To do a certain work, A and B work on alternate days, with B beginning the work on the first day. A can finish the work alone in 48 days. If the work gets completed in `11 1/3` days, then B alone can finish 4 times the same work in:

Options

A. 24 days
B. 32 days
C. 27 days
D. 30 days

Answer

To solve the problem, we need to determine how long it would take B to complete 4 times the work, given that A and B work on alternate days and the total work is completed in 11 1/3 days.

### Step-by-Step Solution:

1. **Determine the Work Done by A in One Day:**
- A can complete the work in 48 days.
- Therefore, the work done by A in one day = \( \frac{1}{48} \) of the work.

**Hint:** To find the work done in one day, take the reciprocal of the total days taken to complete the work.

2. **Determine the Work Done by B in One Day:**
- Let the work done by B in one day be \( \frac{1}{b} \).
- Since A and B work on alternate days, B works on the 1st, 3rd, 5th, etc., days, while A works on the 2nd, 4th, 6th, etc.

3. **Calculate the Total Days Worked:**
- The total time taken to complete the work is \( 11 \frac{1}{3} \) days, which can be converted to an improper fraction:
- \( 11 \frac{1}{3} = \frac{34}{3} \) days.

4. **Determine the Number of Days Each Works:**
- In \( 11 \frac{1}{3} \) days, there are 6 complete days (3 days for A and 3 days for B) and 1 extra day (B works on the 1st day).
- Therefore, A works for 5 days and B works for 6 days.

5. **Set Up the Equation for Total Work Done:**
- Total work done = Work done by A + Work done by B
- \( \text{Total Work} = 5 \times \frac{1}{48} + 6 \times \frac{1}{b} = 1 \) (since they complete the whole work)
- This simplifies to:
\[ \frac{5}{48} + \frac{6}{b} = 1 \]

6. **Solve for B's Work Rate:**
- Rearranging gives:
\[ \frac{6}{b} = 1 - \frac{5}{48} \]
- Convert \( 1 \) to a fraction with a denominator of 48:
\[ 1 = \frac{48}{48} \implies \frac{6}{b} = \frac{48 - 5}{48} = \frac{43}{48} \]
- Cross-multiplying gives:
\[ 6 \times 48 = 43b \implies 288 = 43b \implies b = \frac{288}{43} \]

7. **Calculate the Time B Takes to Complete 4 Times the Work:**
- If B can complete the work in \( \frac{288}{43} \) days, then to complete 4 times the work:
\[ \text{Time for B to complete 4 times the work} = 4 \times \frac{288}{43} = \frac{1152}{43} \text{ days} \]

### Final Answer:
B alone can finish 4 times the same work in \( \frac{1152}{43} \) days, which is approximately \( 26.74 \) days.