Question

If the sum of the fraction and its reciprocal is $2\dfrac{{16}}{{21}}$ , find the fraction.

Answer 1

Hint: We can solve this question by assuming a fraction then showing that the sum of that fraction and its reciprocal is $2\dfrac{{16}}{{21}}$. We must convert this given sum $2\dfrac{{16}}{{21}}$ in regular form in order to simplify the equation formed.

Complete step-by-step answer:
Let us assume that the fraction we need to find is $x$.
Therefore, its reciprocal is $ = \dfrac{1}{x}$.
According to the question, the sum of the fraction and its reciprocal is $2\dfrac{{16}}{{21}}$. That is,
$ \Rightarrow x + \dfrac{1}{x} = 2\dfrac{{16}}{{21}}$
Now if a mixed fraction is like $l\dfrac{m}{n}$ then in regular form it is $ = \dfrac{{\left( {n \times l} \right) + m}}{n}$. Similarly, we will convert $2\dfrac{{16}}{{21}}$ in regular form.
$
   \Rightarrow 2\dfrac{{16}}{{21}} = \dfrac{{\left( {21 \times 2} \right) + 16}}{{21}} \\
   \Rightarrow 2\dfrac{{16}}{{21}} = \dfrac{{\left( {42 + 16} \right)}}{{21}} \\
   \Rightarrow 2\dfrac{{16}}{{21}} = \dfrac{{58}}{{21}} \\
 $
Substituting the value of $2\dfrac{{16}}{{21}}$ in regular form, we get
$
   \Rightarrow x + \dfrac{1}{x} = \dfrac{{58}}{{21}} \\
   \Rightarrow \dfrac{{{x^2} + 1}}{x} = \dfrac{{58}}{{21}} \\
 $
On simplifying this, we will get
$ \Rightarrow 21{x^2} - 58x + 21 = 0$
Now, we have a quadratic equation. Therefore, we will use a quadratic formula to find the roots of this equation.
That is,
\[
   \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\
   \Rightarrow x = \dfrac{{ - \left( { - 58} \right) \pm \sqrt {{{\left( { - 58} \right)}^2} - 4 \times 21 \times } 21}}{{2 \times 21}} \\
   \Rightarrow x = \dfrac{{58 \pm \sqrt {3364 - 1764} }}{{42}} \\
   \Rightarrow x = \dfrac{{58 \pm \sqrt {1600} }}{{42}} \\
   \Rightarrow x = \dfrac{{58 \pm 40}}{{42}} \\
   \Rightarrow x = \dfrac{{29 \pm 20}}{{21}} \\
 \]
Hence, the required fraction is \[x = \dfrac{{29 + 20}}{{21}}\] or $x = \dfrac{{29 - 20}}{{21}}$. That is,
$ \Rightarrow x = \dfrac{{49}}{{21}} = \dfrac{7}{3}$ or $x = \dfrac{9}{{21}} = \dfrac{3}{7}$
Hence, the answer is $x = \dfrac{7}{3},\dfrac{3}{7}$.

Note: The common mistake that we can make in this type of question is that we may assume that the fraction is $\dfrac{a}{b}$. If we assume that the fraction is $\dfrac{a}{b}$, then the reciprocal is $\dfrac{1}{{\left( {\dfrac{a}{b}} \right)}} = \dfrac{b}{a}$. So in this case, the equation formed from the given conditions is $\dfrac{a}{b} + \dfrac{b}{a} = 2\dfrac{{16}}{{21}}$. But we won’t be able to find the value of $a$ and $b$, because here we have only one equation, but two unknown variables $a$ and $b$.