If 2 is added to the numerator of a fraction, it reduces to 1/2 and if 1 is subtracted from the denominator, it reduces to 1/3. Find the fraction.

#### Solution

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

If 2 is added to the numerator of the fraction, it reduces to `1/2`. Thus, we have

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

`⇒ 2(x+2)=y`

`⇒ 2x+4 =y`

`⇒ 2x -y+4=0`

If 1 is subtracted from the denominator, the fraction reduces to `1/3`. Thus, we have

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

`⇒ 3x = y-1`

`⇒ 3x -y +1=0`

So, we have two equations

`2x - y+4=0`

`3x -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(-1)xx4)=(-y)/(2xx1-3xx4)=1/(2xx(-1)-3xx(-1))`

`⇒ x/(-1+4)=(-y)/(2-12)=1/(-2+13)`

`⇒ x/3=-y/-10=1/1`

`⇒ x/3=y/10=1`

`⇒ x/3= y/10=1`

`⇒ x=3,y=10`

Hence, the fraction is`3/10`