The simplified value of $$ \frac{\left(1 + \frac{1}{1 + \frac{1}{100}}\right)\left(1 + \frac{1}{1 + \frac{1}{100}}\right) - \left(1 - \frac{1}{1 + \frac{1}{100}}\right)\left(1 - \frac{1}{1 + \frac{1}{100}}\right)}{\left(1 + \frac{1}{1 + \frac{1}{100}}\right) + \left(1 - \frac{1}{1 + \frac{1}{100}}\right)} $$ is

### Concept & Formula Despite the intimidating nested fractions, this is fundamentally a simple algebraic simplification problem based on the difference of squares identity. The governing formula is: $$ \frac{a^2 - b^2}{a + b} = a - b $$ ### Step-by-Step Solution First, simplify the visual complexity by abstracting the repeating expressions into variables: * Let $ a = 1 + \frac{1}{1 + \frac{1}{100}} $ * Let $ b = 1 - \frac{1}{1 + \frac{1}{100}} $ Rewrite the given expression using these variables. Notice that the numerator consists of two identical terms multiplied together, minus another two identical terms multiplied together: $$ \frac{a \cdot a - b \cdot b}{a + b} $$ $$ = \frac{a^2 - b^2}{a + b} $$ Apply the difference of squares formula to expand the numerator: $$ = \frac{(a - b)(a + b)}{a + b} $$ The $ (a + b) $ terms cancel each other out, leaving only: $$ = a - b $$ Now, substitute the actual expressions back in and evaluate $ a - b $: $$ a - b = \left( 1 + \frac{1}{1 + \frac{1}{100}} \right) - \left( 1 - \frac{1}{1 + \frac{1}{100}} \right) $$ Distribute the negative sign: $$ = 1 + \frac{1}{1 + \frac{1}{100}} - 1 + \frac{1}{1 + \frac{1}{100}} $$ The 1s cancel out, leaving two identical fractions: $$ = 2 \times \left( \frac{1}{1 + \frac{1}{100}} \right) $$ Simplify the inner denominator: $$ 1 + \frac{1}{100} = \frac{101}{100} $$ Substitute this back: $$ = 2 \times \left( \frac{1}{\frac{101}{100}} \right) $$ $$ = 2 \times \frac{100}{101} $$ $$ = \frac{200}{101} $$ ### Exam Strategy & Shortcut Don't let nested fractions bait you into immediate calculation. Notice the $ \frac{x^2 - y^2}{x + y} $ pattern right away. This allows you to deduce that the answer is simply $ x - y $. When calculating $ x - y $, the "1"s cancel out immediately, leaving $ 2 \times \text{the nested fraction} $. ### Common Pitfall Many students attempt to fully evaluate the value of $ 1 + \frac{1}{1 + 1/100} $ and square it before looking at the rest of the equation. This leads to squaring fractions like $ 201/101 $, which is an enormous waste of time. Always look for algebraic structure before doing arithmetic. ### Final Answer **Therefore, the correct answer is $ \frac{200}{101} $.**