Question

If one is added to both numerator and denominator of a fraction, it becomes `1/2`. If 2 is added to the denominator of the fraction, it becomes `1/3`. Find the fraction.

Answer 1

Given:

• If 1 is added to the numerator and denominator of a fraction, the fraction becomes `1/2`.
• If 2 is added to the denominator of the fraction, it becomes `1/3`.

Step-wise calculation:

1. Let the original fraction be `x/y`.

2. According to the first condition,

`(x + 1)/(y + 1) = 1/2`

Cross multiply to get:

2(x + 1) = y + 1

2x + 2 = y + 1

2x – y = –1   ...(Equation 1)

3. According to the second condition,

`x/(y + 2) = 1/3` 

Cross multiply to get:

3x = y + 2 

3x – y = 2   ...(Equation 2)

4. Subtract Equation 1 from Equation 2:

(3x – y) – (2x – y) = 2 – (–1) 

3x – y – 2x + y = 3 

x = 3

5. Substitute (x = 3) into Equation 1: 

2(3) – y = –1 

6 – y = –1 

y = 7

The fraction is `3/7`.