Question

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes `6/5`. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes `2/5`. Find the fraction.

Answer 1

Given: Let the fraction be `x/y`.

From "numerator × 2, denominator –5 gives `6/5`":

`(2x)/(y - 5) = 6/5` 

⇒ 10x = 6(y – 5) 

⇒ 10x = 6y – 30

⇒ 5x – 3y = –15

From "numerator +8, denominator × 2 gives `2/5`":

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

⇒ 5(x + 8) = 4y 

⇒ 5x + 40 = 4y

⇒ 5x – 4y = –40

Subtract the second equation from the first:

(5x – 3y) – (5x – 4y)

= –15 – (–40) 

⇒ y = 25

Substitute y = 25 into 5x – 3y = –15: 

5x – 75 = –15 

⇒ 5x = 60

⇒ x = 12

The fraction is `12/25`.