Both the numerator and the denominator of $\dfrac{3}{4}$ are increased by a positive integer, $x,$ and those of $\dfrac{15}{17}$ are decreased by the same integer. This operation results in the same value for both the fractions.

What is the value of $x ?$

1. $1$
2. $2$
3. $3$
4. $4$

According to the question, $\frac{3+x}{4+x} = \frac{15-x}{17-x}$

$\Rightarrow (3+x)(17-x) = (4+x)(15-x)$

$\Rightarrow \require{cancel} 51-3x + 17x – {\color{Red}{\cancel{x^{2}}}} = 60 + 15x-4x – {\color{Red}{\cancel{x^{2}}}}$

$\Rightarrow 14x-11x = 60-51$

$\Rightarrow 3x = 9$

$\Rightarrow {\color{Blue}{\boxed{x = 3}}}$

Correct Answer $:\text{C}$
12.0k points