The area of a rectangle is three times the area of a square. The rectangle’s length is 40 cm and its breadth equals (3/2) times the side of the square. Find the side of the square (in cm).

Introduction / Context:This problem links the area of a rectangle to the area of a square and gives a proportional relationship between the rectangle’s breadth and the square’s side. Because a rectangle’s area is length * breadth while a square’s area is side^2, we can form an equation and solve for the unknown side of the square directly.

Given Data / Assumptions:

• Rectangle area A_rect = 3 * A_square.
• Rectangle length L = 40 cm.
• Rectangle breadth B = (3/2) * s, where s is the square’s side.
• Square area A_square = s^2.

Concept / Approach:Use area definitions. Since A_rect = L * B and A_rect = 3 * s^2, substitute L and B in terms of s to obtain a single equation in s. Solve for s, ensuring s > 0.

Step-by-Step Solution:A_rect = L * B = 40 * (3/2 * s) = 60sBut A_rect = 3 * s^2Therefore 3s^2 = 60sFor s > 0, divide by s: 3s = 60 ⇒ s = 20 cm

Verification / Alternative check:Square area = 20^2 = 400 cm^2. Rectangle breadth = (3/2)*20 = 30 cm; rectangle area = 40 * 30 = 1200 cm^2. Ratio 1200 : 400 = 3 : 1, as required.

Why Other Options Are Wrong:15, 30, and 60 cm do not satisfy 3s^2 = 40 * (3/2 s) consistently; they would give a mismatch in the triple-area condition.

Common Pitfalls:Confusing “breadth is (3/2) of side” with “breadth is (2/3) of side,” or erroneously cubing instead of squaring when moving between linear and area measures.

Final Answer:20