$ \frac{\left(3 \frac{2}{3}\right)^2 - \left(2 \frac{1}{2}\right)^2}{\left(4 \frac{3}{4}\right)^2 - \left(3 \frac{1}{3}\right)^2} \div \frac{3 \frac{2}{3} - 2 \frac{1}{2}}{4 \frac{3}{4} - 3 \frac{1}{3}} = x $

### Concept & Formula This complex fraction relies on recognizing that the algebraic structure can be simplified entirely before any arithmetic on the fractions is performed. The primary identity used is the difference of squares: $$ x^2 - y^2 = (x - y)(x + y) $$ ### Step-by-Step Solution First, to make the structure clear, assign variables to the mixed fractions: * Let $ a = 3 \frac{2}{3} $ * Let $ b = 2 \frac{1}{2} $ * Let $ c = 4 \frac{3}{4} $ * Let $ d = 3 \frac{1}{3} $ Substitute these variables into the original equation: $$ \frac{a^2 - b^2}{c^2 - d^2} \div \frac{a - b}{c - d} $$ Expand the squared terms using the difference of squares formula: $$ \frac{(a - b)(a + b)}{(c - d)(c + d)} \div \frac{a - b}{c - d} $$ Recall that dividing by a fraction is the same as multiplying by its reciprocal: $$ \frac{(a - b)(a + b)}{(c - d)(c + d)} \times \frac{c - d}{a - b} $$ Cancel out the common $ (a - b) $ and $ (c - d) $ terms from the numerator and denominator: $$ = \frac{a + b}{c + d} $$ Now, evaluate $ a + b $ and $ c + d $ by converting to improper fractions: $$ a + b = 3 \frac{2}{3} + 2 \frac{1}{2} = \frac{11}{3} + \frac{5}{2} = \frac{22 + 15}{6} = \frac{37}{6} $$ $$ c + d = 4 \frac{3}{4} + 3 \frac{1}{3} = \frac{19}{4} + \frac{10}{3} = \frac{57 + 40}{12} = \frac{97}{12} $$ Divide the sum $ (a + b) $ by the sum $ (c + d) $: $$ \frac{37/6}{97/12} = \frac{37}{6} \times \frac{12}{97} $$ $$ = \frac{37 \times 2}{97} = \frac{74}{97} $$ ### Exam Strategy & Shortcut Whenever you see a structure matching $ \frac{x^2 - y^2}{z^2 - w^2} \div \frac{x - y}{z - w} $, you should immediately recognize that the negative components will cancel out. You can instantly skip all middle steps and jump to evaluating $ \frac{x + y}{z + w} $. ### Common Pitfall The most dangerous pitfall is attempting to square the mixed fractions right at the beginning (e.g., trying to calculate $ (11/3)^2 = 121/9 $). This creates massive, unwieldy fractions that will consume minutes of exam time and almost certainly lead to arithmetic errors. ### Final Answer **Therefore, the correct answer is $ \frac{74}{97} $.**