Question

When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes ${\displaystyle \frac{1}{4}}$. And when 6 is added to the numerator and the denominator is multiplied by 3, it becomes ${\displaystyle \frac{2}{3}}$. Find the fraction.

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 using a substitution method so that we will get the required numerator and denominator.

Complete step-by-step solution -
Let the numerator be $x$
And let the denominator be $y$
Now applying first condition i.e. When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes ${\displaystyle \frac{1}{4}}$,
We get-
${\displaystyle \frac{x-2}{y+3}}={\displaystyle \frac{1}{4}}$
$\Rightarrow 4\left(x-2\right)=\left(y+3\right)$
$\Rightarrow 4x–8=y+3$
$\Rightarrow 4x–11=y$………… (1)
And applying second condition i.e. when 6 is added to numerator and the denominator is multiplied by 3, it becomes ${\displaystyle \frac{2}{3}}$ ,
We get-
${\displaystyle \frac{x+6}{3y}}={\displaystyle \frac{2}{3}}$
Solving equation 1 and equation 2,
$\Rightarrow {\displaystyle \frac{3\left(x+6\right)}{3}}=2y$
$\Rightarrow x+6=2y$…………… (3)
Now put value of equation 1 in equation 3,
We get-
$x+6=2(4x–11)$
$\Rightarrow x+6=8x–22$
$\Rightarrow 6+22=8x–22$
$\Rightarrow 6+22=8x–x$
$\Rightarrow 28=7x$
$\Rightarrow x=4$
Now substituting the value of x in equation 1,
We get-
$\Rightarrow y=4(4)–11$
$\Rightarrow y=5$
This implies-
Required fraction $={\displaystyle \frac{x}{y}}={\displaystyle \frac{4}{5}}$

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.