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

Options

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

Answer

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

### Step 1: Define the work rates
Let the work done by P in one day be \( P \), by Q be \( Q \), and by R be \( R \).

From the problem, we have:
1. P and Q together can complete the work in 12 days:
\[ P + Q = \frac{1}{12} \quad \text{(1)} \]
2. Q and R together can complete the work in 15 days:
\[ Q + R = \frac{1}{15} \quad \text{(2)} \]
3. R and P together can complete the work in 20 days:
\[ R + P = \frac{1}{20} \quad \text{(3)} \]

### Step 2: Add the equations
Now, we will add equations (1), (2), and (3):
\[ (P + Q) + (Q + R) + (R + P) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2P + 2Q + 2R = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Find a common denominator
To add the fractions on the right side, we need a common denominator. The least common multiple of 12, 15, and 20 is 60.

Converting each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Adding these gives:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 4: Simplify the equation
Now we substitute back into our equation:
\[ 2P + 2Q + 2R = \frac{1}{5} \]
Dividing everything by 2:
\[ P + Q + R = \frac{1}{10} \quad \text{(4)} \]

### Step 5: Isolate P
Now we can find P by using equation (4) and one of the previous equations. Let's use equation (2) to isolate P:
\[ P = (P + Q + R) - (Q + R) \]
Substituting from (4) and (2):
\[ P = \frac{1}{10} - \frac{1}{15} \]

### Step 6: Find a common denominator for P
The common denominator for 10 and 15 is 30:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
So,
\[ P = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \]

### Step 7: Calculate the days P alone can finish the work
Since \( P \) represents the work done by P in one day, if \( P = \frac{1}{30} \), it means P can complete the work alone in 30 days.

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

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