P alone can complete the work in 5 days,Q alone can do same work in 6 days and R alone can do the same work in 12 days. They jointly complete the work and earn Rs. 5400. What is the share of R?

Options

A. 1000
B. 1200
C. 1500
D. 1800

Answer

To solve the problem, we need to find out how much work P, Q, and R can do together, and then determine R's share of the total earnings based on the amount of work done by each.

### Step 1: Determine the work done by each person in one day.

- **P's work rate**: P can complete the work in 5 days. Therefore, in one day, P can complete \( \frac{1}{5} \) of the work.
- **Q's work rate**: Q can complete the work in 6 days. Therefore, in one day, Q can complete \( \frac{1}{6} \) of the work.
- **R's work rate**: R can complete the work in 12 days. Therefore, in one day, R can complete \( \frac{1}{12} \) of the work.

### Step 2: Calculate the combined work rate of P, Q, and R.

To find out how much work they can do together in one day, we add their individual work rates:

\[ \text{Total work rate} = \frac{1}{5} + \frac{1}{6} + \frac{1}{12} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 5, 6, and 12 is 60.

- Convert \( \frac{1}{5} \) to \( \frac{12}{60} \)
- Convert \( \frac{1}{6} \) to \( \frac{10}{60} \)
- Convert \( \frac{1}{12} \) to \( \frac{5}{60} \)

Now, add them together:

\[ \text{Total work rate} = \frac{12}{60} + \frac{10}{60} + \frac{5}{60} = \frac{27}{60} \]

### Step 3: Calculate the total time taken to complete the work.

The total work done is 1 (the whole work), and the rate of work done together is \( \frac{27}{60} \) per day. Therefore, the total time taken to complete the work is:

\[ \text{Time} = \frac{1}{\text{Total work rate}} = \frac{1}{\frac{27}{60}} = \frac{60}{27} \approx 2.22 \text{ days} \]

### Step 4: Calculate the share of work done by R.

Now we need to find out how much work R did during the time they worked together. R's work rate is \( \frac{1}{12} \) of the work per day. Therefore, the amount of work done by R in \( \frac{60}{27} \) days is:

\[ \text{Work done by R} = \text{R's rate} \times \text{Time} = \frac{1}{12} \times \frac{60}{27} = \frac{60}{324} = \frac{5}{27} \]

### Step 5: Calculate the total earnings and R's share.

The total earnings for the work done is Rs. 5400. R's share of the earnings is proportional to the work done by R:

\[ \text{R's share} = \text{Total earnings} \times \text{R's work fraction} = 5400 \times \frac{5}{27} \]

Calculating R's share:

\[ \text{R's share} = 5400 \times \frac{5}{27} = 5400 \div 27 \times 5 = 200 \times 5 = 1000 \]

### Final Answer:
R's share of the earnings is Rs. 1000.

---