The length of a rectangle is twice its breadth. If the length is decreased by 5 cm and the breadth is increased by 5 cm, the area increases by 75 cm^2. Find the original length of the rectangle in centimetres.

Introduction / Context: This question tests forming algebraic equations from area changes. When dimensions change, the new area can be expressed in terms of the original breadth. The rectangle relationship (length = 2*breadth) reduces the variables and helps solve the problem efficiently.

Given Data / Assumptions:

• Let breadth = b cm
• Original length = 2b cm
• New length = 2b - 5
• New breadth = b + 5
• Increase in area = 75 cm^2

Concept / Approach: Original area = (2b)*b = 2b^2. New area = (2b - 5)(b + 5). The increase condition gives: (2b - 5)(b + 5) - 2b^2 = 75. Solve for b and then compute length = 2b.

Step-by-Step Solution: Original area = 2b^2 New area = (2b - 5)(b + 5) Increase: (2b - 5)(b + 5) - 2b^2 = 75 Expand: (2b - 5)(b + 5) = 2b^2 + 10b - 5b - 25 = 2b^2 + 5b - 25 So (2b^2 + 5b - 25) - 2b^2 = 75 5b - 25 = 75 5b = 100 => b = 20 cm Original length = 2b = 40 cm

Verification / Alternative check: Original area = 40*20 = 800 cm^2. New dimensions: 35 and 25, new area = 35*25 = 875 cm^2. Increase = 875 - 800 = 75 cm^2, correct.

Why Other Options Are Wrong: 30 cm: would imply b = 15; the area change would not be 75. 50 cm: would imply b = 25; then new dimensions 45 and 30 produce a different increase. 60 cm and 70 cm: imply even larger b values and do not satisfy the exact +75 cm^2 condition.

Common Pitfalls: Applying +5 and -5 to the wrong dimension. Forgetting that length is twice breadth in the original rectangle. Not subtracting the original area when using the 'increase by 75' statement.

Final Answer: Original length = 40 cm