Question

The numerator of a fraction is 4 less than the denominator. If the numerator is decreased by 2 and denominator is increased by 1, then the denominator is eight times the numerator. 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 numerator of the fraction is 4 less the denominator. Thus, we have

`x = y -4`

` ⇒ x -y =-4`

If the numerator is decreased by 2 and denominator is increased by 1, then the denominator is 8 times the numerator. Thus, we have

` y + 1 = 8(x-2)`

`⇒ y + 1= 8x -16`

` ⇒ 8x -y = 1+16`

`⇒ 8x - y =17`

So, we have two equations

` x - y = -4`

` 8x -y =17`

Here x and y are unknowns. We have to solve the above equations for x and y.

Subtracting the second equation from the first equation, we get

`( x -y)-(8x -y)=-4-17`

`⇒ x - y -8x +y =-21`

` ⇒ -7 x =-21`

` ⇒ 7 x =21 `

` ⇒ x =21/7`

` ⇒ x = 3`

Substituting the value of x in the first equation, we have

` 3- y =-4`

`⇒ y = 3+4 `

` ⇒ y = 7`
Hence, the fraction is `3/7`