Question

If 1 is added to both the numerator and the denominator of a fraction, it becomes `4/5`. If, however, 5 is subtracted from both the numerator and the denominator, the fraction becomes `1/2`. Find the fraction.

Answer 1

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

The conditions are:

1. `(x + 1)/(y + 1) = 4/5`

2. `(x - 5)/(y - 5) = 1/2`

From (1): 5(x + 1) = 4(y + 1)

⇒ 5x + 5 = 4y + 4

⇒ 5x – 4y = –1

From (2): 2(x – 5) = (y – 5)

⇒ 2x – 10 = y – 5

⇒ 2x – y = 5

Multiply 2x – y = 5 by 4:

8x – 4y = 20

Subtract (5x – 4y = –1) from (8x – 4y = 20): 

3x = 21 

⇒ x = 7

Substitute x = 7 into 2x – y = 5:

14 – y = 5 

⇒ y = 9

Check: `(7 + 1)/(9 + 1) = 8/10 = 4/5`

And `(7 - 5)/(9 - 5) = 2/4 = 1/2`

The fraction is `7/9`.