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Working at their individual same constant rate, 24 machines can complete a certain production job in 10 hours when they all work together. On a certain day, due to a minor malfunction, 8 of those machines were not operating for the first 2 hours. Compared to normal days, what is the extra time taken to complete the production job on that day?
- a)20 minutes
- b)30 minutes
- c)40 minutes
- d)1 hour
- e)1 hour 20 minutes
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Working at their individual same constant rate, 24 machines can comple...
Understanding the Problem
To determine the extra time taken to complete the production job due to the malfunctioning machines, we first need to calculate the total work done and the effective rate of the machines.
Calculation of Work Done
- **Total Work:**
If 24 machines can complete the job in 10 hours, the total work (W) can be calculated as:
\[ W = \text{Machines} \times \text{Time} \]
\[ W = 24 \times 10 = 240 \text{ machine-hours} \]
- **Rate of One Machine:**
The rate of one machine (R) is:
\[ R = \frac{W}{\text{Total Machines} \times \text{Total Time}} \]
\[ R = \frac{240}{24} = 10 \text{ hours per machine} \]
Thus, the rate of one machine is \( \frac{1}{10} \) of the work per hour.
Work Done in the First 2 Hours
- **Machines Working Initially:**
For the first 2 hours, only 16 machines were operational. The work done (W1) in this time is:
\[ W1 = 16 \text{ machines} \times 2 \text{ hours} = 32 \text{ machine-hours} \]
- **Remaining Work:**
The remaining work after 2 hours is:
\[ W_{\text{remaining}} = 240 - 32 = 208 \text{ machine-hours} \]
Completion of Remaining Work
- **Machines Working After 2 Hours:**
After 2 hours, all 24 machines are operational. The time (T) to finish the remaining work is:
\[ T = \frac{W_{\text{remaining}}}{\text{Total Machines}} = \frac{208}{24} \approx 8.67 \text{ hours} \]
Total Time Taken
- **Total Time on Malfunction Day:**
The total time taken on the malfunction day is:
\[ \text{Total Time} = 2 \text{ hours} + 8.67 \text{ hours} = 10.67 \text{ hours} \]
Comparison to Normal Days
- **Extra Time Taken:**
The normal time is 10 hours. The extra time taken is:
\[ \text{Extra Time} = 10.67 - 10 = 0.67 \text{ hours} \approx 40 \text{ minutes} \]
Thus, the extra time taken to complete the job on that day is **40 minutes**, confirming option 'C'.
To determine the extra time taken to complete the production job due to the malfunctioning machines, we first need to calculate the total work done and the effective rate of the machines.
Calculation of Work Done
- **Total Work:**
If 24 machines can complete the job in 10 hours, the total work (W) can be calculated as:
\[ W = \text{Machines} \times \text{Time} \]
\[ W = 24 \times 10 = 240 \text{ machine-hours} \]
- **Rate of One Machine:**
The rate of one machine (R) is:
\[ R = \frac{W}{\text{Total Machines} \times \text{Total Time}} \]
\[ R = \frac{240}{24} = 10 \text{ hours per machine} \]
Thus, the rate of one machine is \( \frac{1}{10} \) of the work per hour.
Work Done in the First 2 Hours
- **Machines Working Initially:**
For the first 2 hours, only 16 machines were operational. The work done (W1) in this time is:
\[ W1 = 16 \text{ machines} \times 2 \text{ hours} = 32 \text{ machine-hours} \]
- **Remaining Work:**
The remaining work after 2 hours is:
\[ W_{\text{remaining}} = 240 - 32 = 208 \text{ machine-hours} \]
Completion of Remaining Work
- **Machines Working After 2 Hours:**
After 2 hours, all 24 machines are operational. The time (T) to finish the remaining work is:
\[ T = \frac{W_{\text{remaining}}}{\text{Total Machines}} = \frac{208}{24} \approx 8.67 \text{ hours} \]
Total Time Taken
- **Total Time on Malfunction Day:**
The total time taken on the malfunction day is:
\[ \text{Total Time} = 2 \text{ hours} + 8.67 \text{ hours} = 10.67 \text{ hours} \]
Comparison to Normal Days
- **Extra Time Taken:**
The normal time is 10 hours. The extra time taken is:
\[ \text{Extra Time} = 10.67 - 10 = 0.67 \text{ hours} \approx 40 \text{ minutes} \]
Thus, the extra time taken to complete the job on that day is **40 minutes**, confirming option 'C'.
Community Answer
Working at their individual same constant rate, 24 machines can comple...
The rate of 24 machines is 1/10.
On a certain day 16 machines first ran for 2 hours.
The rate of the 16 machines is:
16/n = 24/(1/10)
16/n = 240
16 = 240n
n = 16/240 = 1/15
Thus, when 16 machines work for 2 hours, the fraction of the job completed is 1/15 x 2 = 2/15.
Thus, 13/15 of the job needs to be completed by 24 machines. The time it will take to complete the job is:
(13/15)/(1/10) = 130/15 = 26/3 hours
Therefore, the total time spent on the job is 2 + 26/3 = 32/3 hours = 10 ⅔ hours = 10 hours 40 minutes, which is 40 minutes more than on a normal day.
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Working at their individual same constant rate, 24 machines can complete a certain production job in 10 hours when they all work together. On a certain day, due to a minor malfunction, 8 of those machines were not operating for the first 2 hours. Compared to normal days, what is the extra time taken to complete the production job on that day?a)20 minutesb)30 minutesc)40 minutesd)1 houre)1 hour 20 minutesCorrect answer is option 'C'. Can you explain this answer?
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