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?
6
Solving the Work and Time Problem
This problem involves understanding the concept of work rate and how it changes when the number of workers varies over time. Each person works independently, and their work rate is constant.
Understanding Individual Work Rate
Each of the five men can complete the work independently in 20 days. This means that in one day, a single person can complete $\frac{1}{20}$ of the total work.
Let the total work be represented by $W$ units. The work rate of one person is $\frac{W}{20}$ units per day.
Calculating Work Done Day by Day
The number of people working increases each day for the first four days. From the fifth day onwards, the number of people remains constant at five.
- Day 1: 1 person works. Work done = $1 \times \frac{W}{20} = \frac{W}{20}$ units.
- Day 2: 2 persons work. Work done = $2 \times \frac{W}{20} = \frac{2W}{20}$ units.
- Day 3: 3 persons work. Work done = $3 \times \frac{W}{20} = \frac{3W}{20}$ units.
- Day 4: 4 persons work. Work done = $4 \times \frac{W}{20} = \frac{4W}{20}$ units.
Total Work Done in the First Four Days
The total work done in the first four days is the sum of the work done each day:
Total work (Days 1-4) = $\frac{W}{20} + \frac{2W}{20} + \frac{3W}{20} + \frac{4W}{20} = \frac{(1+2+3+4)W}{20} = \frac{10W}{20} = \frac{W}{2}$ units.
Calculating Remaining Work
After 4 days, half of the total work is completed. The remaining work is:
Remaining work = Total work - Work done (Days 1-4) = $W - \frac{W}{2} = \frac{W}{2}$ units.
Work from the Fifth Day Onwards
From the fifth day, five persons work together. The combined work rate of five persons is:
Combined rate (5 persons) = $5 \times \frac{W}{20} = \frac{5W}{20} = \frac{W}{4}$ units per day.
Days to Complete Remaining Work
The remaining work is $\frac{W}{2}$ units, and the five persons work at a rate of $\frac{W}{4}$ units per day. The number of days required to complete the remaining work is:
Days for remaining work = $\frac{\text{Remaining work}}{\text{Combined rate}} = \frac{W/2}{W/4} = \frac{W}{2} \times \frac{4}{W} = \frac{4W}{2W} = 2$ days.
Total Time to Complete the Work
The work took 4 days for the first phase (with increasing numbers of workers) and 2 days for the second phase (with five workers). The total time to complete the work is:
Total days = Days (Phase 1) + Days (Phase 2) = $4 \text{ days} + 2 \text{ days} = 6 \text{ days}$.
Summary of Work Progress
| Day | Number of Persons | Work Done on Day | Cumulative Work Done |
|---|---|---|---|
| 1 | 1 | $\frac{W}{20}$ | $\frac{W}{20}$ |
| 2 | 2 | $\frac{2W}{20}$ | $\frac{W}{20} + \frac{2W}{20} = \frac{3W}{20}$ |
| 3 | 3 | $\frac{3W}{20}$ | $\frac{3W}{20} + \frac{3W}{20} = \frac{6W}{20}$ |
| 4 | 4 | $\frac{4W}{20}$ | $\frac{6W}{20} + \frac{4W}{20} = \frac{10W}{20} = \frac{W}{2}$ |
| 5 | 5 | $\frac{5W}{20}$ | $\frac{W}{2} + \frac{5W}{20} = \frac{10W + 5W}{20} = \frac{15W}{20} = \frac{3W}{4}$ |
| 6 | 5 | $\frac{5W}{20}$ | $\frac{3W}{4} + \frac{5W}{20} = \frac{15W + 5W}{20} = \frac{20W}{20} = W$ |
As shown in the table, the total work $W$ is completed by the end of Day 6.
Revision Table: Key Concepts in Work and Time Problems
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | The amount of work a person or group can do in one unit of time (e.g., day). | Work Rate = $\frac{\text{Total Work}}{\text{Total Time}}$ |
| Total Work | The entire task to be completed. Can be represented as 1 unit or a specific value. | Total Work = Work Rate $\times$ Time |
| Time Taken | The duration required to complete a certain amount of work. | Time Taken = $\frac{\text{Amount of Work}}{\text{Work Rate}}$ |
| Multiple Workers | If workers have individual rates, their combined rate when working together is the sum of their individual rates (assuming they don't affect each other). | Combined Rate = Sum of Individual Rates |
Additional Information on Work and Time Calculations
Work and time problems often involve calculating rates of work, total work, and the time taken under various conditions, such as changing numbers of workers, varying efficiencies, or different work durations per day.
- It's often helpful to assume the total work as 1 unit or a value that is a multiple of the individual time periods (like LCM) to simplify calculations. In this case, assuming total work as $20 \times 5 = 100$ units (since 5 men could finish it in 20 days if they worked together from the start) would also work. One man's rate would be $100/20 = 5$ units per day.
- Day 1: 1 person, 5 units. Cumulative: 5 units.
- Day 2: 2 persons, $2 \times 5 = 10$ units. Cumulative: $5+10=15$ units.
- Day 3: 3 persons, $3 \times 5 = 15$ units. Cumulative: $15+15=30$ units.
- Day 4: 4 persons, $4 \times 5 = 20$ units. Cumulative: $30+20=50$ units.
- Remaining work: $100 - 50 = 50$ units.
- From Day 5: 5 persons work. Combined rate = $5 \times 5 = 25$ units per day.
- Days to complete remaining 50 units = $50 / 25 = 2$ days.
- Total days = 4 (initial days) + 2 (remaining days) = 6 days.
- Both methods (using fraction of work or total work units) yield the same result.
These problems test your ability to break down the work process into stages and apply the work rate formula correctly for each stage.
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?
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?
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?
