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.
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A20 cm
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B30 cm
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C40 cm
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D50 cm
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E25 cm
Answer
Correct Answer: 40 cm
Explanation
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:
Original area A = L * B = 2B^2.New area A' = (L − 5)(B + 5) = (2B − 5)(B + 5) = 2B^2 + 5B − 25.Increase: A' − A = (2B^2 + 5B − 25) − 2B^2 = 5B − 25 = 75.Solve 5B − 25 = 75 ⇒ 5B = 100 ⇒ B = 20 cm.Original length L = 2B = 40 cm.Verification / Alternative check:
Old area: 40 * 20 = 800 cm^2; new area: 35 * 25 = 875 cm^2; increase = 75 cm^2 (matches).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.