Find the decimal form of 1 11 .A) 0.999999B) 0.090909....C) 0.909090.....D) None of the above

Question

Find the decimal form of       1    11  .A) 0.999999B) 0.090909....C) 0.909090.....D) None of the above

Karli Daniel · Accepted Answer

The correct answer is B 0.090909....Given fraction is       1    11  .      1    11    =      1    11    ×      100    100    =  (      100    11    )  ×      1    100    =  (  9      1    11    )  ×      1    100    =  (  9  +      1    11    )  ×      1    100    =      9    100    +      1    1100    .  .  .  (  1  ) Now,       1    1100    =      1          11      ×      100        =  (      1    11    )  ×      1    100    .  .  .  .  .  .  .  (  2  )From (1),       1    11    =      1    100    +      1    1100  ; substitute it in eq (2).  ⇒      1    1100    =  (      9    100    +      1    1100    )  ×      1    100        9    10000    +      1    110000  Now substituting this value in eq(1), we get      1    11    +      9    100    +  (      9    10000    +      1    110000    )  =  (      9    100    +      9    10000    )  +      1    110000    )Similarly finding the value of       1    11000    =  (      1    11    ×      1    10000    )and on solving that will give       9    1000000    =      1    11000000    ⇒      1    11    =      9    100    +      9    10000    +  (      9    1000000    +      1    11000000    )This cycle will continue and leads to       1    11    =      9    100    +      9    10000    +      9    1000000    +  .  .  .= 0.09 + 0.0009 + 0.000009 + .....&amp;lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;  ∴      1    11    =  0.090909...

Find the decimal form of 1/11

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