Question

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

If 3 is added to the denominator and 2 is subtracted from the numerator, the fraction becomes `1/4`. Thus, we have

`(x-2)/(y+3)=1/4`

`⇒ 4(x-2)=y+3`

`⇒ 4x-8=y+3`

`⇒ 4x-y-11=0`

If 6 is added to the numerator and the denominator is multiplied by 3, the fraction becomes `2/3`. Thus, we have

`(x+6)/(3y)=2/3`

`⇒ 3(x+6)=6y`

`⇒ 3x +18 =6y`

`⇒ 3x-6y+18=0`

`⇒ 3(x-2y+6)=0`

`⇒ x-3y+6=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/((-1)xx6-(-2)xx(-11))=(-y)/(4xx6-1xx(-11))=1/(4xx(-2)-1xx(-1))`

`⇒ x/(-6-22)=-y/(24+11)=1/(-8+1)`

`⇒ x/-28=-y/35=1/-7`

`⇒ x= 28/7,y=35/7`

`⇒ x= 4,y=5`

Hence, the fraction is`4/5`