P can complete a job in 60 days while Q can compelete in 90 days. With the help of R, They completed in 20 days. If they earned a total of Rs. 3600, then find R is share (in Rs.)

Options

A. 1360
B. 1600
C. 1480
D. 1540

Explanation

To solve the problem step by step, we will first determine the work done by P, Q, and R, and then calculate R's share of the earnings.

### Step 1: Calculate the work done by P and Q
- P can complete the job in 60 days. Therefore, P's work per day = 1/60 of the job.
- Q can complete the job in 90 days. Therefore, Q's work per day = 1/90 of the job.

### Step 2: Find the combined work of P and Q
- The combined work of P and Q in one day = (1/60 + 1/90).
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 60 and 90 is 180.
- Convert the fractions:
- P's work = 1/60 = 3/180
- Q's work = 1/90 = 2/180
- Therefore, combined work of P and Q = (3/180 + 2/180) = 5/180 = 1/36 of the job per day.

### Step 3: Calculate the total work done by P, Q, and R together
- Let R's work per day be denoted as R.
- Together, they completed the job in 20 days, so their combined work per day is:
- Total work done in 20 days = 1 job, hence work done per day = 1/20 of the job.
- Therefore, we can set up the equation:
- (1/36 + R) = 1/20.

### Step 4: Solve for R's work
- To solve for R, we need to find a common denominator for 36 and 20, which is 180.
- Rewrite the equation:
- (5/180 + R) = (9/180).
- Subtract (5/180) from both sides:
- R = (9/180 - 5/180) = 4/180 = 1/45 of the job per day.

### Step 5: Calculate the total efficiency of P, Q, and R
- The efficiencies are:
- P's efficiency = 1/60
- Q's efficiency = 1/90
- R's efficiency = 1/45
- Convert these to a common unit:
- P's efficiency = 3 units (in 180)
- Q's efficiency = 2 units (in 180)
- R's efficiency = 4 units (in 180)
- Total efficiency = 3 + 2 + 4 = 9 units.

### Step 6: Calculate R's share of the earnings
- Total earnings = Rs. 3600.
- R's share of the total efficiency = (R's efficiency / Total efficiency) * Total earnings.
- R's share = (4/9) * 3600 = 1600.

### Final Answer
R's share is Rs. 1600.

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