The average of two positive real numbers is equal to their difference. What is the ratio of the larger number to the smaller one?

[i]Author: Ray Li[/i]
1. Let \( a \) and \( b \) be the two positive real numbers, where \( a > b \).
2. According to the problem, the average of the two numbers is equal to their difference. This can be written as:
   \[
   \frac{a + b}{2} = a - b
   \]
3. To eliminate the fraction, multiply both sides of the equation by 2:
   \[
   a + b = 2(a - b)
   \]
4. Distribute the 2 on the right-hand side:
   \[
   a + b = 2a - 2b
   \]
5. To isolate \( a \), subtract \( a \) from both sides:
   \[
   b = 2a - 2b - a
   \]
   Simplifying, we get:
   \[
   b = a - 2b
   \]
6. Add \( 2b \) to both sides to isolate \( a \):
   \[
   a = 3b
   \]
7. The ratio of the larger number \( a \) to the smaller number \( b \) is:
   \[
   \frac{a}{b} = \frac{3b}{b} = 3
   \]

The final answer is \(\boxed{3}\).