Squares — Diagonal Ratio to Area Ratio: The diagonals of two squares are in the ratio 3 : 2. Find the ratio of their areas (larger : smaller).
Aptitude
Area
Difficulty: Easy
Choose an option
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A9 : 4
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B9 : 2
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C9 : 5
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D9 : 7
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E4 : 3
Answer
Correct Answer: 9 : 4
Explanation
Introduction / Context:For squares, the diagonal is proportional to the side and area scales with the square of any linear dimension. This item checks proportional reasoning by connecting a diagonal ratio directly to an area ratio without computing actual side lengths.
Given Data / Assumptions:
- Two squares with diagonal ratio d1 : d2 = 3 : 2
- Square properties: d = s * √2; area A = s^2 = d^2 / 2
- We need the ratio of areas (larger : smaller)
Concept / Approach:Because A ∝ d^2 for squares, the ratio of areas equals the square of the ratio of diagonals. No numeric side computation is necessary; simply square each term in the diagonal ratio and simplify the fraction to obtain the area ratio.
Step-by-Step Solution:
Given d1 : d2 = 3 : 2.Since A ∝ d^2, A1 : A2 = d1^2 : d2^2 = 3^2 : 2^2 = 9 : 4.The larger-to-smaller area ratio is 9 : 4.Verification / Alternative check:
Pick d2 = 2 ⇒ d1 = 3. Then A2 = 2^2/2 = 2; A1 = 3^2/2 = 4.5 ⇒ ratio 4.5 : 2 = 9 : 4.Why Other Options Are Wrong:
- 9 : 2, 9 : 5, 9 : 7 do not arise from squaring 3 : 2.
- 4 : 3 inverts the order (smaller : larger).
Common Pitfalls:
- Using 3 : 2 directly for area instead of squaring.
- Swapping larger and smaller by mistake.
Final Answer:9 : 4.