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 $x$. Then the larger number is $4x$ because the ratio is $1:4$. 3. **Write the equation for the product:** The product of the two numbers is given by $$x \times 4x = 36$$ 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. **Solve for $x$ by taking the square root:** $$x = \pm 3$$ 7. **Interpret the solution:** Since $x$ represents the smaller number, and numbers in ratio problems are usually positive, we take the positive value: $$x = 3$$ **Final answer:** The smaller number is **3**.