Question

A naughty student breaks the pencil in such a way that the ratio of two broken parts is same as that of the original length of the pencil to one of the larger part of the pencil, the ratio of the other part to the original length of the pencil is:
${\text{A}}{\text{. }}$$1:2\sqrt 5 $
${\text{B}}{\text{. 2:}}\left( {3 + \sqrt 5 } \right)$
${\text{C}}{\text{. 2:}}\sqrt 5 $
${\text{D}}{\text{. }}$can’t be determined

Answer 1

Hint: First of all assume the length of the pencil is L. After breaking the pencil in two parts let us assume a larger part length as X and hence smaller part length will be L-X. now equate the ratio as given in the question to proceed further.

Complete step-by-step answer:
We have assumed
Length of pencil L.
After breaking the pencil
Larger part length = X
Smaller part length = L- X
Now as given in the question
 The ratio of two broken parts is the same as that of the original length of the pencil to one of the larger parts of the pencil. That means when we write in mathematical form we get,
$\dfrac{x}{{l - x}} = \dfrac{l}{x}$
On cross multiplying we get
${x^2} = {l^2} - lx$
On rearranging and further solving we get,
$
  {x^2} + lx - {l^2} = 0 \\
  x = \dfrac{{ - l \pm \sqrt {{l^2} + 4{l^2}} }}{2} \\
  x = \dfrac{{ - l \pm \sqrt 5 l}}{2} \\
 $
And we know length can not be negative and hence $x = \dfrac{{ - l + \sqrt 5 l}}{2}$
And smaller part = L-X
Smaller part = that means other part is
 $
  l - \left( {\dfrac{{ - l + \sqrt 5 l}}{2}} \right) \\
   \Rightarrow \dfrac{{2l + l - \sqrt 5 l}}{2} \\
   \Rightarrow \dfrac{{3l - \sqrt 5 l}}{2} \\
 $
Now we have to find ratio of other part to the original length of the pencil is
That mean we have to find $\dfrac{{l - x}}{l}$
On putting the value of L-X we get
$\dfrac{{3l - \sqrt 5 l}}{{\dfrac{2}{l}}} = \dfrac{{3l - \sqrt 5 l}}{{2l}} = \dfrac{{3 - \sqrt 5 }}{2}$
On rationalizing we get
$\dfrac{{3 - \sqrt 5 \left( {3 + \sqrt 5 } \right)}}{{2\left( {3 + \sqrt 5 } \right)}} = \dfrac{{9 - 5}}{{2\left( {3 + \sqrt 5 } \right)}} = \dfrac{4}{{2\left( {3 + \sqrt 5 } \right)}}$
On further cancel out we get,
$
  \dfrac{2}{{\left( {3 + \sqrt 5 } \right)}} \\
   \Rightarrow 2:\left( {3 + \sqrt 5 } \right) \\
$
Hence option B is the correct option.

Note: Whenever we get this type of question the key concept of solving is we have to first assume and just proceed as being told in the question. And things should be noticed that when we have solved but the answer is not matching then if rationalization is possible then rationalize it so that option can be matched. As this happened in this question.