The denominator of a fraction is …

Let x be the numerator .

Since we are given that denominator is 2 more than the numerator .

So, denominator = x+2

So Fraction =  {tex}\frac{x}{x+2}{/tex}

Reciprocal of fraction =  {tex}\frac{x+2}{x}{/tex}

Now we are given that the sum of fraction and its reciprocal is 34/15.

{tex}⇒\frac{x}{x+2}+\frac{x+2}{x} =\frac{34}{15}{/tex}

{tex}⇒\frac{(x*x)+[(x+2)*(x+2)]}{(x+2)(x)}=\frac{34}{15}{/tex}

{tex}⇒\frac{2x^{2}+4x+4}{x^{2}+2x}=\frac{34}{15}{/tex}

{tex}⇒15*(2x^{2}+4x+4)=34*(x^{2}+2x){/tex}

{tex}⇒30x^{2}+60x+60)=34x^{2}+68x{/tex}

{tex}⇒4x^{2}+8x-60=0{/tex}

{tex}⇒x^{2}+2x-15=0{/tex}

{tex}⇒x^{2}+5x-3x-15=0{/tex}

⇒x(x+5)-3(x+5)=0

⇒(x-3)(x+5)=0

⇒x-3=0, x+5=0

⇒ x = 3 , x= -5

So Fraction when x = 3

{tex}\frac{x}{x+2} =\frac{3}{3+2}=\frac{3}{5}{/tex}

Fraction when x = -5

{tex}\frac{x}{x+2} =\frac{-5}{-5+2}=\frac{5}{3}{/tex}