P and Q can do a piece of work in 12 days 16 days, resepetively. With the help of R, they completed the work in 5 days and earn Rs. 912. What is the share of R?

Options

A. Rs. 249
B. Rs. 247
C. Rs. 243
D. Rs. 245

Explanation

To solve the problem, we need to determine the share of R in the total earnings of Rs. 912 after P, Q, and R completed the work together in 5 days. Here’s a step-by-step solution:

### Step 1: Determine the work rates of P and Q
- P can complete the work in 12 days, so the work done by P in one day is:
\[ \text{Work rate of P} = \frac{1}{12} \]
- Q can complete the work in 16 days, so the work done by Q in one day is:
\[ \text{Work rate of Q} = \frac{1}{16} \]

### Step 2: Calculate the combined work rate of P and Q
- The combined work rate of P and Q in one day is:
\[ \text{Combined work rate of P and Q} = \frac{1}{12} + \frac{1}{16} \]
- To add these fractions, we need a common denominator. The least common multiple of 12 and 16 is 48.
\[ \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48} \]
- Therefore,
\[ \text{Combined work rate of P and Q} = \frac{4}{48} + \frac{3}{48} = \frac{7}{48} \]

### Step 3: Calculate the total work done by P and Q in 5 days
- In 5 days, the amount of work done by P and Q together is:
\[ \text{Work done by P and Q in 5 days} = 5 \times \frac{7}{48} = \frac{35}{48} \]

### Step 4: Determine the work done by R
- Since the total work is considered as 1 (or 100%), the work done by R in 5 days is:
\[ \text{Work done by R} = 1 - \frac{35}{48} = \frac{48}{48} - \frac{35}{48} = \frac{13}{48} \]

### Step 5: Calculate the earnings of P, Q, and R
- The total earnings for the work completed is Rs. 912.
- The share of earnings is proportional to the amount of work done by each person.
- The total work done by P and Q together is \(\frac{35}{48}\) and by R is \(\frac{13}{48}\).

### Step 6: Calculate the earnings of P and Q
- The earnings of P and Q together can be calculated as:
\[ \text{Earnings of P and Q} = \frac{35}{48} \times 912 = \frac{35 \times 912}{48} \]
- Simplifying this:
\[ \frac{35 \times 912}{48} = 665 \]

### Step 7: Calculate R's earnings
- R's share of the total earnings is:
\[ \text{Earnings of R} = 912 - 665 = 247 \]

### Final Answer
Thus, R's share of the earnings is Rs. 247.

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