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 of the rectangle increases by 75 sq cm. Find the original length of the rectangle in cm, keeping the “twice” condition and the 75 sq cm increase unchanged.

Introduction / Context:
This question tests algebraic modelling of rectangle area changes. You are given a relationship between length and breadth (length = 2 * breadth) and a change scenario that increases area. The right approach is to represent breadth as b and length as 2b, write original area and new area after modifications, then use the given area increase of 75 sq cm to form and solve an equation.

Given Data / Assumptions:

• Let breadth = b cm
• Length = 2b cm
• New length = 2b - 5
• New breadth = b + 5
• New area - original area = 75 sq cm

Concept / Approach:
Original area = (2b)*b = 2b^2. New area = (2b - 5)(b + 5). Expand, subtract original, set equal to 75, solve for b, then compute length = 2b.

Step-by-Step Solution:

Verification / Alternative check:
Original: l = 40, b = 20, area = 800. New: l = 35, b = 25, area = 875. Increase = 875 - 800 = 75 sq cm, matches exactly, so 40 cm is correct.

Why Other Options Are Wrong:

Common Pitfalls:
Common errors include using (l+5)(b-5) instead of the given changes, expanding incorrectly, or forgetting that length is twice breadth from the start. Another pitfall is solving for length directly without first expressing everything in one variable b.

Final Answer:
40 cm