P, Q and R are employed to do a job in 15 days for a total remuneratuion of rs. 12,000. P alone can do the work in 40 days and Q alone in 60days. If all of them work together and complete the work on time what is R's share?

Options

A. a. 4000
B. b. 4500
C. c. 5000
D. d. 5250

Answer

To solve the problem step by step, we will determine the work efficiencies of P, Q, and R, and then calculate R's share of the total remuneration.

### Step 1: Determine the work efficiencies of P and Q
- P can complete the job in 40 days, so P's work efficiency is:
\[ \text{Efficiency of P} = \frac{1}{40} \text{ (work per day)} \]
- Q can complete the job in 60 days, so Q's work efficiency is:
\[ \text{Efficiency of Q} = \frac{1}{60} \text{ (work per day)} \]

### Step 2: Calculate the combined efficiency of P and Q
- To find the combined efficiency of P and Q, we add their efficiencies:
\[ \text{Combined Efficiency of P and Q} = \frac{1}{40} + \frac{1}{60} \]
- To add these fractions, we need a common denominator, which is 120:
\[ \frac{1}{40} = \frac{3}{120}, \quad \frac{1}{60} = \frac{2}{120} \]
- Therefore,
\[ \text{Combined Efficiency of P and Q} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120} = \frac{1}{24} \]

### Step 3: Calculate the combined efficiency of P, Q, and R
- If P, Q, and R together can complete the job in 15 days, their combined efficiency is:
\[ \text{Combined Efficiency of P, Q, and R} = \frac{1}{15} \]

### Step 4: Find R's efficiency
- We know the combined efficiency of P and Q is \(\frac{1}{24}\), and the combined efficiency of P, Q, and R is \(\frac{1}{15}\). To find R's efficiency, we subtract the efficiency of P and Q from the combined efficiency:
\[ \text{Efficiency of R} = \frac{1}{15} - \frac{1}{24} \]
- To perform this subtraction, we need a common denominator, which is 120:
\[ \frac{1}{15} = \frac{8}{120}, \quad \frac{1}{24} = \frac{5}{120} \]
- Thus,
\[ \text{Efficiency of R} = \frac{8}{120} - \frac{5}{120} = \frac{3}{120} = \frac{1}{40} \]

### Step 5: Calculate the total work done and R's share
- The total remuneration for the job is Rs. 12,000. The total efficiency of all three workers is:
\[ \text{Total Efficiency} = \frac{1}{15} \]
- The total work done in terms of efficiency is:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} = \frac{1}{15} \times 15 = 1 \text{ (complete work)} \]
- Now, we calculate the share of R based on his efficiency:
- Total efficiency of all three = 8 (P: 3, Q: 2, R: 3)
- R's efficiency = 3
- R's share of the remuneration is calculated as follows:
\[ \text{R's Share} = \left(\frac{\text{R's Efficiency}}{\text{Total Efficiency}}\right) \times \text{Total Remuneration} \]
\[ \text{R's Share} = \left(\frac{3}{8}\right) \times 12000 = 4500 \]

### Final Answer
R's share of the remuneration is Rs. 4500.

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