Let's solve the problem step by step.

Step 1: Work done by A in one day

If A can complete the work in 15 days, then A's one-day work is:

$\text{A's one-day work} = \frac{1}{15}$

Step 2: Work done by A and B together in one day

If A and B together can complete the work in 7.5 days, their one-day work is:

$\text{(A + B)'s one-day work} = \frac{1}{7.5} = \frac{2}{15}$

Step 3: Work done by B alone in one day

The one-day work of B alone can be calculated by subtracting A's one-day work from the combined one-day work of A and B:

$\text{B's one-day work} = \text{(A + B)'s one-day work} - \text{A's one-day work}$

Substitute the values:

$\text{B's one-day work} = \frac{2}{15} - \frac{1}{15} = \frac{1}{15}$

Step 4: Time taken by B to complete the work alone

If B's one-day work is $\frac{1}{15}$, then the total time B will take to complete the work is the reciprocal of B's one-day work:

$\text{Time taken by B alone} = \frac{1}{\frac{1}{15}} = 15 \text{ days}$

Final Answer:

B alone will take 15 days to complete the work.

---

Questions for Better Understanding:

1. Can you explain why the one-day work method is effective for such problems?
2. What happens if we change the time for A to 20 days instead of 15 days?
3. What if A and B together took 10 days instead of 7.5 days? How would it affect B's time?
4. How can we solve this problem using the concept of rates or algebra?
5. What are the key steps to ensure accuracy in work and time problems?

---

Tip:

Always write the given information clearly and calculate one-day work for each entity. This method simplifies complex work-and-time problems.