P, Q and R can do a piece of work in 60 days, 100 days and 80 days respectively. They together work to finish the work and receive Rs. 2256. Then P will get

Options

A. Rs 576
B. Rs 752
C. Rs 960
D. Rs 564

Answer

To solve the problem step by step, we will first determine the work done by P, Q, and R in one day, then calculate their efficiencies, and finally distribute the total payment based on their efficiencies.

### Step 1: Determine the work done by P, Q, and R in one day.
- P can complete the work in 60 days, so in one day, P does:
\[ \text{Work done by P in one day} = \frac{1}{60} \]
- Q can complete the work in 100 days, so in one day, Q does:
\[ \text{Work done by Q in one day} = \frac{1}{100} \]
- R can complete the work in 80 days, so in one day, R does:
\[ \text{Work done by R in one day} = \frac{1}{80} \]

### Step 2: Find the total work done by P, Q, and R in one day.
To find the total work done by all three in one day, we need to add their daily work contributions:
\[ \text{Total work in one day} = \frac{1}{60} + \frac{1}{100} + \frac{1}{80} \]

To add these fractions, we first find the Least Common Multiple (LCM) of the denominators (60, 100, and 80). The LCM is 600.

Now we convert each fraction:
- For P:
\[ \frac{1}{60} = \frac{10}{600} \]
- For Q:
\[ \frac{1}{100} = \frac{6}{600} \]
- For R:
\[ \frac{1}{80} = \frac{7.5}{600} = \frac{15}{1200} = \frac{5}{400} = \frac{7.5}{600} \]

Now adding these:
\[ \text{Total work in one day} = \frac{10 + 6 + 7.5}{600} = \frac{23.5}{600} \]

### Step 3: Calculate the total work.
Since they complete the work together, we can set the total work to be equal to the LCM we calculated:
\[ \text{Total Work} = 600 \text{ units} \]

### Step 4: Calculate the efficiency of each worker.
- P's efficiency = 10 units
- Q's efficiency = 6 units
- R's efficiency = 7.5 units

### Step 5: Calculate the total efficiency.
\[ \text{Total efficiency} = 10 + 6 + 7.5 = 23.5 \text{ units} \]

### Step 6: Calculate the share of each worker.
Now, we need to distribute the total payment of Rs. 2256 based on their efficiencies.

- P's share:
\[ \text{P's share} = \left(\frac{10}{23.5}\right) \times 2256 \]

Calculating this:
\[ \text{P's share} = \frac{10 \times 2256}{23.5} = \frac{22560}{23.5} \approx 960 \]

### Conclusion:
Thus, P will receive Rs. 960.