Question

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. 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 denominator of the fraction is 12. Thus, we have

`x+y=12`

`⇒ x+y-12=0`

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

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

`⇒ 2x=y+3`

`⇒ 2x -y -3=0`

So, we have two equations

`x+y-12=0`

`2x-y-3=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(-3)-(-1)xx(-12))=(-y)/(1xx(-3)-2xx(-12))=1/(1xx(-1)-2xx1)`

`⇒ x/(-3-12)=(-y)/(-3+24)=1/(-1-2)`

`⇒ x/-15=(-y)/21=1/-3`

`⇒ x/15=y/21=1/3`

`⇒ x=15/3,y =21/3`

`⇒ x=5,y=7`

Hence, the fraction is `5/7`