Simplified value of $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, - \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$ $$ ÷ $$ $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, + \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$ = ?
A. $$\frac{{20}}{{101}}$$
B. $$\frac{{100}}{{101}}$$
C. 2
D. $$\frac{{90}}{{101}}$$
Answer: Option A
Solution(By Examveda Team)
$$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, - \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$ $$ ÷ $$ $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, + \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$$$ {\text{Let }}\left[ {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right] = a,$$ $${\text{ }}\left[ {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right] = b $$
$$\eqalign{ & \Rightarrow \left( {{a^2} - {b^2}} \right) \div \left( {a + b} \right) = a - b = ? \cr & \Rightarrow a = 1 + \frac{{10}}{{101}} \cr & \Rightarrow a = \frac{{111}}{{101}} \cr & \Rightarrow b = 1 - \frac{{10}}{{101}} \cr & \Rightarrow b = \frac{{91}}{{101}} \cr & \Rightarrow a - b = \frac{{111}}{{101}} - \frac{{91}}{{101}} \cr & \Rightarrow a - b = \frac{{20}}{{101}} \cr} $$
This Question Belongs to Arithmetic Ability >> Algebra
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