The value of $$ \frac{(67.542)^2 - (32.458)^2}{75.458 - 40.374} $$ is

### Concept & Formula This problem requires evaluating both the numerator and the denominator independently. The numerator uses the standard difference of squares formula, while the denominator requires straightforward subtraction. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * **Step 1: Simplify the Numerator** * $a = 67.542$ and $b = 32.458$. * Apply $(a - b)(a + b)$: $(67.542 - 32.458)(67.542 + 32.458)$. * Calculate the sum: $67.542 + 32.458 = 100$. * Calculate the difference: $67.542 - 32.458 = 35.084$. * Numerator becomes: $35.084 \times 100$. * **Step 2: Simplify the Denominator** * Subtract the given values: $75.458 - 40.374 = 35.084$. * **Step 3: Solve the Fraction** * Substitute back into the expression: $$ \frac{35.084 \times 100}{35.084} $$ * The $35.084$ terms cancel out cleanly. * You are left with $100$. ### Exam Strategy & Shortcut When dealing with large decimal numbers in exam questions, look for sums that round to clean multiples of $10$, $100$, or $1000$. The sum of the numerator terms $(67.542 + 32.458)$ equals exactly $100$. Once you notice the denominator evaluates to the exact difference $(35.084)$, you can immediately deduce the answer is the sum factor, $100$. ### Common Pitfall Getting overwhelmed by the large decimals in the denominator and trying to find a complex algebraic link between the numerator's terms and the denominator's terms, instead of just performing the simple subtraction first. ### Final Answer Therefore, the correct answer is 100.