A and B can do `(4)/(5)`th and `(3)/(5)`th of a piece of work in 15 days and 10 days respectively. In how many days can A and B working together complete the work, if B worked for 5 days without A ?

Options

A. `6(2)/(17)`
B. `6(10)/(17)`
C. `6(3)/(17)`
D. `6(12)/(17)`

Explanation

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

### Step 1: Calculate A's one-day work
A can complete \( \frac{4}{5} \) of the work in 15 days. Therefore, the total work can be calculated as:
\[ \text{Total work} = 15 \times \frac{5}{4} = \frac{75}{4} \]
Now, A's one-day work is:
\[ A's \text{ one-day work} = \frac{1}{\text{Total days}} = \frac{4}{75} \]

### Step 2: Calculate B's one-day work
B can complete \( \frac{3}{5} \) of the work in 10 days. Therefore, the total work can be calculated as:
\[ \text{Total work} = 10 \times \frac{5}{3} = \frac{50}{3} \]
Now, B's one-day work is:
\[ B's \text{ one-day work} = \frac{1}{\text{Total days}} = \frac{3}{50} \]

### Step 3: Calculate the work done by B in 5 days
B works alone for 5 days. The work done by B in 5 days is:
\[ \text{Work done by B} = 5 \times B's \text{ one-day work} = 5 \times \frac{3}{50} = \frac{15}{50} = \frac{3}{10} \]

### Step 4: Calculate the remaining work
The total work is considered as 1 unit. The remaining work after B has worked for 5 days is:
\[ \text{Remaining work} = 1 - \frac{3}{10} = \frac{10}{10} - \frac{3}{10} = \frac{7}{10} \]

### Step 5: Calculate the combined one-day work of A and B
Now, we need to find out how much work A and B can do together in one day:
\[ \text{Combined one-day work} = A's \text{ one-day work} + B's \text{ one-day work} = \frac{4}{75} + \frac{3}{50} \]
To add these fractions, we need a common denominator. The LCM of 75 and 50 is 150:
\[ \frac{4}{75} = \frac{4 \times 2}{75 \times 2} = \frac{8}{150} \]
\[ \frac{3}{50} = \frac{3 \times 3}{50 \times 3} = \frac{9}{150} \]
Now, adding these:
\[ \text{Combined one-day work} = \frac{8}{150} + \frac{9}{150} = \frac{17}{150} \]

### Step 6: Calculate the number of days to complete the remaining work
Now, we need to find out how many days A and B will take to complete the remaining \( \frac{7}{10} \) of the work:
\[ \text{Days to complete remaining work} = \frac{\text{Remaining work}}{\text{Combined one-day work}} = \frac{\frac{7}{10}}{\frac{17}{150}} = \frac{7}{10} \times \frac{150}{17} = \frac{7 \times 150}{10 \times 17} = \frac{1050}{170} = \frac{105}{17} \]
This can be simplified to:
\[ \frac{105}{17} \approx 6.176 \text{ days} \]

### Final Answer
Thus, A and B together can complete the remaining work in approximately \( 6 \frac{1}{17} \) days.

---