A and B can complete a task in 30 days when working together after A and B have been working together for 11 days, B is called away and A, all by himself completes the task in the next 28 days. Had A been working alone, the number of days taken by him to complete the task would have been :

Options

A. `33(3)/(19)`
B. `19(6)/2`
C. `44(4)/(19)`
D. none of these

Answer

To solve the problem step by step, we will follow the logic presented in the transcript and break it down into clear steps.

### Step 1: Determine the combined work rate of A and B
A and B can complete a task together in 30 days. Therefore, their combined work rate is:
\[ \text{Work rate of A + B} = \frac{1 \text{ task}}{30 \text{ days}} = \frac{1}{30} \text{ tasks per day} \]

### Step 2: Calculate the work done by A and B in 11 days
In 11 days, the amount of work completed by A and B together is:
\[ \text{Work done in 11 days} = 11 \times \frac{1}{30} = \frac{11}{30} \text{ tasks} \]

### Step 3: Determine the remaining work after 11 days
The total work is 1 task. The remaining work after A and B have worked together for 11 days is:
\[ \text{Remaining work} = 1 - \frac{11}{30} = \frac{30 - 11}{30} = \frac{19}{30} \text{ tasks} \]

### Step 4: Calculate the time taken by A to complete the remaining work
After 11 days, B leaves, and A completes the remaining work in 28 days. Therefore, A's work rate can be calculated as:
\[ \text{Work rate of A} = \frac{\text{Remaining work}}{\text{Time taken by A}} = \frac{\frac{19}{30}}{28} = \frac{19}{30 \times 28} = \frac{19}{840} \text{ tasks per day} \]

### Step 5: Calculate A's work rate in terms of total work
Since we know A's work rate, we can express it in terms of the total work done:
\[ \text{Total work done by A alone} = \text{Work rate of A} \times \text{Time taken by A alone} \]
Let \( x \) be the number of days A would take to complete the entire task alone. Therefore:
\[ \frac{1 \text{ task}}{x \text{ days}} = \frac{19}{840} \text{ tasks per day} \]
This implies:
\[ x = \frac{1}{\frac{19}{840}} = \frac{840}{19} \text{ days} \]

### Step 6: Calculate the value of \( x \)
Now we compute \( \frac{840}{19} \):
\[ 840 \div 19 = 44.21 \text{ days} \quad \text{(approximately)} \]

### Conclusion
Thus, the number of days taken by A to complete the task alone is approximately:
\[ \boxed{44 \frac{4}{19}} \text{ days} \]