Answer

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**Hint:**First, find the equation from the data provided in the question. Then solve further, a quadratic equation will be formed. Now solve the equation by either factoring or by quadratic formula $\left( x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \right)$ to get the fraction. The fraction is the desired result.

**Complete step by step answer:**

Given: Difference between the reciprocal and the fraction is $\dfrac{9}{20}$.

Let the required fraction be $x$.

Since the difference between the reciprocal and the fraction is $\dfrac{9}{20}$.

$\Rightarrow \dfrac{1}{x}-x=\dfrac{9}{20}$

Take LCM on the left side,

$\Rightarrow \dfrac{1-{{x}^{2}}}{x}=\dfrac{9}{20}$

Cross multiply the term,

$\Rightarrow 20-20{{x}^{2}}=9x$

Move all terms to one side,

$\Rightarrow 20{{x}^{2}}+9x-20=0$

Factorization can be done in 2 ways.

Method 1:

$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

Put $a=20$, $b=9$ and $c=-20$,

$\Rightarrow x=\dfrac{-9\pm \sqrt{{{9}^{2}}-4\times 20\times (-20)}}{2\times 20}$

Multiply and square the terms and add them in the square root,

$\Rightarrow x=\dfrac{-9\pm \sqrt{1681}}{40}$

Then,

$\Rightarrow x=\dfrac{-9\pm 41}{40}$

Then,

$\Rightarrow x=-\dfrac{50}{40},\dfrac{32}{40}$

Since there is no option with a negative value. So,

$\Rightarrow x=\dfrac{32}{40}$

Cancel out the common factors,

$\Rightarrow x=\dfrac{4}{5}$

**Hence, the number of oranges is 12.**

**Additional information:**

A quadratic equation is a polynomial equation of degree 2. A quadratic equation has two solutions. Either two distinct real solutions, one double real solution, or two imaginary solutions.

There are several methods you can use to solve a quadratic equation:

- Factoring

- Completing the Square

- Quadratic Formula

- Graphing

All methods start with setting the equation equal to zero.

**Note:**

Alternative method:

$\Rightarrow 20{{x}^{2}}+9x-20=0$

We can write 9 as \[\left( 25-16 \right)\],

$\Rightarrow 20{{x}^{2}}+\left( 25-16 \right)x-20=0$

Open the brackets and multiply the terms,

$\Rightarrow 20{{x}^{2}}+25x-16x-20=0$

Take common factors from the equation,

$\Rightarrow 5x\left( 4x+5 \right)-4\left( 4x+5 \right)=0$

Take common factors from the equation,

$\Rightarrow \left( 4x+5 \right)\left( 5x-4 \right)=0$

Then,

$\Rightarrow x=-\dfrac{5}{4},\dfrac{4}{5}$

Since there is no option with a negative value. So,

$\Rightarrow x=\dfrac{4}{5}$

**Hence, option (C) is the correct answer.**

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