P, Q and R together can complete a work in 36 days. If the ratio of efficiency of P, Q and R is `3 : 2 : 1` respectively, then in how many days R alone can complete the same work?

Options

A. 108
B. 216
C. 72
D. 144

Answer

To solve the problem, we need to find out how many days R alone can complete the work, given that P, Q, and R together can complete it in 36 days and their efficiencies are in the ratio of 3:2:1.

### Step-by-Step Solution:

1. **Understand the given information:**
- P, Q, and R together can complete the work in 36 days.
- The ratio of their efficiencies is 3:2:1.

2. **Calculate the total efficiency of P, Q, and R:**
- Let the efficiencies of P, Q, and R be represented as:
- Efficiency of P = 3x
- Efficiency of Q = 2x
- Efficiency of R = 1x
- Therefore, the total efficiency of P, Q, and R combined is:
\[ \text{Total Efficiency} = 3x + 2x + 1x = 6x \]

3. **Calculate the total work done:**
- Since they can complete the work in 36 days, the total work (W) can be calculated as:
\[ W = \text{Total Efficiency} \times \text{Number of Days} = 6x \times 36 = 216x \]

4. **Calculate R's efficiency:**
- R's efficiency is 1x.

5. **Calculate the number of days R alone can complete the work:**
- The number of days R alone can complete the work is given by:
\[ \text{Days taken by R} = \frac{\text{Total Work}}{\text{R's Efficiency}} = \frac{216x}{1x} = 216 \text{ days} \]

Thus, R alone can complete the work in **216 days**.