Question

A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator. It 1 is added to both the numerator and denominator, it 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`

If 1 is subtracted from both numerator and the denominator, the fraction becomes `1/3`. Thus, we have

` (x-1)/(y-1)=1/3`

` ⇒ 3(x-1)=y-1`

`⇒ 3x-3=y-1 `

`⇒ 3x -y -2=0`

If 1 is added to both numerator and the denominator, the fraction becomes `1/2`. Thus, we have

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

`⇒ 2(x+1)=y+1`

` ⇒ 2x +2 =y +1`

` ⇒ 2x -y +1=0`

So, we have two equations

` 3x -y -2=0`

` 2x -y +1=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)xx1-(-1)xx(-2)=` ` (-y)/(3xx1-2xx2(-2))=` `1/(3xx(-1)-2xx(-1))`

`⇒ x/(-1-2) = (-y)/(3+4)=1/(-3+2) `

`⇒x/-3= (-y)/7 =1/-1`

`⇒ x/3= y/7=1`

` ⇒ x=3,y = 7`

Hence, the fraction is `3/7`