Question

When 3 is added to the denominator and 2 is subtracted from the numerator a fraction becomes `1/4`. And, when 6 is added to numerator and the denominator is multiplied by 3, it becomes `2/3`. Find the fraction.

Answer 1

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

When 2 is subtracted from numerator and 3 is added to denominator: `(x - 2)/(y + 3) = 1/4`.

When 6 is added to numerator and denominator is multiplied by 3: `(x + 6)/(3y) = 2/3`.

Step-wise calculation:

1. From `(x - 2)/(y + 3) = 1/4`:

4(x – 2) = y + 3

⇒ 4x – 8 = y + 3

⇒ 4x – y = 11   ...(1)

2. From `(x + 6)/(3y) = 2/3`:

3(x + 6) = 2 × (3y) 

⇒ 3x + 18 = 6y

⇒ x – 2y = –6   ...(2)

3. Solve (1) and (2): From (1): y = 4x – 11.

Substitute into (2): x – 2(4x – 11) = –6 

⇒ x – 8x + 22 = –6 

⇒ –7x = –28 

⇒ x = 4

Then y = 4x – 11

= 16 – 11

= 5

The required fraction is `4/5`.