Explanation
Let $$\dfrac {x}{y}$$ be the fraction, where $$x$$ and $$y$$ are positive integers.
$$\Rightarrow \dfrac {x+2}{y+2}=\dfrac {9}{11}$$ ..(1)$$\Rightarrow \dfrac {x+3}{y+3}=\dfrac {5}{6}$$ ..(2)
From (1) we get $$\Rightarrow 11x+22 = 9y+18$$$$\Rightarrow 11x-9y = -4$$ ...(3)
From (2) we get $$\Rightarrow 6x +18 = 5y +15$$$$\Rightarrow 6x-5y = -3$$ ...(4)
From (4) will get$$y = \dfrac{3+6x}{5}$$
Now substituting $$y$$ in (3),
$$\Rightarrow 11x-\dfrac{9\left ( 3+6x \right )}{5} = -4$$
$$\Rightarrow 55x-27-54x = -20$$
$$\Rightarrow 55x-54x = -20+27$$
$$\Rightarrow x = 7$$
Now substituting $$x$$ in (4)
$$\Rightarrow 6\times 7 -5y = -3$$
$$\Rightarrow 42-5y = -3$$
$$\Rightarrow -5y = -3 - 42$$
$$\Rightarrow -5y = -45$$
$$\Rightarrow y = 9$$
Hence, $$ x = 7$$ and $$ y = 9$$.
And fraction $$\dfrac{x}{y}=\dfrac{7}{9}$$
$${\textbf{Step -1: Framing equations}}$$
$$\text{Without loss of generality, suppose,}$$ $$ x$$ $${\text{and}}$$ $$ y$$ $${\text{be two numbers and }} x > y.$$
$${\text{Given that, sum of two numbers is 35}}{\text{.}}$$
$$ \Rightarrow x + y = 35$$ $$\ldots \left( 1 \right)$$
$${\text{Difference between these two numbers is 13}}{\text{.}}$$
$$ \Rightarrow x - y = 13$$ $$ \ldots \left( 2 \right) $$
$${\textbf{Step -2: Adding equation }}\left( \mathbf 1 \right)$$ $${\textbf{and equation }}\left(\mathbf 2 \right)\textbf .$$
$$ \Rightarrow x + y + \left( {x - y} \right) = 35 + 13$$
$$ \Rightarrow 2x = 48$$
$$ \Rightarrow x = 24$$
$${\text{Substituting the value of}}$$ $$x$$ $${\text{in equation }}\left( 1 \right)$$ $${\text{we get,}}$$
$$ \Rightarrow 24 + y = 35$$
$$ \Rightarrow y = 35 - 24$$
$$ \Rightarrow y = 11$$
$${\text{Hence, these two numbers are 24 and 11}}{\text{.}}$$
$${\textbf{ Hence, option (C) is correct answer.}}$$
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