A can build up a wall in 8 days while B can break it in 3 days. A has worked for 4 days and then B joined to work with A for another 2 days only. In how many days will A alone build up the remaining part of wall?
(a)`13 (1)/(3)` days
(b)`7 (1)/(3)` days
(c)`6 (1)/(3)` days
(d)7 days

Options

A. `13 (1)/(3)` days
B. `7 (1)/(3)` days
C. `6 (1)/(3)` days
D. 7 days

Answer

To solve the problem step by step, we will follow the given information and perform calculations accordingly.

### Step 1: Determine the work done by A and B
- A can build the wall in 8 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{8} \text{ of the wall} \]
- B can break the wall in 3 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in 1 day} = -\frac{1}{3} \text{ of the wall} \]
(Note: B's work is negative because he is breaking the wall.)

### Step 2: Calculate the total work in terms of units
To make calculations easier, we can consider the total work to build the wall as 24 units (the least common multiple of 8 and 3).
- Work done by A in one day:
\[ \text{A's work in one day} = \frac{24}{8} = 3 \text{ units} \]
- Work done by B in one day:
\[ \text{B's work in one day} = -\frac{24}{3} = -8 \text{ units} \]

### Step 3: Calculate the work done by A in the first 4 days
A works alone for 4 days:
\[ \text{Work done by A in 4 days} = 4 \times 3 = 12 \text{ units} \]

### Step 4: Calculate the work done by A and B together for 2 days
When B joins A, they work together for 2 days. The combined work done by A and B in one day is:
\[ \text{Combined work in one day} = 3 - 8 = -5 \text{ units} \]
Thus, in 2 days, the work done is:
\[ \text{Work done by A and B in 2 days} = 2 \times (-5) = -10 \text{ units} \]

### Step 5: Calculate the total work done so far
Now, we can calculate the total work done:
\[ \text{Total work done} = \text{Work done by A in 4 days} + \text{Work done by A and B in 2 days} \]
\[ \text{Total work done} = 12 + (-10) = 2 \text{ units} \]

### Step 6: Calculate the remaining work
The total work required to build the wall is 24 units. Therefore, the remaining work is:
\[ \text{Remaining work} = 24 - 2 = 22 \text{ units} \]

### Step 7: Calculate the time taken by A to finish the remaining work
Now, we need to find out how many days A will take to complete the remaining 22 units of work:
\[ \text{Days taken by A} = \frac{\text{Remaining work}}{\text{Work done by A in one day}} = \frac{22}{3} \]
Calculating this gives:
\[ \frac{22}{3} = 7 \frac{1}{3} \text{ days} \]

### Final Answer
Thus, A will take \( 7 \frac{1}{3} \) days to build the remaining part of the wall.