A completes `(2)/(3)` of a certain job in 6 days. B can complete `(1)/(3)` of the same job in 8 days and C can complete `(3)/(4)` of the work in 12 days. All of them work together for 4 days and then A and C quit. How long will it take for B to complete the remaining work alone?

Options

A. 3.8 days
B. 3.33 days
C. 2.22 days
D. 4.3 days

Answer

To solve the problem step by step, we will first determine the work rates of A, B, and C, then calculate the total work done when they work together, and finally find out how long it takes for B to finish the remaining work alone.

### Step 1: Calculate the total time taken by A, B, and C to complete the entire job.

1. **A's work rate**:
- A completes \( \frac{2}{3} \) of the job in 6 days.
- Therefore, to complete the entire job, A would take:
\[ \text{Total time for A} = 6 \times \frac{3}{2} = 9 \text{ days} \]

2. **B's work rate**:
- B completes \( \frac{1}{3} \) of the job in 8 days.
- Therefore, to complete the entire job, B would take:
\[ \text{Total time for B} = 8 \times 3 = 24 \text{ days} \]

3. **C's work rate**:
- C completes \( \frac{3}{4} \) of the job in 12 days.
- Therefore, to complete the entire job, C would take:
\[ \text{Total time for C} = 12 \times \frac{4}{3} = 16 \text{ days} \]

### Step 2: Calculate the work done by A, B, and C in one day.

1. **A's work per day**:
\[ \text{A's work per day} = \frac{1}{9} \text{ of the job} \]

2. **B's work per day**:
\[ \text{B's work per day} = \frac{1}{24} \text{ of the job} \]

3. **C's work per day**:
\[ \text{C's work per day} = \frac{1}{16} \text{ of the job} \]

### Step 3: Calculate the total work done by A, B, and C together in 4 days.

1. **Combined work rate**:
\[ \text{Combined work rate} = \text{A's work per day} + \text{B's work per day} + \text{C's work per day} \]
\[ = \frac{1}{9} + \frac{1}{24} + \frac{1}{16} \]

To add these fractions, find a common denominator. The least common multiple (LCM) of 9, 24, and 16 is 144.

- Convert each fraction:
\[ \frac{1}{9} = \frac{16}{144}, \quad \frac{1}{24} = \frac{6}{144}, \quad \frac{1}{16} = \frac{9}{144} \]

- Now add them:
\[ \text{Combined work rate} = \frac{16 + 6 + 9}{144} = \frac{31}{144} \]

2. **Work done in 4 days**:
\[ \text{Work done in 4 days} = \text{Combined work rate} \times 4 = \frac{31}{144} \times 4 = \frac{124}{144} = \frac{31}{36} \text{ of the job} \]

### Step 4: Calculate the remaining work.

1. **Remaining work**:
\[ \text{Remaining work} = 1 - \frac{31}{36} = \frac{5}{36} \text{ of the job} \]

### Step 5: Calculate how long it will take for B to complete the remaining work alone.

1. **B's work rate**:
\[ \text{B's work per day} = \frac{1}{24} \text{ of the job} \]

2. **Time taken by B to complete the remaining work**:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{B's work per day}} = \frac{\frac{5}{36}}{\frac{1}{24}} = \frac{5}{36} \times 24 = \frac{120}{36} = \frac{10}{3} \text{ days} \approx 3.33 \text{ days} \]

### Final Answer:
B will take approximately \( 3.33 \) days to complete the remaining work alone.

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