P and Q can do a piece of work in 10 days, Q and R can do same work in 15 days, R and P can do the same work in 20 days. Then in how many days R will complete it alone.

Options

A. 115 days
B. 110 days
C. 130 days
D. 120 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 about their combined work rates.

### Step 1: Determine the total work
Let the total work be represented by the least common multiple (LCM) of the days taken by the pairs to complete the work. The pairs are:
- P and Q can do the work in 10 days.
- Q and R can do the work in 15 days.
- R and P can do the work in 20 days.

To find the LCM of 10, 15, and 20:
- The prime factorization of 10 is \(2 \times 5\).
- The prime factorization of 15 is \(3 \times 5\).
- The prime factorization of 20 is \(2^2 \times 5\).

The LCM is calculated by taking the highest power of each prime:
- For 2: \(2^2\)
- For 3: \(3^1\)
- For 5: \(5^1\)

Thus, \(LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\).

### Step 2: Calculate efficiencies
Now we can calculate the efficiencies of each pair:
- Efficiency of P and Q = Total work / Time = \(60 / 10 = 6\) units/day.
- Efficiency of Q and R = Total work / Time = \(60 / 15 = 4\) units/day.
- Efficiency of R and P = Total work / Time = \(60 / 20 = 3\) units/day.

Let:
- Efficiency of P = \(P\)
- Efficiency of Q = \(Q\)
- Efficiency of R = \(R\)

From the above, we can set up the following equations:
1. \(P + Q = 6\) (Equation 1)
2. \(Q + R = 4\) (Equation 2)
3. \(R + P = 3\) (Equation 3)

### Step 3: Solve the equations
Now we will add all three equations together:
\[ (P + Q) + (Q + R) + (R + P) = 6 + 4 + 3 \]
This simplifies to:
\[ 2P + 2Q + 2R = 13 \]
Dividing by 2:
\[ P + Q + R = \frac{13}{2} = 6.5 \quad (Equation 4) \]

### Step 4: Find individual efficiencies
Now we can find the individual efficiencies:
- From Equation 1: \(Q = 6 - P\)
- Substitute \(Q\) in Equation 2:
\[ (6 - P) + R = 4 \implies R = 4 - (6 - P) = P - 2 \]
- Substitute \(R\) in Equation 3:
\[ (P - 2) + P = 3 \implies 2P - 2 = 3 \implies 2P = 5 \implies P = 2.5 \]
- Now substitute \(P\) back to find \(Q\) and \(R\):
\[ Q = 6 - 2.5 = 3.5 \]
\[ R = P - 2 = 2.5 - 2 = 0.5 \]

### Step 5: Calculate the time taken by R to complete the work alone
To find the time taken by R to complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of R}} = \frac{60}{0.5} = 120 \text{ days} \]

### Final Answer
R will complete the work alone in **120 days**.

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