Question

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2:3. Determine 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 denominator be $y$.
Since we are given that the sum of the numerator and denominator of a fraction is 4 more than twice the numerator.
So the equation is-
$x + y = 2x + 4$
$\Rightarrow x – y = -4 $…………… (1)
Also if the numerator and denominator are increased by 3, they are in the ratio 2:3.
So the equation is-
$\dfrac{{x + 3}}{{y + 3}} = \dfrac{2}{3}$
By cross multiplication,
We get-
$3(x + 3) = 2(y + 3)$
$\Rightarrow 3x + 9 = 2y + 6$
$\Rightarrow 3x – 2y = -3$ …………. (2)
Now solving equation 1 and equation 2,
We will solve these linear equations in 2 variables by using substitution method-
Firstly, we will take equation 1 and form equation 3.
$\Rightarrow x – y = -4 $ …………. (1)
$\Rightarrow x + 4 = y $ ………… (3)
now substitute the value of equation 3 in equation 2, i.e. $3x – 2y = -3$,
we get-
$\Rightarrow 3x – 2(x + 4) = -3$
$\Rightarrow 3x – 2x – 8 = -3$
$\Rightarrow x – 8 = -3$
$\Rightarrow x = 5$
Substituting the value of x in equation 1 i.e. $x – y = -4$,
We get-
$\Rightarrow 5 – y = -4$
$\Rightarrow {–y = -4 – 5} $
$\Rightarrow {–y = -9}$
$\Rightarrow y = 9$
Hence the numerator required fraction be 5 and denominator be 9.
$\therefore \dfrac{5}{9}$ is the required fraction.

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.