The ratio of the ages of A to B two years ago was $13:15$. After $5$ years from now, B's age will be $4$ years more than A's age. What is the present age of B (in years)?

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**Solution:**

1. Let the present ages of A and B be $x$ and $y$ respectively.

2. Two years ago, their ages were $(x - 2)$ and $(y - 2)$.

3. Given the ratio of their ages two years ago:
   $$\frac{x - 2}{y - 2} = \frac{13}{15}$$
   Cross-multiplying:
   $$15(x - 2) = 13(y - 2)$$
   $$15x - 30 = 13y - 26$$
   Rearranging:
   $$15x - 13y = 4 \quad \quad (1)$$

4. After 5 years, B's age will be $y + 5$ and A's age will be $x + 5$.

5. Given that B's age will be 4 years more than A's age at that time:
   $$y + 5 = (x + 5) + 4$$
   Simplifying:
   $$y + 5 = x + 9$$
   $$y - x = 4 \quad \quad (2)$$

6. From equation (2), express $y$ in terms of $x$:
   $$y = x + 4$$

7. Substitute $y = x + 4$ into equation (1):
   $$15x - 13(x + 4) = 4$$
   $$15x - 13x - 52 = 4$$
   $$2x - 52 = 4$$
   $$2x = 56$$
   $$x = 28$$

8. Using $y = x + 4$:
   $$y = 28 + 4 = 32$$

**Answer:** The present age of B is $32$ years.

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