Question

The denominator of a fraction is $3$ more than its numerator. The sum of the fraction and its reciprocal is $2\frac{9}{10}$. Find the fraction

Answer 1

Let the dfraction is $\frac{x}{y}$

Then, according to given question

$y=x+3.......\left(1\right)$

Again,

$\frac{x}{y}+\frac{y}{x}=2\frac{9}{10}$

$\frac{x^2+y^2}{xy}=\frac{29}{10}$

$10x^2+10y^2=29xy.......\left(2\right)$

Put the value of y in equation (2).

$10x^2+10{(x+3)}^2=29x(x+3)$

$\Rightarrow 10x^2+10(x^2+9+6x)=29x^2+87x$

$\Rightarrow 10x^2+10x^2+90+60x=29x^2+87x$

$\Rightarrow 20x^2-29x^2+60x-87x+90=0$

$\Rightarrow -9x^2-27x+90=0$

$\Rightarrow -9(x^2+3x-10)=0$

$\Rightarrow x^2+3x-10=0$

$\Rightarrow (x+5)(x-2)=0byfactorize$

$\Rightarrow x+5=0x-2=0$

$\Rightarrow x=-5x=2$

Put $x=2$ In equation (1)

$y=2+3=5$

fraction $\frac{x}{y}=\frac{2}{5}$ $ans.$