Question

The sum of numerator and denominator of a fraction is 18. If 3 is added to both numerator and denominator of the fraction it becomes `1/2`. Find the fraction.

Answer 1

Given:

• The sum of numerator and denominator of a fraction is 18.
• If 3 is added to both numerator and denominator, the fraction becomes `1/2`.

Let the numerator be (x) and the denominator be (y).

Then, we have the following:

Step 1: Write the sum equation:

x + y = 18

Step 2: Write the equation for the fraction after adding 3 to numerator and denominator:

`(x + 3)/(y + 3) = 1/2`

Cross-multiplying:

2(x + 3) = y + 3

2x + 6 = y + 3

2x – y = –3

Step 3: We now have the system of linear equations:

x + y = 18

2x – y = –3

Step 4: Add the two equations to eliminate (y):

(x + y) + (2x – y) = 18 + (–3) 

3x = 15 

x = 5

Step 5: Substitute (x = 5) into the first equation to find (y):

5 + y = 18

⇒ y = 13

The fraction initially is `5/13`.