1. **State the problem:** We are given two numbers whose product is 36 and their ratio is 1:4. We need to find the smaller number. 2. **Set variables:** Let the smaller number be $x$. Since the ratio is 1:4, the larger number is $4x$. 3. **Write the equation for the product:** $$x \times 4x = 36$$ 4. **Simplify the equation:** $$4x^2 = 36$$ 5. **Divide both sides by 4:** $$\cancel{4}x^2 = \frac{36}{\cancel{4}}$$ $$x^2 = 9$$ 6. **Take the square root of both sides:** $$x = \pm 3$$ 7. **Interpret the result:** Since $x$ represents a number in a ratio and is positive, we take $x = 3$. 8. **Answer:** The smaller number is **3**.