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?

Options

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

Explanation

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

### Step 1: Determine the work done by A in 4 days
A can build a wall in 8 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in 1 day} = \frac{1}{8} \]
In 4 days, the work done by A is:
\[ \text{Work done by A in 4 days} = 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]
**Hint:** Remember that work done is a fraction of the total work based on the number of days worked.

### Step 2: Determine the work done by B in one day
B can break the wall in 3 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in 1 day} = -\frac{1}{3} \]
(The negative sign indicates that B is breaking down the wall.)

### Step 3: Calculate the combined work done by A and B in one day
The combined work done by A and B in one day is:
\[ \text{Work done by A and B in 1 day} = \frac{1}{8} - \frac{1}{3} \]
To perform this subtraction, we need a common denominator, which is 24:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{3} = \frac{8}{24} \]
Thus,
\[ \text{Work done by A and B in 1 day} = \frac{3}{24} - \frac{8}{24} = -\frac{5}{24} \]

### Step 4: Calculate the work done by A and B together in 2 days
Now, we calculate the work done by A and B together in 2 days:
\[ \text{Work done by A and B in 2 days} = 2 \times \left(-\frac{5}{24}\right) = -\frac{10}{24} = -\frac{5}{12} \]

### Step 5: Calculate the total work done after 6 days
After A works alone for 4 days and then A and B work together for 2 days, the total work done is:
\[ \text{Total work done} = \frac{1}{2} - \frac{5}{12} \]
To perform this subtraction, we convert \(\frac{1}{2}\) to have a common denominator of 12:
\[ \frac{1}{2} = \frac{6}{12} \]
Thus,
\[ \text{Total work done} = \frac{6}{12} - \frac{5}{12} = \frac{1}{12} \]

### Step 6: Determine the remaining work
The total work of the wall is considered as 1 (the whole wall). Therefore, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{12} = \frac{11}{12} \]

### Step 7: Calculate the time A will take to complete the remaining work
A can complete \(\frac{1}{8}\) of the wall in one day. To find out how many days A will take to complete \(\frac{11}{12}\) of the wall:
\[ \text{Days required by A} = \frac{\text{Remaining work}}{\text{Work done by A in 1 day}} = \frac{\frac{11}{12}}{\frac{1}{8}} = \frac{11}{12} \times 8 = \frac{88}{12} = \frac{22}{3} \]
This means A will take \(\frac{22}{3}\) days, which is approximately 7 days and 1 hour.

### Final Answer
A will take \(\frac{22}{3}\) days to build the remaining part of the wall.

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