Ram, Rohit and Rahul can do a work in 18 days, 12 days and 15 days respectively. In how many days will they finish the work?

Answer

To solve the problem step by step, we will first determine the individual work rates of Ram, Rohit, and Rahul, and then calculate their combined work rate to find out how many days they will take to finish the work together.

### Step-by-Step Solution:

1. **Calculate Individual Work Rates**:
- Ram can complete the work in 18 days.
- Work rate of Ram = 1 work / 18 days = \( \frac{1}{18} \) work/day
- Rohit can complete the work in 12 days.
- Work rate of Rohit = 1 work / 12 days = \( \frac{1}{12} \) work/day
- Rahul can complete the work in 15 days.
- Work rate of Rahul = 1 work / 15 days = \( \frac{1}{15} \) work/day

2. **Find a Common Denominator**:
- The least common multiple (LCM) of 18, 12, and 15 is 180. This will help us find a common basis for their work rates.

3. **Convert Work Rates to a Common Basis**:
- Ram's work rate in terms of 180:
- \( \frac{180}{18} = 10 \) (units of work per day)
- Rohit's work rate in terms of 180:
- \( \frac{180}{12} = 15 \) (units of work per day)
- Rahul's work rate in terms of 180:
- \( \frac{180}{15} = 12 \) (units of work per day)

4. **Calculate Combined Work Rate**:
- Combined work rate = Ram's rate + Rohit's rate + Rahul's rate
- Combined work rate = 10 + 15 + 12 = 37 units of work per day

5. **Calculate Total Time to Complete the Work**:
- Total work = 180 units
- Time taken to complete the work together = Total work / Combined work rate
- Time = \( \frac{180}{37} \) days

6. **Final Answer**:
- The time taken by Ram, Rohit, and Rahul to finish the work together is \( \frac{180}{37} \) days, which can be approximated to about 4.86 days.