A does `66(2)/(3)%` of work in 8 days and is replaced by B and B completes the remaining work. If whole work is completed in 14 days, then find the time taken by A and B together to complete the work.

Options

A. 3.6 days
B. 5.4 days
C. 6.8 days
D. 7.2 days

Answer

To solve the problem step-by-step, we will break down the information given and calculate the required values systematically.

### Step 1: Understand the Work Done by A
A does \(66 \frac{2}{3}\%\) of the work in 8 days. First, we convert this percentage into a fraction:
\[ 66 \frac{2}{3}\% = \frac{200}{3}\% = \frac{200}{3 \times 100} = \frac{2}{3} \]
This means A completes \(\frac{2}{3}\) of the work in 8 days.

### Step 2: Calculate the Total Time for A to Complete the Work Alone
If A completes \(\frac{2}{3}\) of the work in 8 days, we can find out how long it would take A to complete the entire work (1 unit of work) using the unitary method:
\[ \text{Time taken by A to complete 1 work} = 8 \text{ days} \times \frac{3}{2} = 12 \text{ days} \]

### Step 3: Calculate the Remaining Work Done by B
Since the total work is completed in 14 days, and A worked for 8 days, the time B worked is:
\[ \text{Time taken by B} = 14 \text{ days} - 8 \text{ days} = 6 \text{ days} \]
B completes the remaining work, which is:
\[ 1 - \frac{2}{3} = \frac{1}{3} \]

### Step 4: Calculate the Time for B to Complete the Whole Work Alone
If B completes \(\frac{1}{3}\) of the work in 6 days, we can find out how long it would take B to complete the entire work:
\[ \text{Time taken by B to complete 1 work} = 6 \text{ days} \times 3 = 18 \text{ days} \]

### Step 5: Calculate the Total Work in Terms of Efficiency
Now we know:
- A can complete the work in 12 days (efficiency = \(\frac{1}{12}\))
- B can complete the work in 18 days (efficiency = \(\frac{1}{18}\))

To find the combined efficiency of A and B:
\[ \text{Efficiency of A} = \frac{1}{12}, \quad \text{Efficiency of B} = \frac{1}{18} \]
To combine these, we find a common denominator:
\[ \text{LCM of 12 and 18} = 36 \]
Thus:
\[ \text{Efficiency of A} = \frac{3}{36}, \quad \text{Efficiency of B} = \frac{2}{36} \]
So,
\[ \text{Total Efficiency} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \]

### Step 6: Calculate the Time Taken by A and B Together to Complete the Work
Now, we can find the time taken by A and B together to complete the whole work:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{1}{\frac{5}{36}} = \frac{36}{5} = 7.2 \text{ days} \]

### Final Answer
The time taken by A and B together to complete the work is **7.2 days**.