Difficulty: Easy
Correct Answer: 15 : 16
Explanation:
Introduction / Context:
This problem tests understanding of how ratios of areas of similar shapes relate to ratios of their linear dimensions. For squares, the area is proportional to the square of the side length. Therefore, if you know the ratio of areas, you can determine the ratio of sides, and since perimeter is directly proportional to side length, the perimeter ratio is the same as the side ratio.
Given Data / Assumptions:
Concept / Approach:
If a^2 : b^2 = 225 : 256, then taking square roots of both sides gives a : b = sqrt(225) : sqrt(256). Once you find the side ratio, the perimeter ratio will be exactly the same, because perimeter is 4 times the side length. This is a standard similarity based reasoning step.
Step-by-Step Solution:
Given a^2 : b^2 = 225 : 256
Take square roots: a : b = sqrt(225) : sqrt(256)
sqrt(225) = 15, sqrt(256) = 16
So side ratio a : b = 15 : 16
Perimeter of first square P1 = 4a
Perimeter of second square P2 = 4b
P1 : P2 = 4a : 4b = a : b = 15 : 16
Verification / Alternative check:
Assume side lengths are 15 units and 16 units. Then areas are 225 and 256 respectively, which matches the given area ratio. Perimeters are 60 and 64, giving a ratio 60 : 64, which simplifies to 15 : 16. This confirms that the perimeter ratio is indeed 15 : 16.
Why Other Options Are Wrong:
25 : 16: Comes from misreading the area ratio directly as a perimeter ratio without taking square roots.
16 : 15: Reverses the correct ratio, suggesting the larger perimeter belongs to the smaller area.
17 : 16 and 3 : 4: Do not relate correctly to the square root relationship of 225 and 256.
Common Pitfalls:
Using area ratio directly as perimeter ratio without square root.
Incorrect square root calculation for 225 or 256.
Reversing the order of the ratio by mistake.
Final Answer:
The ratio of the perimeters of the two squares is 15 : 16
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