Question

If a proper fraction in its lower form is subtracted from its reciprocal, the value of difference is $\frac{45}{14}$. What is the value of the square of the proper fraction?

• A. $-\frac{7}{2}$
  
• B. $\frac{2}{7}$
  
• C. $\frac{49}{4}$
  
• D. $\frac{4}{49}$
  

Answer 1

The correct option is D $\frac{4}{49}$


Let us assume the value of the proper fraction be $x$.
Then, the reciprocal of the proper fraction would be $\frac{1}{x}$.

According to the question,
$\frac{1}{x}-x=\frac{45}{14}$

$\Rightarrow \frac{1-x^2}{x}=\frac{45}{14}$

$\Rightarrow 14\times (1-x^2)=45\times x$
[Cross multiplication]

$\Rightarrow 14-14x^2=45x$

$\Rightarrow 14x^2+45x-14=0$
Splitting the middle term, we get

$\Rightarrow 14x^2+49x-4x-14=0$

$\Rightarrow 7x(2x+7)-2(2x+7)=0$

$\Rightarrow (2x+7)(7x-2)=0$

$\Rightarrow x=-\frac{7}{2},\frac{2}{7}$

As we know that for a proper fraction, the numerator is less than the denominator, and both the numerator and denominator belong to natural numbers. So, we can reject $x=-\frac{7}{2}$.

As a result, we get the proper fraction as $x=\frac{2}{7}.$

We need to find the square of this proper fraction.
$\therefore \underset{–}{(\frac{2}{7}{)}^2=\frac{2}{7}\times \frac{2}{7}=\frac{2\times 2}{7\times 7}=\frac{4}{49}}$

Thus, the value of square of the proper fraction $\frac{2}{7}$ would be $\frac{4}{49}$.

Therefore, option (d.) is the correct answer.