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
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
