The area of a rectangle is thrice that of a square. If the length of the rectangle is 40 cm and its breadth is $\frac{3}{2}$ times that of the side of the square, then the side of the square is

### Concept & Algebraic Formulation of Areas By defining variables for the unknown side lengths, we can translate the relationships between the areas and dimensions of the square and rectangle into an algebraic equation. $$ \text{Area of Rectangle} = 3 \times \text{Area of Square} $$ ### Step-by-Step Solution * Let the side of the square be $x$ cm. Its area is $x^2 \text{ cm}^2$. * The length of the rectangle is given as $l = 40$ cm. * The breadth of the rectangle is $\frac{3}{2}$ times the side of the square: $b = \frac{3}{2}x$ cm. * The area of the rectangle is $l \times b = 40 \times \frac{3}{2}x = 60x \text{ cm}^2$. * According to the problem, the area of the rectangle is thrice the area of the square: $60x = 3(x^2)$. * Simplify the equation: $60x = 3x^2 \Rightarrow 20x = x^2$. * Since $x$ representing a length cannot be 0, divide both sides by $x$: $x = 20$ cm. ### Exam Strategy & Shortcut Instead of writing out full algebraic equations, you can quickly test the given options. If the side is 20 (Option b), the square area is 400. The rectangle breadth is $\frac{3}{2}(20) = 30$. The rectangle area is $40 \times 30 = 1200$. Since 1200 is exactly $3 \times 400$, Option (b) is verified instantly. ### Common Pitfall Forgetting to square the side of the square when representing its area, thereby writing $3x$ instead of $3x^2$, is a frequent and fatal calculation error. ### Final Answer Therefore, the correct answer is **20 cm**.