A can do a piece of work in 12 days. B can do the same work in 8 days and C can do the same job in `4/5`th time required by both A and B. A and B work together for 3 days. Then C completes the job. How many complete days did C work?

Options

A. 8
B. 6
C. 3
D. None of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work rates of A and B
- A can complete the work in 12 days, so A's work rate is:
\[ \text{A's work rate} = \frac{1}{12} \text{ (work per day)} \]
- B can complete the work in 8 days, so B's work rate is:
\[ \text{B's work rate} = \frac{1}{8} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- The combined work rate of A and B is:
\[ \text{A + B's work rate} = \frac{1}{12} + \frac{1}{8} \]
- To add these fractions, find a common denominator (which is 24):
\[ \text{A + B's work rate} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} \]

### Step 3: Calculate the work done by A and B in 3 days
- In 3 days, A and B together will complete:
\[ \text{Work done by A and B in 3 days} = 3 \times \frac{5}{24} = \frac{15}{24} = \frac{5}{8} \]

### Step 4: Determine the remaining work
- The total work is considered as 1 (the whole job), so the remaining work after A and B worked for 3 days is:
\[ \text{Remaining work} = 1 - \frac{5}{8} = \frac{3}{8} \]

### Step 5: Determine the time taken by C to complete the remaining work
- C can do the work in \( \frac{4}{5} \) of the time required by both A and B. First, we need to find the time required by A and B together:
\[ \text{Time required by A and B together} = \frac{1}{\text{A + B's work rate}} = \frac{1}{\frac{5}{24}} = \frac{24}{5} \text{ days} \]
- Therefore, C's time to complete the work is:
\[ \text{C's time} = \frac{4}{5} \times \frac{24}{5} = \frac{96}{25} \text{ days} \]
- C's work rate is:
\[ \text{C's work rate} = \frac{1}{\frac{96}{25}} = \frac{25}{96} \text{ (work per day)} \]

### Step 6: Calculate the time taken by C to complete the remaining work
- To find out how many days C worked to complete the remaining \( \frac{3}{8} \) of the work:
\[ \text{Time taken by C} = \frac{\text{Remaining work}}{\text{C's work rate}} = \frac{\frac{3}{8}}{\frac{25}{96}} = \frac{3}{8} \times \frac{96}{25} = \frac{288}{200} = \frac{72}{50} = \frac{36}{25} \text{ days} \]
- Converting \( \frac{36}{25} \) to a mixed number gives:
\[ \frac{36}{25} = 1 \frac{11}{25} \text{ days} \]

### Final Answer
C worked for 1 complete day.