Rectangle with Length Twice Breadth — Area Change Scenario: 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.

Introduction / Context:
This is a classic algebraic geometry problem linking dimensional changes to area differences. The relationship among original dimensions, their changes, and the resulting area increase yields a linear equation for the unknown breadth, and hence the original length (given length is twice breadth).

Given Data / Assumptions:

• Original length L = 2B (B = original breadth)
• Modified dimensions: (L − 5) by (B + 5)
• Area increase = 75 cm^2
• All dimensions in centimetres

Concept / Approach:
Compute the new area and subtract the old area. The difference equals 75. Substitute L = 2B to reduce to a single variable equation in B. Solve B, then obtain L = 2B. This approach avoids guessing and ensures consistency with the percentage-free, exact arithmetic setup.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 20 cm is the breadth, not the length.
• 30 cm and 50 cm do not satisfy the derived relationship.
• 25 cm would imply non-integer breadth inconsistent with the solved equation.

Common Pitfalls:

• Expanding (2B − 5)(B + 5) incorrectly; watch signs in the cross term.
• Forgetting L = 2B when setting up the equation for B.

Final Answer:
40 cm.