A and B can do a certain piece of work in 8 days, B and C can do it in 12 days and C and A can do it in 24 days. How long would B take separately to do it ?

Answer

To solve the problem, we need to find out how long B would take to complete the work alone. We will start by determining the work rates of A, B, and C based on the information provided.

### Step 1: Determine the work rates of A, B, and C

1. **A and B together can complete the work in 8 days.**
- Work done by A and B in one day = \( \frac{1}{8} \) of the work.

2. **B and C together can complete the work in 12 days.**
- Work done by B and C in one day = \( \frac{1}{12} \) of the work.

3. **C and A together can complete the work in 24 days.**
- Work done by C and A in one day = \( \frac{1}{24} \) of the work.

### Step 2: Set up equations based on the work rates

Let:
- Work rate of A = \( a \)
- Work rate of B = \( b \)
- Work rate of C = \( c \)

From the information above, we can set up the following equations:

1. \( a + b = \frac{1}{8} \) (Equation 1)
2. \( b + c = \frac{1}{12} \) (Equation 2)
3. \( c + a = \frac{1}{24} \) (Equation 3)

### Step 3: Solve the equations

We can add all three equations together:

\[ (a + b) + (b + c) + (c + a) = \frac{1}{8} + \frac{1}{12} + \frac{1}{24} \]

This simplifies to:

\[ 2a + 2b + 2c = \frac{1}{8} + \frac{1}{12} + \frac{1}{24} \]

### Step 4: Find a common denominator

The least common multiple of 8, 12, and 24 is 24. We convert each fraction:

\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{12} = \frac{2}{24}, \quad \frac{1}{24} = \frac{1}{24} \]

Now, we can add them:

\[ \frac{3}{24} + \frac{2}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4} \]

### Step 5: Substitute back to find \( a + b + c \)

From the equation \( 2a + 2b + 2c = \frac{1}{4} \), we divide by 2:

\[ a + b + c = \frac{1}{8} \]

### Step 6: Find individual work rates

Now we can express \( c \) in terms of \( a \) and \( b \):

From Equation 1:
\[ c = \frac{1}{12} - b \]

Substituting \( c \) into Equation 3:

\[ \frac{1}{12} - b + a = \frac{1}{24} \]

Rearranging gives us:

\[ a - b = \frac{1}{24} - \frac{1}{12} = \frac{1}{24} - \frac{2}{24} = -\frac{1}{24} \]

So, \( a = b - \frac{1}{24} \).

### Step 7: Substitute \( a \) into Equation 1

Substituting \( a \) back into Equation 1:

\[ (b - \frac{1}{24}) + b = \frac{1}{8} \]

This simplifies to:

\[ 2b - \frac{1}{24} = \frac{3}{24} \]

Adding \( \frac{1}{24} \) to both sides:

\[ 2b = \frac{4}{24} = \frac{1}{6} \]

Dividing by 2:

\[ b = \frac{1}{12} \]

### Step 8: Find the time taken by B alone

Since \( b \) is the work rate of B, the time taken by B to complete the work alone is the reciprocal of \( b \):

\[ \text{Time taken by B} = \frac{1}{b} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Final Answer

B would take **12 days** to complete the work alone.

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