To solve the problem step by step, we will first determine the work efficiencies of Panu, Jiban, and Jalil, then calculate how much work is done over a period of days, and finally find out how many days it will take to complete the task.
### Step 1: Determine the work efficiencies of Panu, Jiban, and Jalil.
- Panu can complete the task in 30 days, so his work efficiency is \( \frac{1}{30} \) of the task per day.
- Jiban can complete the task in 20 days, so his work efficiency is \( \frac{1}{20} \) of the task per day.
- Jalil can complete the task in 60 days, so his work efficiency is \( \frac{1}{60} \) of the task per day.
### Step 2: Calculate the combined efficiency.
To find the combined efficiency of Panu, Jiban, and Jalil, we first need to express their efficiencies with a common denominator. The least common multiple of 30, 20, and 60 is 60.
- Panu's efficiency: \( \frac{1}{30} = \frac{2}{60} \)
- Jiban's efficiency: \( \frac{1}{20} = \frac{3}{60} \)
- Jalil's efficiency: \( \frac{1}{60} = \frac{1}{60} \)
Now, we can express their efficiencies as a ratio:
- Panu : Jiban : Jalil = 2 : 3 : 1
### Step 3: Calculate the total work.
The total work can be assumed to be 60 units (as it is the least common multiple of their individual work days).
### Step 4: Calculate the work done in the first two days.
Jiban works alone for the first two days:
- Work done by Jiban in 2 days = \( 2 \times \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \) of the task.
### Step 5: Calculate the work done every third day.
On the third day, Jiban, Panu, and Jalil work together:
- Combined work done by Jiban, Panu, and Jalil in one day = \( \frac{1}{20} + \frac{1}{30} + \frac{1}{60} \)
- To combine these, we find a common denominator (which is 60):
- Jiban: \( \frac{3}{60} \)
- Panu: \( \frac{2}{60} \)
- Jalil: \( \frac{1}{60} \)
Total work done on the third day:
- \( \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{6}{60} = \frac{1}{10} \)
### Step 6: Calculate the total work done in a cycle of 3 days.
In 3 days:
- Work done = Work done in first 2 days + Work done on the third day
- Total work done in 3 days = \( \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \)
### Step 7: Calculate how many cycles are needed to complete the task.
To complete the entire task (which we assumed to be 60 units):
- Total cycles needed = Total work / Work done in 3 days
- Total work = 1 (the whole task), and work done in 3 days = \( \frac{1}{5} \)
- Number of cycles = \( 1 \div \frac{1}{5} = 5 \)
### Step 8: Calculate the total number of days.
Since each cycle takes 3 days:
- Total days = Number of cycles × Days per cycle
- Total days = \( 5 \times 3 = 15 \) days
Thus, it will take **15 days** to complete the task.
