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Hint- In order to solve such a question with multiple lines of cases consider the numerator and the denominator of the fraction in terms of some unknown variables and then try to make an algebraic equation with the help of given cases in question.

Let $a$ be the numerator of the fraction and $b$ be the denominator of the given fraction.

So the fraction becomes $\dfrac{a}{b}$

According to the first statement of the problem:

If we add 1 to the numerator and subtract 1 from the denominator a fraction reduces to 1

$

\Rightarrow \dfrac{{a + 1}}{{b - 1}} = 1 \\

\Rightarrow a + 1 = b - 1 \\

\Rightarrow a + 2 = b \\

\Rightarrow b = a + 2.............(1) \\

$

According to the second statement:

If we only add 1 to the denominators, it becomes $\dfrac{1}{2}$.

$

\Rightarrow \dfrac{a}{{b + 1}} = \dfrac{1}{2} \\

\Rightarrow 2a = b + 1 \\

\Rightarrow 2a - 1 = b \\

\Rightarrow b = 2a - 1..............(2) \\

$

From equations (1) and (2), equating the value of $b$ in both the cases.

$

\Rightarrow 2a - 1 = a + 2 \\

\Rightarrow 2a - a = 2 + 1 \\

\Rightarrow a = 3 \\

$

Now substituting the value of $a$ in equation (1)

$

\Rightarrow b = 3 + 2 \\

\Rightarrow b = 5 \\

$

Finally the fraction $\dfrac{a}{b} = \dfrac{3}{5}$ .

Hence, the fraction is $\dfrac{3}{5}$ .

Note- We know that fraction is defined as the ratio of numerator and denominator. In order to solve such questions algebraic method is the only way to proceed for an easy and fast solution. It consists of considering the numerator and denominator in terms of some unknown variables and then satisfying the given conditions on the basis of variables to find the equation.

Let $a$ be the numerator of the fraction and $b$ be the denominator of the given fraction.

So the fraction becomes $\dfrac{a}{b}$

According to the first statement of the problem:

If we add 1 to the numerator and subtract 1 from the denominator a fraction reduces to 1

$

\Rightarrow \dfrac{{a + 1}}{{b - 1}} = 1 \\

\Rightarrow a + 1 = b - 1 \\

\Rightarrow a + 2 = b \\

\Rightarrow b = a + 2.............(1) \\

$

According to the second statement:

If we only add 1 to the denominators, it becomes $\dfrac{1}{2}$.

$

\Rightarrow \dfrac{a}{{b + 1}} = \dfrac{1}{2} \\

\Rightarrow 2a = b + 1 \\

\Rightarrow 2a - 1 = b \\

\Rightarrow b = 2a - 1..............(2) \\

$

From equations (1) and (2), equating the value of $b$ in both the cases.

$

\Rightarrow 2a - 1 = a + 2 \\

\Rightarrow 2a - a = 2 + 1 \\

\Rightarrow a = 3 \\

$

Now substituting the value of $a$ in equation (1)

$

\Rightarrow b = 3 + 2 \\

\Rightarrow b = 5 \\

$

Finally the fraction $\dfrac{a}{b} = \dfrac{3}{5}$ .

Hence, the fraction is $\dfrac{3}{5}$ .

Note- We know that fraction is defined as the ratio of numerator and denominator. In order to solve such questions algebraic method is the only way to proceed for an easy and fast solution. It consists of considering the numerator and denominator in terms of some unknown variables and then satisfying the given conditions on the basis of variables to find the equation.