A work can be completed by P and Q in 12 days, and R in 15 days, R and P in 20 days. In how many days P alone can finish the work?

Options

A. 10 days
B. 20 days
C. 30 days
D. 60 days

Answer

To solve the problem step by step, we will first determine the efficiencies of P, Q, and R based on the information given.

### Step 1: Determine the total work in units
The work can be completed by:
- P and Q in 12 days
- Q and R in 15 days
- R and P in 20 days

To find the total work, we can take the least common multiple (LCM) of the days taken by each pair.

The LCM of 12, 15, and 20 is 60. This means the total work can be considered as 60 units.

### Step 2: Calculate the efficiencies of each pair
Now, we can calculate the efficiencies of the pairs:
- Efficiency of P and Q = Total work / Days taken = 60 / 12 = 5 units/day
- Efficiency of Q and R = Total work / Days taken = 60 / 15 = 4 units/day
- Efficiency of R and P = Total work / Days taken = 60 / 20 = 3 units/day

### Step 3: Set up equations for individual efficiencies
Let the efficiencies of P, Q, and R be represented as P, Q, and R respectively. We can set up the following equations based on the efficiencies calculated:
1. P + Q = 5 (from P and Q)
2. Q + R = 4 (from Q and R)
3. R + P = 3 (from R and P)

### Step 4: Solve the equations
Now, we can solve these equations to find the individual efficiencies.

From equation 1:
- Q = 5 - P

Substituting Q in equation 2:
- (5 - P) + R = 4
- R = 4 - (5 - P) = P - 1

Now substituting R in equation 3:
- (P - 1) + P = 3
- 2P - 1 = 3
- 2P = 4
- P = 2

### Step 5: Find efficiencies of Q and R
Now that we have P's efficiency:
- Q = 5 - P = 5 - 2 = 3
- R = P - 1 = 2 - 1 = 1

### Step 6: Calculate the time taken by P to finish the work alone
Now, we know P's efficiency is 2 units/day. To find out how many days P alone can finish the work:
- Total work = 60 units
- Time taken by P = Total work / P's efficiency = 60 / 2 = 30 days

Thus, P alone can finish the work in **30 days**.

### Final Answer
P alone can finish the work in **30 days**.

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