Question

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

If the numerator is multiplied by 2 and the denominator is reduced by 5, the fraction becomes `6/5`. Thus, we have

`(2x)/(y-5)=6/5`

`⇒ 10x=6(y-5)`

`⇒ 10x=6y-30`

`⇒ 10x-6y+30 =0`

`⇒ 2(5x-3y+15)=0`

`⇒ 5x - 3y+15=0`

If the denominator is doubled and the numerator is increased by 8, the fraction becomes `2/5`. Thus, we have

`(x+8)/(2y)=2/5`

`⇒ 5(x+8)=4y`

`⇒ 5x+40=4y`

`⇒ 5x-4y+40=0`

So, we have two equations

`5x-3y+15=0`

`5x-4y+40=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/((-3)xx40-(-4)xx15)=-y/(5xx40-5xx15)=1/(5xx(-4)-5xx(-3))`

`⇒ x/(-120+60)=(-y)/(200-75)=1/(-20+15)`

`⇒x/(-60)=-y/125``=1/-5`

`⇒ x= 60/5,y=125/5`

`⇒ x=12,y=25`
Hence, the fraction is `12/25`