Recognize a perfect-square structure: Evaluate (3.65^2 + 2.35^2 − 2 × 2.35 × 3.65) / 1.69 by converting the numerator into a binomial square.
Aptitude
Decimal Fraction
Difficulty: Easy
Choose an option
-
A1.69
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B2.35
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C3.65
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D1
Answer
Correct Answer: 1
Explanation
Introduction / Context:This is a direct application of the identity x^2 + y^2 − 2xy = (x − y)^2. Spotting this pattern allows you to reduce the problem to a simple square and then divide by a given constant. It is a staple technique in simplifying quadratic-looking expressions quickly.
Given Data / Assumptions:
- Numerator: 3.65^2 + 2.35^2 − 2 × 3.65 × 2.35.
- Denominator: 1.69.
Concept / Approach:Apply (x − y)^2 when you see x^2 + y^2 − 2xy. Here x = 3.65 and y = 2.35, so x − y = 1.30. Hence the numerator is simply 1.30^2. Then divide by 1.69 to finish. The numbers are intentionally chosen so that the final ratio is exact.
Step-by-Step Solution:
Compute x − y: 3.65 − 2.35 = 1.30.Therefore numerator = (1.30)^2 = 1.69.Quotient = 1.69 / 1.69 = 1.Verification / Alternative check:
Note that 1.3^2 = 1.69 exactly, so the cancellation is perfect.Why Other Options Are Wrong:
- 1.69: That is the numerator before division, not the final value.
- 2.35 and 3.65: Misread constants or skipped squaring/identity step.
Common Pitfalls:
- Changing the sign in −2xy and producing (x + y)^2 instead.
- Rounding 1.3^2 incorrectly; it is exactly 1.69.
Final Answer:
1