A and B can do a work in 10 days .B and C can do the same work in 15 days . C and A can complete the same work in 20 days . In how many days can C alone complete the work ?

Options

A. 120 days
B. 118 days
C. 115 days
D. 110 days

Answer

To solve the problem, we need to find out how many days C alone can complete the work given the efficiencies of A, B, and C working together in pairs. Let's break it down step by step.

### Step 1: Understand the given information
- A and B can complete the work in 10 days.
- B and C can complete the work in 15 days.
- C and A can complete the work in 20 days.

### Step 2: Calculate the total work
We can assume the total work to be a common multiple of the days taken by each pair. The least common multiple (LCM) of 10, 15, and 20 is 60. Thus, we can say:
- Total work = 60 units.

### Step 3: Calculate the efficiencies of each pair
1. **Efficiency of A and B**:
\[ A + B = \frac{60 \text{ units}}{10 \text{ days}} = 6 \text{ units/day} \]

2. **Efficiency of B and C**:
\[ B + C = \frac{60 \text{ units}}{15 \text{ days}} = 4 \text{ units/day} \]

3. **Efficiency of C and A**:
\[ C + A = \frac{60 \text{ units}}{20 \text{ days}} = 3 \text{ units/day} \]

### Step 4: Set up the equations
From the above calculations, we have:
1. \( A + B = 6 \) (Equation 1)
2. \( B + C = 4 \) (Equation 2)
3. \( C + A = 3 \) (Equation 3)

### Step 5: Solve the equations
To find the individual efficiencies, we can add all three equations:
\[ (A + B) + (B + C) + (C + A) = 6 + 4 + 3 \]
This simplifies to:
\[ 2A + 2B + 2C = 13 \implies A + B + C = \frac{13}{2} = 6.5 \text{ units/day} \]

### Step 6: Find the efficiency of C
Now, we can find C's efficiency using the total efficiency:
\[ C = (A + B + C) - (A + B) = 6.5 - 6 = 0.5 \text{ units/day} \]

### Step 7: Calculate the time taken by C alone
To find the time taken by C to complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of C}} = \frac{60 \text{ units}}{0.5 \text{ units/day}} = 120 \text{ days} \]

### Conclusion
C alone can complete the work in **120 days**.

---