Question

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is `2 9/10` Find the fraction. 

Answer 1

Let the numerator be x. 

∴Denominator =`x+3` 

∴ Original number =`x/(x+3)` 

According to the question: 

`x/(x+3)+1/(x/(x+3))=2 9/10` 

⇒`x/(x+3)+(x+3)/x=29/10` 

⇒`(x^2+(x+3)^2)/(x(x+3))=29/10` 

⇒`(x^2+x^2+6x+9)/(x^2+3x)=29/10` 

⇒`(2x^2+6x+9)/(x^2+3x)=29/10` 

⇒`29x^2+87x=20x^2+60x+90` 

⇒`9x^2+27x-90=0` 

⇒`9(x^2+3x-10)=0` 

⇒`x^2+3x-10=0` 

⇒`x^2+5x-2x-10=0` 

⇒`x(x+5)-2(x+5)=0` 

⇒`(x-2)(x+5)=0` 

⇒`x-2=0  or  x+5=0` 

⇒`x=2 or  x=-5` (rejected) 

So, number =` x = 2  `  

denominator =`x+3=2+3=5` 

So, required fraction =`2/5`