Tushar, Megha, Punam and Richa can complete a piece of work in 10 days, 12 days, 15 days and 18 days, respectively. In how many days will the work be completed if each of them work on alternate days starting with Megha on first day, Punam on second day, Richa on third day, Tushar on fourth day and then again Megha on fifth day and so on.
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Understanding the Work and Time Problem
This problem involves four individuals, Tushar, Megha, Punam, and Richa, working on a task on alternate days following a specific sequence. We are given the time each person takes to complete the entire work individually and need to find the total time taken when they work in a rotating pattern.
Calculating Individual Work Rates (Efficiency)
To solve problems involving work and time, it's helpful to determine the amount of work each person can complete in a single day. This is often called their work rate or efficiency. We can assume a total amount of work that is a common multiple of all the individual times. The Least Common Multiple (LCM) is a convenient choice.
- Time taken by Tushar = 10 days
- Time taken by Megha = 12 days
- Time taken by Punam = 15 days
- Time taken by Richa = 18 days
Let's find the LCM of 10, 12, 15, and 18.
- Prime factorization of 10 = 2 × 5
- Prime factorization of 12 = 2² × 3
- Prime factorization of 15 = 3 × 5
- Prime factorization of 18 = 2 × 3²
LCM(10, 12, 15, 18) = 2² × 3² × 5 = 4 × 9 × 5 = 180.
Let the total work be 180 units.
Now, we can calculate the daily work rate for each person:
- Tushar's daily work rate = Total work / Time taken = \(180 / 10\) = 18 units/day
- Megha's daily work rate = Total work / Time taken = \(180 / 12\) = 15 units/day
- Punam's daily work rate = Total work / Time taken = \(180 / 15\) = 12 units/day
- Richa's daily work rate = Total work / Time taken = \(180 / 18\) = 10 units/day
Work Done in One Cycle of Alternate Days
The individuals work on alternate days in a specific sequence:
- Day 1: Megha
- Day 2: Punam
- Day 3: Richa
- Day 4: Tushar
This sequence of 4 days constitutes one complete cycle. After Day 4, the sequence repeats starting with Megha on Day 5.
Let's calculate the total work done in one 4-day cycle:
- Work done on Day 1 (Megha) = 15 units
- Work done on Day 2 (Punam) = 12 units
- Work done on Day 3 (Richa) = 10 units
- Work done on Day 4 (Tushar) = 18 units
Total work done in 1 cycle (4 days) = \(15 + 12 + 10 + 18 = 55\) units.
Calculating the Total Number of Days
The total work to be completed is 180 units. In each 4-day cycle, 55 units of work are completed. We need to find how many cycles are required to complete 180 units.
Number of full cycles = Total work / Work per cycle
Number of full cycles = \(180 / 55\)
\(180 \div 55 \approx 3.27\). This means 3 full cycles are completed, and some work remains.
Work done in 3 full cycles = 3 cycles × 55 units/cycle = 165 units.
Days taken for 3 full cycles = 3 cycles × 4 days/cycle = 12 days.
Remaining work = Total work - Work done in 3 cycles
Remaining work = \(180 - 165 = 15\) units.
After 12 days (3 full cycles), the next day is Day 13, and the sequence restarts with Megha.
On Day 13, Megha works. Her daily work rate is 15 units/day.
The remaining work is 15 units.
Time taken by Megha to complete the remaining 15 units = Remaining work / Megha's daily rate
Time taken = \(15 \text{ units} / 15 \text{ units/day} = 1\) day.
So, Megha completes the remaining work on Day 13.
Total number of days to complete the work = Days for full cycles + Days for remaining work
Total number of days = 12 days + 1 day = 13 days.
| Day | Worker | Work Done (units) | Cumulative Work (units) |
|---|---|---|---|
| 1 | Megha | 15 | 15 |
| 2 | Punam | 12 | \(15 + 12 = 27\) |
| 3 | Richa | 10 | \(27 + 10 = 37\) |
| 4 | Tushar | 18 | \(37 + 18 = 55\) |
| 5 | Megha | 15 | \(55 + 15 = 70\) |
| ... | ... | ... | ... |
| 12 (End of Cycle 3) | Tushar | 18 | \(55 \times 3 = 165\) |
| 13 (Start of Cycle 4) | Megha | 15 | \(165 + 15 = 180\) |
The work is completed exactly on Day 13.
Conclusion
By calculating the individual work rates, the work done in one complete cycle, and determining the remaining work after full cycles, we found that the total time taken to complete the work when they work on alternate days in the given sequence is 13 days.
Revision Table: Work and Time Concepts
| Concept | Description | Formula |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Work Rate = Total Work / Time Taken |
| Total Work | The total amount of task to be completed. Often assumed as the LCM of individual times. | - |
| Time Taken | The duration required to complete the work. | Time Taken = Total Work / Work Rate |
| Work Done in 'n' Days | If a person's daily rate is 'r', work done in 'n' days is \(n \times r\). | Work Done = Rate × Time |
| Alternate Days Work | Individuals work on different days or in a specific sequence. Calculate work done in one complete cycle of the sequence. | Sum of work rates of individuals in one cycle duration. |
Additional Information: Efficiency and LCM in Work Problems
The concept of efficiency (work rate) is fundamental in solving work and time problems. Efficiency is inversely proportional to the time taken; more efficient individuals take less time. Using the LCM of the given times as the total work simplifies calculations, as it results in integer values for daily work rates.
In alternate day problems, identifying the repeating cycle of workers and calculating the work done within that cycle is crucial. This allows you to determine how many cycles are needed to complete most of the work and then calculate the time required for the remaining work based on the sequence of workers after the last full cycle.
Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
