Arun is three years older than Varun. Eight years ago, $\frac{5}{6}$ th of Arun's age exceeded $\frac{3}{5}$ th of Varun's
age by 6 years. If the present age of Varun is x years, then the value of x can be determined by
solving the equation :

Question

Accepted Answer

To determine the correct equation for finding Varun's present age, let's analyze the problem step by step and see why Option 1 is the correct choice.

### Problem Analysis

1. **Given Information:**
   - Arun is three years older than Varun.
   - Eight years ago, $\frac{5}{6}$ of Arun's age exceeded $\frac{3}{5}$ of Varun's age by 6 years.

2. **Variables:**
   - Let Varun's present age be $x$ years.
   - Therefore, Arun's present age is $x + 3$ years.

3. **Eight Years Ago:**
   - Varun's age was $x - 8$.
   - Arun's age was $(x + 3) - 8 = x - 5$.

4. **Equation Formation:**
   - Eight years ago, $\frac{5}{6}$ of Arun's age exceeded $\frac{3}{5}$ of Varun's age by 6 years.
   - This translates to the equation:
     $$
     \frac{5}{6}(x - 5) - \frac{3}{5}(x - 8) = 6
     $$

### Explanation of Option 1

- **Option 1:** $\frac{5}{6}(x-5)-\frac{3}{5}(x – 8) = 6$

This equation correctly represents the situation described:
  - $\frac{5}{6}(x - 5)$ calculates $\frac{5}{6}$ of Arun's age eight years ago.
  - $\frac{3}{5}(x - 8)$ calculates $\frac{3}{5}$ of Varun's age eight years ago.
  - The difference between these two values is given as 6, which matches the problem statement.

### Explanation of Why Other Options are Incorrect

- **Option 2:** $\frac{5}{6}(x-8)-\frac{3}{5}(x - 5) = 6$

This option incorrectly swaps the ages in the fractions:
  - $\frac{5}{6}(x-8)$ implies calculating $\frac{5}{6}$ of Varun's age, which is incorrect.
  - $\frac{3}{5}(x - 5)$ implies calculating $\frac{3}{5}$ of Arun's age, which is incorrect.

- **Option 3:** $\frac{3}{5}(x + 5)-\frac{5}{6}(x - 8) = 6$

This option uses incorrect expressions for both ages:
  - $\frac{3}{5}(x + 5)$ suggests calculating $\frac{3}{5}$ of an incorrect age for Varun.
  - $\frac{5}{6}(x - 8)$ incorrectly uses Varun's age instead of Arun's.

- **Option 4:** $\frac{3}{5}(x + 3)-\frac{5}{6}(x - 8) = 6$

This option also uses incorrect expressions:
  - $\frac{3}{5}(x + 3)$ incorrectly calculates $\frac{3}{5}$ of Arun's current age.
  - $\frac{5}{6}(x - 8)$ incorrectly uses Varun's age instead of Arun's.

### Conclusion

Option 1 is the only equation that correctly models the relationship described in the problem. It accurately represents the ages of Arun and Varun eight years ago and the condition that $\frac{5}{6}$ of Arun's age exceeded $\frac{3}{5}$ of Varun's age by 6 years.