A and B can complete a job in 24 days working together. A alone can complete it in 32 days. Both of them worked together for 8 days and then A left. The number of days B will take to complete the remaining job is:

64 days

The correct answer is Option (2) → 64 days

1. Calculate the Work Capacity of A and B

• A and B together can complete the job in 24 days.
• Their combined 1-day work = $\frac{1}{24}$
• A alone can complete the job in 32 days.
• A's 1-day work = $\frac{1}{32}$

2. Find B's 1-Day Work Capacity

To find how much work B does in a single day, subtract A's capacity from their combined capacity:

$\text{B's 1-day work} = \frac{1}{24} - \frac{1}{32}$

Finding the least common multiple (LCM) of 24 and 32 (which is 96):

$\text{B's 1-day work} = \frac{4}{96} - \frac{3}{96} = \frac{1}{96}$

This means B alone would take 96 days to complete the entire job.

3. Calculate the Work Done and Remaining Work

• Both worked together for 8 days.
• Work done in 8 days = $8 \times \frac{1}{24} = \frac{1}{3}$
• Remaining work = $1 - \frac{1}{3} = \frac{2}{3}$

4. Calculate the Time Taken by B for the Remaining Work

Now, we find how long it takes B to complete the remaining $\frac{2}{3}$ of the job:

$\text{Days} = \frac{\text{Remaining Work}}{\text{B's 1-day work}}$

$\text{Days} = \frac{2/3}{1/96}$

$\text{Days} = \frac{2}{3} \times 96 = 2 \times 32 = \mathbf{64 \text{ days}}$