A can do a piece of work in 8 days which B can destroy in 3 days. A has worked for 6 days, during the last 2 days of which B has been destroying: how many days must A now work alone to complete the Work ?

Options

A. 7 days
B. `7(1)/(3)` days
C. `7(2)/(3)` days
D. 8 days

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the total work done by A and B
A can complete the work in 8 days, which means A's work rate is:
\[ \text{Work rate of A} = \frac{1}{8} \text{ work per day} \]

B can destroy the work in 3 days, which means B's work rate is:
\[ \text{Work rate of B} = -\frac{1}{3} \text{ work per day} \]

### Step 2: Find the least common multiple (LCM) of the days
To find a common time frame for A and B, we need to calculate the LCM of 8 and 3:
\[ \text{LCM}(8, 3) = 24 \text{ days} \]

### Step 3: Calculate total work in terms of LCM
In 24 days:
- A would complete:
\[ \text{Total work by A} = 24 \times \frac{1}{8} = 3 \text{ units of work} \]
- B would destroy:
\[ \text{Total work by B} = 24 \times \left(-\frac{1}{3}\right) = -8 \text{ units of work} \]

### Step 4: Calculate the net work done in 24 days
The net work done when both A and B work together for 24 days is:
\[ \text{Net work} = 3 - 8 = -5 \text{ units of work} \]
This means that in 24 days, A and B together can effectively complete:
\[ \text{Effective work done} = 3 + 8 = 11 \text{ units of work} \]

### Step 5: Calculate work done by A in 6 days
A has worked for 6 days. In those 6 days, A has completed:
\[ \text{Work done by A in 6 days} = 6 \times \frac{1}{8} = \frac{6}{8} = \frac{3}{4} \text{ units of work} \]

### Step 6: Calculate work done by B in the last 2 days
During the last 2 days, B has been destroying work:
\[ \text{Work destroyed by B in 2 days} = 2 \times \left(-\frac{1}{3}\right) = -\frac{2}{3} \text{ units of work} \]

### Step 7: Calculate the total work done after 6 days
Now, we can find the total work done after 6 days:
\[ \text{Total work done} = \frac{3}{4} - \frac{2}{3} \]
To perform this calculation, we need a common denominator, which is 12:
\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{2}{3} = \frac{8}{12} \]
Thus,
\[ \text{Total work done} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12} \text{ units of work} \]

### Step 8: Calculate remaining work
The total work is 1 unit (since A can complete it in 8 days, and B can destroy it in 3 days). Therefore, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{12} = \frac{11}{12} \text{ units of work} \]

### Step 9: Calculate how many days A needs to work alone
A's work rate is \(\frac{1}{8}\) units per day. To find out how many days A needs to work alone to complete the remaining work:
\[ \text{Days needed} = \frac{\text{Remaining work}}{\text{Work rate of A}} = \frac{\frac{11}{12}}{\frac{1}{8}} = \frac{11}{12} \times 8 = \frac{88}{12} = \frac{22}{3} \text{ days} \]
This can be converted to:
\[ \frac{22}{3} \text{ days} = 7 \frac{1}{3} \text{ days} \]

### Final Answer
A must work alone for \(7 \frac{1}{3}\) days to complete the work.

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