40 men can complete a work in 15 days. Three days after they started working, 20 more men joined them. In how many days the total work will be completed?
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Understanding the Work and Time Problem
This problem involves the concept of work and time, specifically how the number of workers affects the time taken to complete a task. The core idea is that the total amount of work remains constant, regardless of the number of workers or the time taken, assuming the work rate per person is constant.
Step-by-Step Solution for Work Completion
1. Calculate the Total Work
The total work required to complete the task can be calculated by multiplying the number of men initially assigned by the number of days they would take alone.
Initial number of men = 40
Days to complete work by 40 men = 15 days
Total Work = Number of men $\times$ Number of days
Total Work = $40 \times 15 = 600$ units of work
So, the total work is 600 units.
2. Calculate Work Done in the First 3 Days
The 40 men worked for the first 3 days before more men joined. We need to calculate how much work was completed during this initial period.
Number of men in the first 3 days = 40
Days worked initially = 3 days
Work done in first 3 days = Number of men $\times$ Number of days worked
Work done in first 3 days = $40 \times 3 = 120$ units of work
After 3 days, 120 units of work have been completed.
3. Calculate the Remaining Work
Subtract the work done in the first 3 days from the total work to find the work that still needs to be completed.
Remaining Work = Total Work - Work done in first 3 days
Remaining Work = $600 - 120 = 480$ units of work
There are 480 units of work remaining.
4. Calculate the New Number of Men
After 3 days, 20 more men joined the initial group of 40 men.
Initial men = 40
Men joined = 20
New number of men = Initial men + Men joined
New number of men = $40 + 20 = 60$ men
The remaining work will be done by 60 men.
5. Calculate Days to Complete Remaining Work
Now, the 480 units of remaining work will be completed by the new team of 60 men. Assuming each man works at the same rate, the time taken will be the remaining work divided by the new number of men.
Days to complete remaining work = Remaining Work / New number of men
Days to complete remaining work = $480 / 60 = 8$ days
The remaining work will take 8 more days to complete.
6. Calculate the Total Days to Complete the Work
The total time taken to complete the entire work is the sum of the days worked initially and the days taken to complete the remaining work.
Total Days = Days worked initially + Days to complete remaining work
Total Days = $3 \text{ days} + 8 \text{ days} = 11 \text{ days}
The total work will be completed in 11 days.
| Step | Description | Calculation | Result |
|---|---|---|---|
| 1 | Total Work | $40 \times 15$ | 600 units |
| 2 | Work Done in 3 Days | $40 \times 3$ | 120 units |
| 3 | Remaining Work | $600 - 120$ | 480 units |
| 4 | New Number of Men | $40 + 20$ | 60 men |
| 5 | Days for Remaining Work | $480 / 60$ | 8 days |
| 6 | Total Days | $3 + 8$ | 11 days |
Therefore, the total work will be completed in 11 days.
Revision Table: Work and Time Concepts
| Concept | Formula/Relationship | Explanation |
|---|---|---|
| Total Work | Work = Men $\times$ Days | Assuming a constant work rate per man, total work is proportional to the number of men and the time they work. |
| Work Rate | Work Rate per Man = Total Work / (Men $\times$ Days) | The amount of work one man can do in one day. Often assumed constant in these problems. |
| Effect of More Men | If Men increase, Days decrease (for the same work) | If the number of men increases, they can complete the same amount of work in less time, assuming the same work rate. |
| Partial Work | Work Done = Men $\times$ Days Worked | Calculates the portion of total work completed in a specific period by a specific number of men. |
Additional Information: Direct and Inverse Proportion
Problems involving work and time often relate to direct and inverse proportion:
- Men and Work: If the number of days is constant, the amount of work done is directly proportional to the number of men. More men means more work done in the same time. ($W \propto M$)
- Days and Work: If the number of men is constant, the amount of work done is directly proportional to the number of days. More days means more work done by the same group. ($W \propto D$)
- Men and Days: If the total work is constant, the number of men is inversely proportional to the number of days. More men means fewer days needed to complete the same work. ($M \propto 1/D$) or ($M \times D = \text{Constant (Total Work)}$)
In this problem, when more men join, the remaining work is completed faster due to the inverse relationship between the number of men and the days required for a fixed amount of work.
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?
