If the ratio of areas of two squares is 225 : 256, then the ratio of their perimeters is

### Concept & Linear Ratios The ratio of the perimeters of two squares is exactly the same as the ratio of their sides. The ratio of their sides is the square root of the ratio of their areas. ### Step-by-Step Solution * **Step 1:** Let the areas of the two squares be $A_1$ and $A_2$. We are given $\frac{A_1}{A_2} = \frac{225}{256}$. * **Step 2:** The area of a square is the side squared ($s^2$). So, $\frac{s_1^2}{s_2^2} = \frac{225}{256}$. * **Step 3:** Take the square root of both sides to find the ratio of their sides: $\frac{s_1}{s_2} = \frac{\sqrt{225}}{\sqrt{256}} = \frac{15}{16}$. * **Step 4:** The perimeter of a square is $4s$. The ratio of their perimeters is $\frac{P_1}{P_2} = \frac{4s_1}{4s_2} = \frac{s_1}{s_2}$. * **Step 5:** Therefore, the ratio of the perimeters is also $\frac{15}{16}$, or $15 : 16$. ### Exam Strategy & Shortcut Recognize immediately that perimeter is a 1D measure and area is a 2D measure. Simply take the square root of the area ratio ($\sqrt{225} : \sqrt{256}$) to get the 1D ratio directly. ### Common Pitfall A common mistake is forgetting to take the square root and selecting $225 : 256$, or reversing the order and selecting $16 : 15$. ### Final Answer Therefore, the correct answer is **15 : 16**.