Arun is three years older than Varun. Eight years ago, 5/6th of Arun’s age…

To turn an age word problem into an equation: assign a variable to one person's present age, express the other person's present age using the given age difference, and express each person's age at any earlier time as (present age − years elapsed). Then translate a stated fraction-of-age comparison ("5/6th of one age exceeds 3/5th of another by 6") directly into an equation of that same form.

1. Let Varun's present age be x years. Since Arun is three years older than Varun, Arun's present age is (x + 3) years.

2. Eight years ago, Varun's age was (x − 8) years, and Arun's age was (x + 3) − 8 = (x − 5) years.

3. "5/6th of Arun's age (eight years ago) exceeded 3/5th of Varun's age (eight years ago) by 6" translates directly to: 5/6 (x − 5) − 3/5 (x − 8) = 6.

4. Clearing denominators (multiply throughout by 30): 25(x − 5) − 18(x − 8) = 180, i.e. 25x − 125 − 18x + 144 = 180, i.e. 7x + 19 = 180, i.e. 7x = 161, so x = 23.

Cross-check: with x = 23, Varun's present age is 23 and Arun's is 26; eight years ago they were 15 and 18. 5/6 of 18 is 15, and 3/5 of 15 is 9, and 15 − 9 = 6 — exactly the excess the problem states, confirming both the equation and x = 23.

So the equation that determines x is 5/6 (x − 5) − 3/5 (x − 8) = 6.