$\frac{\left[\left(1 + \frac{1}{10 + \frac{1}{10}}\right)\left(1 + \frac{1}{10 + \frac{1}{10}}\right) - \left(1 - \frac{1}{10 + \frac{1}{10}}\right)\left(1 - \frac{1}{10 + \frac{1}{10}}\right)\right]}{\left[\left(1 + \frac{1}{10 + \frac{1}{10}}\right) + \left(1 - \frac{1}{10 + \frac{1}{10}}\right)\right]}$ simplifies to

### Concept & Strategy This is a classic **Algebraic Identity** simplification problem masked by bulky fractional terms. The core strategy is to replace the repeating complex expressions with simple variables to reveal a fundamental algebraic structure, specifically the difference of squares: $a^2 - b^2$. ### Step-by-Step Solution * **Step 1: Abstract the complex terms** Let $a = 1 + \frac{1}{10 + \frac{1}{10}}$ Let $b = 1 - \frac{1}{10 + \frac{1}{10}}$ * **Step 2: Rewrite the expression** Substituting $a$ and $b$ into the original expression gives: $$\frac{[a \cdot a - b \cdot b]}{[a + b]}$$ $$\frac{a^2 - b^2}{a + b}$$ * **Step 3: Apply the algebraic identity** We know that $a^2 - b^2 = (a - b)(a + b)$. Substitute this into the numerator: $$\frac{(a - b)(a + b)}{a + b}$$ The $(a + b)$ terms cancel out, leaving simply: $$a - b$$ * **Step 4: Substitute the original values back** Now we evaluate $a - b$: $$a - b = \left(1 + \frac{1}{10 + \frac{1}{10}}\right) - \left(1 - \frac{1}{10 + \frac{1}{10}}\right)$$ The $1$ and $-1$ cancel out. The fractional parts add together: $$a - b = \frac{2}{10 + \frac{1}{10}}$$ * **Step 5: Final Evaluation** Simplify the denominator: $$10 + \frac{1}{10} = \frac{100 + 1}{10} = \frac{101}{10}$$ Now divide 2 by this result: $$\frac{2}{\frac{101}{10}} = 2 \times \frac{10}{101} = \frac{20}{101}$$ ### Exam Strategy & Shortcut Train your eyes to see the pattern $\frac{(x^2 - y^2)}{(x + y)}$. The moment you spot this, you know the entire massive expression immediately reduces to just $(x - y)$. You can skip all the intermediate writing and directly calculate the difference of the two bracketed terms in your head. ### Common Pitfall A major trap is attempting to simplify the inner fraction ($10 + 1/10$) right at the beginning and substituting $101/10$ into every single instance across the huge numerator. This leads to heavy, error-prone arithmetic ($1 + 10/101$, squaring it, etc.). Always simplify the algebraic structure *before* evaluating the numbers. ### Final Answer Therefore, the correct answer is **\frac{20}{101}**.