Let's solve this step by step.

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

A and B together can complete the work in 3 days. Therefore, their combined work rate is:

$\text{Work rate of A and B together} = \frac{1}{3} \text{ work per day}$

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

Since A and B work together for 2 days, the amount of work completed in these 2 days is:

$\text{Work done by A and B in 2 days} = 2 \times \frac{1}{3} = \frac{2}{3} \text{ of the work}$

Step 3: Determine the remaining work.

After 2 days, $\frac{2}{3}$ of the work is completed. Therefore, the remaining work is:

$\text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3}$

Step 4: Determine the work rate of A alone.

Let's assume A can complete the work in $a$ days. Therefore, A's work rate is:

$\text{Work rate of A} = \frac{1}{a} \text{ work per day}$

Step 5: Calculate the time taken by A to complete the remaining work.

The remaining $\frac{1}{3}$ of the work is completed by A in 2 days after B left. Therefore:

$\frac{1}{3} = 2 \times \frac{1}{a}$

Solving for $a$:

$\frac{1}{3} = \frac{2}{a} \implies a = 6 \text{ days}$

So, A alone can complete the entire work in 6 days.

Step 6: Determine B's work rate.

Let $b$ be the number of days B alone would take to complete the work. The combined work rate of A and B is:

$\frac{1}{a} + \frac{1}{b} = \frac{1}{3}$

Substitute $a = 6$ into the equation:

$\frac{1}{6} + \frac{1}{b} = \frac{1}{3}$

Step 7: Solve for $b$.

$\frac{1}{b} = \frac{1}{3} - \frac{1}{6}$ $\frac{1}{b} = \frac{2 - 1}{6} = \frac{1}{6}$ $b = 6 \text{ days}$

Therefore, B alone could do the work in 6 days.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:

1. How long would it take A alone to complete half of the work?
2. If B worked alone for 2 days, how much of the work would be left?
3. What if A and B started together, but B left after 1 day? How many more days would A need to complete the work?
4. If both A and B work on the task, but A is twice as efficient as B, how long would they take to complete the work together?
5. How would the result change if B left after only 1 day instead of 2 days?

Tip: When solving work problems, it's often helpful to express work rates as fractions of the total work per unit time (e.g., work per day) to make the calculations easier.