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 6/25`
C. `44 4/19`
D. none of these

Answer

To solve the problem, we will first determine the work rates of A and B, then find out how long A would take to complete the task alone.

### Step 1: Determine the combined work rate of A and B

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

### Step 2: Calculate the work done by A and B in 11 days

In 11 days, A and B together would complete:
\[ \text{Work done in 11 days} = 11 \times \frac{1}{30} = \frac{11}{30} \text{ of the task} \]

### Step 3: Determine the remaining work after 11 days

The remaining work after A and B have worked together for 11 days is:
\[ \text{Remaining work} = 1 - \frac{11}{30} = \frac{30 - 11}{30} = \frac{19}{30} \text{ of the task} \]

### Step 4: Calculate the work done by A alone in 28 days

A completes the remaining \(\frac{19}{30}\) of the task in 28 days. Therefore, A's work rate can be calculated as:
\[ \text{Work rate of A} = \frac{\frac{19}{30}}{28} = \frac{19}{840} \text{ (tasks per day)} \]

### Step 5: Determine the work rate of B

Since we know the combined work rate of A and B is \(\frac{1}{30}\), we can find B's work rate:
\[ \text{Work rate of B} = \text{Work rate of A and B} - \text{Work rate of A} \]
\[ \text{Work rate of B} = \frac{1}{30} - \frac{19}{840} \]
To perform this subtraction, we need a common denominator. The least common multiple of 30 and 840 is 840.
\[ \frac{1}{30} = \frac{28}{840} \]
Now, substituting:
\[ \text{Work rate of B} = \frac{28}{840} - \frac{19}{840} = \frac{9}{840} = \frac{1}{93.33} \text{ (tasks per day)} \]

### Step 6: Calculate the number of days A would take to complete the task alone

To find out how long A would take to complete the entire task alone, we use A's work rate:
\[ \text{Days taken by A alone} = \frac{1 \text{ task}}{\frac{19}{840} \text{ tasks per day}} = \frac{840}{19} \approx 44.21 \text{ days} \]

Thus, the number of days taken by A to complete the task alone is approximately 44.21 days.

### Final Answer:
A would take approximately 44.21 days to complete the task alone.

---