The problem is to simplify the following expression: $(\frac{a-3}{a} - \frac{a-3}{a^2-2a}) : \frac{a-3}{a}$.

1. Problem Description

The problem is to simplify the following expression:

$(\frac{a-3}{a} - \frac{a-3}{a^2-2a}) : \frac{a-3}{a}$.

2. Solution Steps

First, we simplify the expression inside the parenthesis.

$\frac{a-3}{a} - \frac{a-3}{a^2-2a} = \frac{a-3}{a} - \frac{a-3}{a(a-2)}$

Find a common denominator for the two fractions:

$\frac{(a-3)(a-2)}{a(a-2)} - \frac{a-3}{a(a-2)} = \frac{(a-3)(a-2) - (a-3)}{a(a-2)}$

Factor out $(a-3)$ in the numerator:

$\frac{(a-3)(a-2-1)}{a(a-2)} = \frac{(a-3)(a-3)}{a(a-2)} = \frac{(a-3)^2}{a(a-2)}$

Now, we divide this by $\frac{a-3}{a}$:

$\frac{(a-3)^2}{a(a-2)} : \frac{a-3}{a} = \frac{(a-3)^2}{a(a-2)} \cdot \frac{a}{a-3}$

Cancel out the common factors:

$\frac{(a-3)(a-3) \cdot a}{a(a-2)(a-3)} = \frac{a-3}{a-2}$

3. Final Answer

The simplified expression is $\frac{a-3}{a-2}$.