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Algebraic Identities

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Question

$⎡ ⎣ ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ \times ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ - ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ \times ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ ⎤ ⎦$ $\div ⎡ ⎣ ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ + ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ ⎤ ⎦$ simplifies to

A

\frac{100}{101}

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B

\frac{90}{101}

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C

\frac{20}{101}

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D

\frac{101}{100}

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Solution

The correct option is
B
$\frac{20}{101}$

$\Rightarrow \frac{{⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠}^{2} - {⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠}^{2}}{⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ + ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠}$ ..... $(a^{2} - b^{2}) = (a + b) (a - b)$
$\Rightarrow \frac{⎡ ⎣ ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ + ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ ⎤ ⎦ ⎡ ⎣ ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ - ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ ⎤ ⎦}{⎡ ⎣ ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ + ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ ⎤ ⎦}$
$\Rightarrow ⎛ ⎝ 1 + \frac{1}{10 + \frac{1}{10}} ⎞ ⎠ - ⎛ ⎝ 1 - \frac{1}{10 + \frac{1}{10}} ⎞ ⎠$
$\Rightarrow ⎛ ⎜ ⎜ ⎜ ⎝ 1 + \frac{1}{\frac{101}{10}} ⎞ ⎟ ⎟ ⎟ ⎠ - ⎛ ⎜ ⎜ ⎜ ⎝ 1 - \frac{1}{\frac{101}{10}} ⎞ ⎟ ⎟ ⎟ ⎠$
$\Rightarrow (1 + \frac{10}{101}) - (1 - \frac{10}{101})$
$\Rightarrow (\frac{101 + 10}{101}) - (\frac{101 - 10}{101}) = (\frac{111}{101}) - (\frac{91}{101}) = \frac{20}{101}$

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