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 $$ 3, $$ then are in the ratio 2: 3 . Determine the fraction.

Answer 1

Let the numerator $$ =x $$ and the denominator $$ =y $$
So, the fraction $$ =\dfrac{x}{y} $$
According to the question,
Condition I:
$$ x+y=2 x+4 $$
$$ \Rightarrow x+y-2 x=4 $$
$$ \Rightarrow-x+y=4 $$
$$ \Rightarrow \mathrm{y}=4+\mathrm{x} \ldots(\mathrm{i}) $$
Condition II:
$$ \dfrac{x+3}{y+3}=\dfrac{2}{3} $$
$$ \Rightarrow 3(x+3)=2(y+3) $$
$$ \Rightarrow 3 x+9=2 y+6 $$
$$ \Rightarrow 3 x-2 y=-3 ....(ii)$$
On putting the value of y in Eq.(ii), we get
$$ 3 x-2(4+x)=-3 $$
$$ \Rightarrow 3 x-8-2 x=-3 $$
$$ \Rightarrow x=5 $$
On putting the value of $$ x $$ in Eq. (i), we get
$$ y=4+5 $$
$$ \Rightarrow \mathrm{y}=9 $$
So, the numerator is 5 and the denominator is 9
Hence, the fraction is $$ \dfrac{5}{9} $$