Bob produces \(\frac{x}{3}\) widgets in 8 minutes.
Bob's Rate = \(\frac{\text{Work}}{\text{Time}} = \frac{x/3 \text{ widgets}}{8 \text{ minutes}} = \frac{x}{24}\) widgets per minute.
Next, let's calculate the production rate for Jack.
Jack produces \(2x\) widgets in 40 minutes.
Jack's Rate = \(\frac{\text{Work}}{\text{Time}} = \frac{2x \text{ widgets}}{40 \text{ minutes}} = \frac{x}{20}\) widgets per minute.
Now, we calculate the time for Quantity A.
Quantity A: Time for Bob to produce \(5x\) widgets.
Time = \(\frac{\text{Work}}{\text{Rate}} = \frac{5x \text{ widgets}}{x/24 \text{ widgets/minute}} = 5x \times \frac{24}{x} = 120\) minutes.
Finally, we calculate the time for Quantity B.
Quantity B: Time for Jack to produce \(6x\) widgets.
Time = \(\frac{\text{Work}}{\text{Rate}} = \frac{6x \text{ widgets}}{x/20 \text{ widgets/minute}} = 6x \times \frac{20}{x} = 120\) minutes.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 120 minutes.
Quantity B = 120 minutes.
The two quantities are equal.
This is a work-rate problem. The key is to first determine the rate of production for both Bob and Jack, and then use that rate to find the time required to produce a different quantity of widgets.
Step 2: Key Formula or Approach:
The fundamental relationship is: \(\text{Rate} = \frac{\text{Work}}{\text{Time}}\). Consequently, \(\text{Time} = \frac{\text{Work}}{\text{Rate}}\). We can also solve this using proportions.
Step 3: Detailed Explanation:
First, let's calculate the production rate for Bob.
Bob produces \(\frac{x}{3}\) widgets in 8 minutes.
Bob's Rate = \(\frac{\text{Work}}{\text{Time}} = \frac{x/3 \text{ widgets}}{8 \text{ minutes}} = \frac{x}{24}\) widgets per minute.
Next, let's calculate the production rate for Jack.
Jack produces \(2x\) widgets in 40 minutes.
Jack's Rate = \(\frac{\text{Work}}{\text{Time}} = \frac{2x \text{ widgets}}{40 \text{ minutes}} = \frac{x}{20}\) widgets per minute.
Now, we calculate the time for Quantity A.
Quantity A: Time for Bob to produce \(5x\) widgets.
Time = \(\frac{\text{Work}}{\text{Rate}} = \frac{5x \text{ widgets}}{x/24 \text{ widgets/minute}} = 5x \times \frac{24}{x} = 120\) minutes.
Finally, we calculate the time for Quantity B.
Quantity B: Time for Jack to produce \(6x\) widgets.
Time = \(\frac{\text{Work}}{\text{Rate}} = \frac{6x \text{ widgets}}{x/20 \text{ widgets/minute}} = 6x \times \frac{20}{x} = 120\) minutes.
Step 4: Final Answer:
Comparing the two quantities:
Quantity A = 120 minutes.
Quantity B = 120 minutes.
The two quantities are equal.
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