Question

The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.

Answer 1

Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is`x/y`

The sum of the numerator and the denominator of the fraction is `18`. Thus, we have

`x+y=18`

`⇒ x+y-18=0`

If the denominator is increased by 2, the fraction reduces to `1/3`. Thus, we have

`x/(y+2)=1/3`

`⇒ 3x = y+2`

`⇒ 3x -y -2 =0`

So, we have two equations

`x+y -18=0`

`3x -y -2=0`

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

`x/(1xx(-2)-(-1)xx(-18))=-y/(1xx(-2)-3xx(-18))=1/(1xx(-1)-3xx1)`

`⇒ x/(-2-18)=(-y)/(-2+54)=1/(-1-3)`

`⇒ x/-20=(-y)/(52)=1/4`

`⇒ x/20 = y/52 =1/4`

`⇒ x =20/4,y=52/4`

`⇒ x =5 ,y =13`

Hence, the fraction is `5/13`