The area of a rectangle is four times the area of a square. The length of the rectangle is 90 cm and the breadth of the rectangle is $\frac{2}{3}\text{rd}$ the side of the square. What is the side of the square? (Bank Recruitment, 2008)

### Concept & Setting up Algebraic Ratios Translate the word problem into mathematical expressions by assigning a variable to the side of the square and expressing all other given values in terms of this variable to form an equation. $$ \text{Area of Rectangle} = 4 \times \text{Area of Square} $$ ### Step-by-Step Solution * Let the side of the square be $x$ cm. The area of the square is $x^2 \text{ cm}^2$. * The length of the rectangle is $l = 90$ cm. * The breadth of the rectangle is $\frac{2}{3}$ the side of the square: $b = \frac{2}{3}x$ cm. * The area of the rectangle is $l \times b = 90 \times \frac{2}{3}x = 60x \text{ cm}^2$. * We are given that the area of the rectangle is four times the area of the square: $60x = 4x^2$. * Simplify the equation by dividing both sides by $4x$ (since length $x \neq 0$): $15 = x$. * The side of the square is 15 cm. * Checking the given options: (a) 9 cm, (b) 10 cm, (c) 20 cm, (d) Cannot be determined. None of these match the calculated side of 15 cm. ### Exam Strategy & Shortcut You can quickly solve the mental math: $90 \times \frac{2}{3}x = 4x^2 \Rightarrow 60x = 4x^2 \Rightarrow x = 15$. Scanning the options, 15 is evidently missing, making "None of these" the clear and obvious choice. Do not doubt your calculation just because the explicit number isn't listed. ### Common Pitfall Students often second-guess themselves when their calculated answer is not among the explicit numeric options and might pick a closely related incorrect option or mistakenly choose "Cannot be determined" out of confusion. ### Final Answer Therefore, the correct answer is **None of these**.