1. **State the problem:** We are given the first two terms of a geometric progression (GP): $a_1 = \frac{1}{4}$ and $a_2 = \frac{15}{2}$. We need to find the common ratio $r$ of this progression. 2. **Formula used:** In a geometric progression, each term is obtained by multiplying the previous term by the common ratio $r$. Thus, $$a_2 = a_1 \times r$$ 3. **Apply the formula:** Substitute the known values: $$\frac{15}{2} = \frac{1}{4} \times r$$ 4. **Solve for $r$:** Multiply both sides by 4 to isolate $r$: $$4 \times \frac{15}{2} = \cancel{4} \times \frac{1}{\cancel{4}} \times r$$ $$2 \times 15 = r$$ $$30 = r$$ 5. **Conclusion:** The common ratio of the geometric progression is $\boxed{30}$.