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

Answer

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

### Step 1: Determine the work done by A and B
- **A's work rate**: A can build the wall in 8 days, so in one day, A can build \( \frac{1}{8} \) of the wall.
- **B's work rate**: B can break the wall in 3 days, so in one day, B can break \( \frac{1}{3} \) of the wall.

### Step 2: Calculate the work done by A in 4 days
- In 4 days, A will build:
\[ 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \]
So, A has built half of the wall.

### Step 3: Calculate the work done by A and B together in 2 days
- When B joins A, their combined work rate per day is:
\[ A's \ work \ rate + B's \ work \ rate = \frac{1}{8} - \frac{1}{3} \]
To combine these fractions, we need a common denominator, which is 24:
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{3} = \frac{8}{24} \]
So,
\[ A's \ work \ rate - B's \ work \ rate = \frac{3}{24} - \frac{8}{24} = -\frac{5}{24} \]
This means that in one day, they effectively remove \( \frac{5}{24} \) of the wall.

### Step 4: Calculate the work done by A and B together in 2 days
- In 2 days, they will remove:
\[ 2 \times -\frac{5}{24} = -\frac{10}{24} = -\frac{5}{12} \]
So, after 2 days of working together, they have removed \( \frac{5}{12} \) of the wall.

### Step 5: Calculate the total work done after 6 days
- Total work done after 6 days (4 days by A and 2 days by A and B together):
\[ \text{Work done by A} + \text{Work done by A and B} = \frac{1}{2} - \frac{5}{12} \]
To combine these fractions, we need a common denominator, which is 12:
\[ \frac{1}{2} = \frac{6}{12} \]
So,
\[ \frac{6}{12} - \frac{5}{12} = \frac{1}{12} \]
Thus, after 6 days, \( \frac{1}{12} \) of the wall remains.

### Step 6: Calculate how long it will take A to finish the remaining work
- A's work rate is \( \frac{1}{8} \) of the wall per day. To find out how many days A needs to finish \( \frac{1}{12} \) of the wall:
\[ \text{Let } x \text{ be the number of days.} \]
\[ x \times \frac{1}{8} = \frac{1}{12} \]
To solve for \( x \):
\[ x = \frac{1/12}{1/8} = \frac{8}{12} = \frac{2}{3} \]
So, A will take \( \frac{2}{3} \) of a day to finish the remaining part of the wall.

### Final Answer
A will take \( \frac{2}{3} \) of a day to build the remaining part of the wall.

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