A can build up a structure in 8 days and 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 builed up the remaining part of the structure ?

Options

A. 10 days
B. 9 days
C. `7 1/3` days
D. 13 days

Answer

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

### Step 1: Determine the work done by A and B
- A can complete the structure in 8 days, so A's work rate is \( \frac{1}{8} \) of the structure per day.
- B can break the structure in 3 days, so B's work rate is \( -\frac{1}{3} \) of the structure per day (negative because B is breaking it down).

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

### Step 3: Calculate the work done by A and B together in 2 days
- The combined work rate of A and B is:
\[ \text{Work rate of A and B} = \frac{1}{8} - \frac{1}{3} \]
To combine these fractions, find a common denominator (which is 24):
\[ \frac{1}{8} = \frac{3}{24}, \quad \frac{1}{3} = \frac{8}{24} \]
Thus,
\[ \text{Work rate of A and B} = \frac{3}{24} - \frac{8}{24} = -\frac{5}{24} \]
- In 2 days, A and B together will complete:
\[ \text{Work done by A and B} = 2 \times -\frac{5}{24} = -\frac{10}{24} = -\frac{5}{12} \]
This means they have broken down \( \frac{5}{12} \) of the structure.

### Step 4: Calculate the remaining structure
- Initially, A built \( \frac{1}{2} \) of the structure. After A and B worked together, the total structure completed is:
\[ \text{Total work done} = \frac{1}{2} - \frac{5}{12} \]
Convert \( \frac{1}{2} \) to a fraction with a denominator of 12:
\[ \frac{1}{2} = \frac{6}{12} \]
Now calculate:
\[ \text{Remaining work} = \frac{6}{12} - \frac{5}{12} = \frac{1}{12} \]

### Step 5: Calculate the time A will take to complete the remaining work
- A's work rate is \( \frac{1}{8} \) of the structure per day. To find out how many days A will take to complete \( \frac{1}{12} \) of the structure, set up the equation:
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Work rate of A}} = \frac{\frac{1}{12}}{\frac{1}{8}} = \frac{1}{12} \times \frac{8}{1} = \frac{8}{12} = \frac{2}{3} \]
Thus, A will take \( \frac{2}{3} \) of a day to complete the remaining part of the structure.

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

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