If each side of a square is increased by 50%, the ratio of the area of the resulting square to that of the given square is

### Concept & Proportional Area Scaling If the side of a square is multiplied by a certain factor, the area of the square is multiplied by the square of that factor. ### Step-by-Step Solution * **Step 1:** Let the original side of the square be $s$. The original area is $A_1 = s^2$. * **Step 2:** The side is increased by $50\%$. The new side is $s + 0.5s = 1.5s$. * **Step 3:** Calculate the area of the resulting square: $A_2 = (1.5s)^2 = 2.25s^2$. * **Step 4:** Find the requested ratio (Resulting Area to Given Area): $\frac{A_2}{A_1} = \frac{2.25s^2}{s^2} = 2.25$. * **Step 5:** Convert the decimal to a fraction: $2.25 = \frac{225}{100}$. * **Step 6:** Simplify the fraction by dividing numerator and denominator by 25: $\frac{9}{4}$. * **Step 7:** The ratio is $9 : 4$. ### Exam Strategy & Shortcut Assume the original side is $2$. Original area $= 4$. A $50\%$ increase on $2$ adds $1$, so the new side is $3$. New area $= 3^2 = 9$. The ratio of the new area to the old area is immediately $9 : 4$. ### Common Pitfall Choosing $4 : 9$ because of reading the ratio order backward. The question asks for the "resulting square" first, then the "given square". ### Final Answer Therefore, the correct answer is **9 : 4**.