12 men can complete a work in 6 days working 4 hours per days. If 4 men work 8 hours per day, then in how many days the same work will be completed?
9 days
Solving Men Work Days Hours Work Problems
This problem involves the concept of work rate, where the total work done is proportional to the number of workers, the number of days they work, and the number of hours they work per day. For the same work, the product of these quantities remains constant.
Understanding the Work Formula
We can represent the total work done as the product of Men, Days, and Hours per day. When the total work is the same in two different scenarios, we can use the formula:
\(M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2\)
Where:
- \(M_1\) = Number of men in the first scenario
- \(D_1\) = Number of days taken in the first scenario
- \(H_1\) = Number of hours per day worked in the first scenario
- \(M_2\) = Number of men in the second scenario
- \(D_2\) = Number of days taken in the second scenario
- \(H_2\) = Number of hours per day worked in the second scenario
Applying the Formula to the Given Problem
From the question, we have the following information:
Scenario 1:
- Number of men (\(M_1\)): 12 men
- Number of days (\(D_1\)): 6 days
- Hours per day (\(H_1\)): 4 hours/day
Scenario 2:
- Number of men (\(M_2\)): 4 men
- Number of days (\(D_2\)): Unknown (what we need to find)
- Hours per day (\(H_2\)): 8 hours/day
Calculation Steps
Substitute the known values into the formula:
\(M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2\)
\(12 \times 6 \times 4 = 4 \times D_2 \times 8\)
Calculate the product on the left side:
\(72 \times 4 = 32 \times D_2\)
\(288 = 32 \times D_2\)
Now, isolate \(D_2\) by dividing both sides by 32:
\(D_2 = \frac{288}{32}\)
\(D_2 = 9\)
So, in the second scenario, it will take 9 days to complete the same work.
Conclusion
If 4 men work 8 hours per day, the same work will be completed in 9 days.
Revision Table: Work Problem Formula
| Concept | Formula | Description |
|---|---|---|
| Work Done | Work \(\propto\) Men \(\times\) Days \(\times\) Hours | Total work is proportional to the number of workers, days, and hours worked daily. |
| Constant Work | \(M_1D_1H_1 = M_2D_2H_2\) | Used when the same amount of work is done under different conditions. |
| Units | Consistent Units Required | Ensure men, days, and hours are in consistent units for comparison. |
Additional Information: Time and Work Concepts
Time and work problems often involve understanding inverse and direct proportions:
- Men and Days (Inverse Proportion): If the number of men increases, the number of days required to complete the work decreases (assuming hours/day is constant).
- Hours/Day and Days (Inverse Proportion): If the number of hours worked per day increases, the number of days required decreases (assuming the number of men is constant).
- Men and Work (Direct Proportion): If the number of men increases, the amount of work done in a fixed time increases.
The formula \(M \times D \times H = \text{Work}\) captures these relationships. When Work is constant, any increase in M or H must result in a proportional decrease in D to keep the product constant.
This type of problem is fundamental in understanding efficiency and resource allocation in various tasks.
Important Questions from Time and Work
Surbhi can do a piece of work in 24 days. She completed 3/8 of the work and then left it. Amit can complete the remaining work in 10 days. Working together, they will complete 125% of the same work in:
Rama and Hari can together finish a piece of work in 15 day. Rama works twice as fast as Hari, then Hari alone can finish work in :
Anil, Deepak and Dinesh together can complete a work in 35 days. Anil and Dinesh together can complete the same work in 60 days. In how many days Deepak alone can complete the same work?
Anu is four times as good as Binni in completing a task. Together they finish the same task in 7 hours. In how many hours will Anu alone complete the task?
P, Q and R can complete a work in 10 days, 20 days and 30 days, respectively, working alone. How soon can the work be completed if P is assisted by Q and R on alternate days?
