Squares with Diagonals in Ratio — Compare Areas: The ratio concerns two squares where the diagonal of one is double the diagonal of the other. What is the ratio of their areas (larger : smaller)?
Aptitude
Area
Difficulty: Easy
Choose an option
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A2:1
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B3:1
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C3:2
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D4:1
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E5:2
Answer
Correct Answer: 4:1
Explanation
Introduction / Context:For squares, both side and diagonal are linear measures. Because area scales with the square of a linear measure, doubling a linear dimension (like the diagonal) multiplies area by the square of that factor. This is a quick test of proportional reasoning rather than computation-heavy arithmetic.
Given Data / Assumptions:
- Square 1 diagonal d1
- Square 2 diagonal d2 = 2 * d1
- Area proportionality: A ∝ (diagonal)^2 for squares
Concept / Approach:Since A ∝ d^2 for a square (because s = d/√2 and A = s^2 = d^2/2), the ratio of areas equals the ratio of squared diagonals. With d2 = 2d1, A2/A1 = (2d1)^2 / d1^2 = 4. Therefore, the larger-to-smaller area ratio is 4:1.
Step-by-Step Solution:
Express A1 = k * d1^2 and A2 = k * d2^2 where k = 1/2.Substitute d2 = 2d1 ⇒ A2/A1 = (k * 4d1^2) / (k * d1^2) = 4.Hence the ratio (larger : smaller) = 4 : 1.Verification / Alternative check:
Numeric example: Let d1 = 10 ⇒ A1 = 10^2/2 = 50. With d2 = 20 ⇒ A2 = 20^2/2 = 200 ⇒ ratio = 200:50 = 4:1.Why Other Options Are Wrong:
- 2:1 assumes linear, not quadratic, scaling.
- 3:1 and 3:2 arise from ad-hoc, unsupported factors.
- 5:2 ≈ 2.5:1 is also inconsistent with quadratic scaling.
Common Pitfalls:
- Confusing side or diagonal doubling with doubling area; area quadruples when a linear dimension doubles.
Final Answer:4:1.