Question

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes$\dfrac{1}{2}$. Find the fraction.
$
  A\dfrac{3}{7} \\
  B\dfrac{2}{7} \\
  C\dfrac{4}{7} \\
  D\dfrac{5}{7} \\
 $

Answer 1

Hint- Here we will proceed by assuming the numerator and denominator be x and y respectively. Then we will use given conditions to form linear equations in 2 variables and use a substitution method so that we will get the required numerator and denominator.
Let the required fraction be $\dfrac{x}{y}$.

Complete step-by-step solution -
According to first condition i.e. The sum of the numerator and denominator of a fraction is 12.
We get-
$x + y = 12 $ …………. (1)
and using second condition i.e. If the denominator is increased by 3, the fraction becomes$\dfrac{1}{2}$.
We get-
$\dfrac{x}{{y + 3}} = \dfrac{1}{2}$
$\Rightarrow 2x – y = 3 $…………. (2)

Now adding equation 1 and equation 2,
$(x + y = 12) + (2x – y = 3)$
We get-
$\Rightarrow x = 5$
on putting x = 5 in equation 1,
we get-
$\Rightarrow y = 7 $
Therefore, the required fraction is $\dfrac{5}{7}$.
Hence option D is right.

Note- While solving this question, we can assume any variables instead of x and y. As here we used a substitution method to solve these linear equations in 2 variables, we can also solve these linear equations in 2 variables using elimination method.