A and B can do piece of work in 10h. B and C can do it in 15h , while A and C take 12h to complete the work .B independently can complete the work in

Options

A. 12h
B. 16h
C. 20h
D. 24h

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C based on the information given.

### Step-by-Step Solution:

1. **Determine the work rates of A + B, B + C, and A + C:**
- A + B can complete the work in 10 hours, so their combined work rate is:
\[ \text{Rate of A + B} = \frac{1}{10} \text{ work/hour} \]
- B + C can complete the work in 15 hours, so their combined work rate is:
\[ \text{Rate of B + C} = \frac{1}{15} \text{ work/hour} \]
- A + C can complete the work in 12 hours, so their combined work rate is:
\[ \text{Rate of A + C} = \frac{1}{12} \text{ work/hour} \]

2. **Add the three equations:**
\[ (A + B) + (B + C) + (A + C) = \frac{1}{10} + \frac{1}{15} + \frac{1}{12} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{10} + \frac{1}{15} + \frac{1}{12} \]

3. **Find a common denominator for the right side:**
The least common multiple (LCM) of 10, 15, and 12 is 60. We convert each fraction:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Thus,
\[ \frac{1}{10} + \frac{1}{15} + \frac{1}{12} = \frac{6 + 4 + 5}{60} = \frac{15}{60} = \frac{1}{4} \]

4. **Substituting back into the equation:**
\[ 2A + 2B + 2C = \frac{1}{4} \]
Dividing everything by 2 gives:
\[ A + B + C = \frac{1}{8} \]

5. **Now, we can express the individual work rates:**
- From \(A + B = \frac{1}{10}\), we can express C:
\[ C = (A + B + C) - (A + B) = \frac{1}{8} - \frac{1}{10} \]
- Finding a common denominator (40):
\[ \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{10} = \frac{4}{40} \]
\[ C = \frac{5}{40} - \frac{4}{40} = \frac{1}{40} \]

6. **Now substitute C back to find A and B:**
- From \(B + C = \frac{1}{15}\):
\[ B + \frac{1}{40} = \frac{1}{15} \]
\[ B = \frac{1}{15} - \frac{1}{40} \]
- Finding a common denominator (120):
\[ \frac{1}{15} = \frac{8}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
\[ B = \frac{8}{120} - \frac{3}{120} = \frac{5}{120} = \frac{1}{24} \]

7. **Calculate the time taken by B to complete the work alone:**
Since B's work rate is \(\frac{1}{24}\) work/hour, the time taken by B to complete the work alone is:
\[ \text{Time} = \frac{1}{\text{Rate}} = \frac{1}{\frac{1}{24}} = 24 \text{ hours} \]

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