If A and B together can complete a work in 18 days, A and C together in 12 days, and B and C together in 9 days, then B alone can do the work in:

Options

A. 18 days
B. 24 days
C. 30 days
D. 40 days

Answer

To solve the problem, we need to find out how long B alone can complete the work given the information about A and B, A and C, and B and C. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Work Rates**:
- Let the total work be represented as 1 unit of work.
- If A and B together can complete the work in 18 days, their combined work rate is:
\[ \text{Rate of (A + B)} = \frac{1}{18} \text{ work per day} \]

- If A and C together can complete the work in 12 days, their combined work rate is:
\[ \text{Rate of (A + C)} = \frac{1}{12} \text{ work per day} \]

- If B and C together can complete the work in 9 days, their combined work rate is:
\[ \text{Rate of (B + C)} = \frac{1}{9} \text{ work per day} \]

2. **Setting Up the Equations**:
- Let the work rates of A, B, and C be represented as \( a, b, \) and \( c \) respectively. We can write the following equations based on the combined work rates:
\[ a + b = \frac{1}{18} \quad \text{(1)} \]
\[ a + c = \frac{1}{12} \quad \text{(2)} \]
\[ b + c = \frac{1}{9} \quad \text{(3)} \]

3. **Adding the Equations**:
- Now, we will add all three equations:
\[ (a + b) + (a + c) + (b + c) = \frac{1}{18} + \frac{1}{12} + \frac{1}{9} \]
- This simplifies to:
\[ 2a + 2b + 2c = \frac{1}{18} + \frac{1}{12} + \frac{1}{9} \]

4. **Finding a Common Denominator**:
- The least common multiple (LCM) of 18, 12, and 9 is 36. We convert each fraction:
\[ \frac{1}{18} = \frac{2}{36}, \quad \frac{1}{12} = \frac{3}{36}, \quad \frac{1}{9} = \frac{4}{36} \]
- Adding these gives:
\[ \frac{2}{36} + \frac{3}{36} + \frac{4}{36} = \frac{9}{36} = \frac{1}{4} \]

5. **Simplifying the Equation**:
- Now we have:
\[ 2a + 2b + 2c = \frac{1}{4} \]
- Dividing everything by 2:
\[ a + b + c = \frac{1}{8} \quad \text{(4)} \]

6. **Finding Individual Work Rates**:
- Now we can find the individual work rates:
- From equation (1):
\[ c = \frac{1}{8} - (a + b) = \frac{1}{8} - \frac{1}{18} \]
To subtract, we need a common denominator (LCM of 8 and 18 is 72):
\[ c = \frac{9}{72} - \frac{4}{72} = \frac{5}{72} \]

- Substitute \( c \) back into equation (3):
\[ b + \frac{5}{72} = \frac{1}{9} \quad \Rightarrow \quad b = \frac{1}{9} - \frac{5}{72} \]
Again, using LCM of 9 and 72 (which is 72):
\[ b = \frac{8}{72} - \frac{5}{72} = \frac{3}{72} = \frac{1}{24} \]

7. **Finding the Time for B Alone**:
- Since \( b = \frac{1}{24} \), this means B can complete the work alone in:
\[ \text{Time taken by B} = \frac{1}{b} = 24 \text{ days} \]

### Final Answer:
B alone can complete the work in **24 days**.