Apply the difference of squares identity with decimals: Evaluate (.538^2 − .462^2) / (1 − 0.924) by recognizing algebraic factorization patterns and simplifying cleanly.
Aptitude
Decimal Fraction
Difficulty: Easy
Choose an option
-
A.076
-
B1.042
-
C1
-
D2
Answer
Correct Answer: 1
Explanation
Introduction / Context:This question highlights the algebraic identity a^2 − b^2 = (a − b)(a + b). Recognizing this pattern avoids tedious decimal squaring. Pairing that with a straightforward subtraction in the denominator gives a clean simplification to an integer answer.
Given Data / Assumptions:
- Numerator: 0.538^2 − 0.462^2.
- Denominator: 1 − 0.924.
Concept / Approach:Use the identity a^2 − b^2 = (a − b)(a + b). Compute (a − b) and (a + b) once; then multiply. For the denominator, simple subtraction suffices. Compare numerator and denominator to see if cancellation occurs.
Step-by-Step Solution:
Set a = 0.538 and b = 0.462.a − b = 0.538 − 0.462 = 0.076.a + b = 0.538 + 0.462 = 1.000.So numerator = (a − b)(a + b) = 0.076 × 1.000 = 0.076.Denominator: 1 − 0.924 = 0.076.Quotient = 0.076 / 0.076 = 1.Verification / Alternative check:
Because a + b = 1 exactly, the numerator equals a − b. Direct subtraction confirms both numerator and denominator equal 0.076.Why Other Options Are Wrong:
- .076: That is the numerator alone, not the final ratio.
- 1.042 or 2: These result from arithmetic or identity misuse.
Common Pitfalls:
- Squaring decimals individually, increasing error risk.
- Forgetting the identity and missing the perfect cancellation.
Final Answer:
1