Two natural numbers are in the ratio $3 : 5$ and their product is $2160$. The smaller of the numbers is

### Concept & Products of Ratios When multiplying numbers represented by a ratio (e.g., $ax$ and $bx$), the product will be in terms of $x^2$. Do not forget to take the square root to find the base multiplier. ### Step-by-Step Solution Given: * Ratio of two numbers = $3 : 5$ * Product of the numbers = $2160$ 1. Let the two numbers be $3x$ and $5x$. 2. Form the equation based on their product: $3x \times 5x = 2160$ 3. Simplify the equation: $15x^2 = 2160$ 4. Solve for $x^2$: $x^2 = \frac{2160}{15}$ $x^2 = 144$ 5. Solve for $x$: $x = \sqrt{144} = 12$ 6. Find the smaller number ($3x$): Smaller number = $3 \times 12 = 36$ ### Exam Strategy & Shortcut Divide the product by the product of the ratio parts: $\frac{2160}{3 \times 5} = \frac{2160}{15} = 144$. Then immediately take the square root ($\sqrt{144} = 12$) to find the multiplier. The smaller number is $3 \times 12 = 36$. ### Common Pitfall Setting up the product as $15x = 2160$ instead of $15x^2 = 2160$. This leads to $x = 144$ and absurdly large incorrect numbers. ### Final Answer Therefore, the correct answer is **36**.