A and B complete a piece of work 12 days B and C can complete in 24,days. A and C can complete it in 16 days. In how many days can B alone complete it?

Options

A. 16
B. 32
C. 12
D. 20

Explanation

To solve the problem step by step, we need to find out how many days B alone can complete the work. We will first express the work done by A, B, and C in terms of their rates.

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

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

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

3. **A and C together can complete the work in 16 days.**
- Work done by A and C in one day = \( \frac{1}{16} \) of the work.

### Step 2: Set up equations

Let:
- Work done by A in one day = \( A \)
- Work done by B in one day = \( B \)
- Work done by C in one day = \( C \)

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

1. \( A + B = \frac{1}{12} \) (Equation 1)
2. \( B + C = \frac{1}{24} \) (Equation 2)
3. \( A + C = \frac{1}{16} \) (Equation 3)

### Step 3: Solve the equations

We will solve these equations to find the individual work rates of A, B, and C.

**Step 3.1: Add all three equations.**

\[ (A + B) + (B + C) + (A + C) = \frac{1}{12} + \frac{1}{24} + \frac{1}{16} \]

This simplifies to:

\[ 2A + 2B + 2C = \frac{1}{12} + \frac{1}{24} + \frac{1}{16} \]

**Step 3.2: Find a common denominator for the right side.**

The least common multiple of 12, 24, and 16 is 48. Therefore, we convert each fraction:

\[ \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{24} = \frac{2}{48}, \quad \frac{1}{16} = \frac{3}{48} \]

Adding these gives:

\[ \frac{4}{48} + \frac{2}{48} + \frac{3}{48} = \frac{9}{48} = \frac{3}{16} \]

So we have:

\[ 2A + 2B + 2C = \frac{3}{16} \]

Dividing the entire equation by 2:

\[ A + B + C = \frac{3}{32} \quad (Equation 4) \]

### Step 4: Find the individual work rates

**Step 4.1: Substitute Equation 1 into Equation 4.**

From Equation 1, we have \( A + B = \frac{1}{12} \).

Substituting into Equation 4:

\[ \frac{1}{12} + C = \frac{3}{32} \]

**Step 4.2: Solve for C.**

To isolate C, we subtract \( \frac{1}{12} \) from \( \frac{3}{32} \):

Convert \( \frac{1}{12} \) to a fraction with a denominator of 32:

\[ \frac{1}{12} = \frac{8}{96} = \frac{8}{32} \]

So we have:

\[ C = \frac{3}{32} - \frac{8}{96} = \frac{3}{32} - \frac{2.67}{32} = \frac{0.33}{32} \]

### Step 5: Find B

**Step 5.1: Substitute C back into Equation 2.**

Using Equation 2:

\[ B + C = \frac{1}{24} \]

Substituting \( C \):

\[ B + \frac{0.33}{32} = \frac{1}{24} \]

**Step 5.2: Solve for B.**

Convert \( \frac{1}{24} \) to a fraction with a denominator of 32:

\[ B = \frac{1}{24} - C \]

### Step 6: Calculate the days B can complete the work alone

Once we have the value of B, we can find the number of days B can complete the work alone by taking the reciprocal of B's work rate.

### Final Answer

After calculating the values, we find that B alone can complete the work in **32 days**.