Ram can complete a piece work in 15 days, Rohan in 25 days, and Rohit in 30 days. Rohan and Rohit worked together for 2 days and then Rohit was replaced by Ram. In how many days altogether was the work completed?
10 days
Understanding the Work and Time Problem
This question involves a classic work and time scenario where multiple people work at different rates, and the team composition changes over time. To solve this, we first need to determine the individual work rate or efficiency of each person. We can do this by assuming a total amount of work, which is typically the least common multiple (LCM) of the number of days each person takes to complete the work individually.
Calculating Total Work and Individual Efficiencies
The given times to complete the work are:
- Ram: 15 days
- Rohan: 25 days
- Rohit: 30 days
Let's find the LCM of 15, 25, and 30. The LCM of these numbers is 150. We assume the total work is 150 units.
Now, we can calculate the efficiency (units of work per day) for each person:
- Ram's efficiency = Total work / Days taken by Ram = $150 \text{ units} / 15 \text{ days} = 10 \text{ units/day}$
- Rohan's efficiency = Total work / Days taken by Rohan = $150 \text{ units} / 25 \text{ days} = 6 \text{ units/day}$
- Rohit's efficiency = Total work / Days taken by Rohit = $150 \text{ units} / 30 \text{ days} = 5 \text{ units/day}$
Work Done in the First Phase (Rohan and Rohit)
Rohan and Rohit worked together for the first 2 days. We need to calculate their combined efficiency and the total work done during this period.
- Combined efficiency of Rohan and Rohit = Rohan's efficiency + Rohit's efficiency = $6 \text{ units/day} + 5 \text{ units/day} = 11 \text{ units/day}$
- Work done in the first 2 days = Combined efficiency $\times$ Number of days = $11 \text{ units/day} \times 2 \text{ days} = 22 \text{ units}$
Remaining Work After the First Phase
The total work is 150 units, and 22 units have been completed. The remaining work is:
- Remaining work = Total work - Work done in first phase = $150 \text{ units} - 22 \text{ units} = 128 \text{ units}$
Work Done in the Second Phase (Rohan and Ram)
After 2 days, Rohit is replaced by Ram. Now, Rohan and Ram work together to complete the remaining work. We calculate their combined efficiency.
- Combined efficiency of Rohan and Ram = Rohan's efficiency + Ram's efficiency = $6 \text{ units/day} + 10 \text{ units/day} = 16 \text{ units/day}$
The time taken by Rohan and Ram to complete the remaining 128 units of work is:
- Time taken by Rohan and Ram = Remaining work / Combined efficiency of Rohan and Ram = $128 \text{ units} / 16 \text{ units/day} = 8 \text{ days}$
Total Time to Complete the Work
The work was done in two phases. The first phase lasted 2 days, and the second phase lasted 8 days. The total time taken to complete the work is the sum of the durations of these two phases.
- Total time = Time in first phase + Time in second phase = $2 \text{ days} + 8 \text{ days} = 10 \text{ days}$
Summary of Calculations
| Worker | Days to complete alone | Efficiency (Units/day) |
|---|---|---|
| Ram | 15 | 10 |
| Rohan | 25 | 6 |
| Rohit | 30 | 5 |
| Phase | Workers | Combined Efficiency (Units/day) | Days worked | Work done (Units) |
|---|---|---|---|---|
| Phase 1 | Rohan & Rohit | 11 | 2 | $11 \times 2 = 22$ |
| Phase 2 | Rohan & Ram | 16 | Calculated below | Remaining work = $150 - 22 = 128$ |
Time taken in Phase 2 = Remaining Work / Combined Efficiency of Rohan & Ram = $128 / 16 = 8$ days.
Total time = Time in Phase 1 + Time in Phase 2 = $2 + 8 = 10$ days.
Therefore, the work was completed in a total of 10 days.
Revision Table: Work and Time Concepts
| Concept | Explanation |
|---|---|
| Individual Efficiency | The amount of work a person can do in one unit of time (e.g., one day). It's calculated as Total Work / Time taken. |
| Total Work (Assumed) | Often taken as the LCM of the individual times to complete the work. This makes calculations with efficiencies easier as they become integers. |
| Combined Efficiency | When multiple people work together, their efficiencies are added to find their combined efficiency per unit of time. |
| Time Taken | Calculated as Total Work / Combined Efficiency. If work is done in phases, calculate time for each phase and sum them up for total time. |
Additional Information: Work and Time Problems
Work and time problems often involve understanding inverse proportionality. If a person takes more time, their work rate (efficiency) is lower, and vice versa. When solving these problems, consistency in units (e.g., units of work per day) is crucial. Problems can include scenarios with varying numbers of workers, workers leaving or joining, or workers working for different durations. The key is to break down the problem into steps, calculate the work done in each phase, find the remaining work, and then calculate the time needed for subsequent phases based on the new team's efficiency. Remember to always account for the time already spent when calculating the total time for the project.
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