Simplify the expression $\frac{\frac{a}{a-3}-1}{\frac{a-2}{a-3}}\cdot\frac{a}{a-3}$.

1. Problem Description

Simplify the expression $\frac{\frac{a}{a-3}-1}{\frac{a-2}{a-3}}\cdot\frac{a}{a-3}$.

2. Solution Steps

First, we simplify the numerator of the first fraction:

$\frac{a}{a-3} - 1 = \frac{a}{a-3} - \frac{a-3}{a-3} = \frac{a-(a-3)}{a-3} = \frac{a-a+3}{a-3} = \frac{3}{a-3}$.

Now, we can rewrite the expression as:

$\frac{\frac{3}{a-3}}{\frac{a-2}{a-3}}\cdot\frac{a}{a-3}$.

We simplify the first part of the expression by dividing the two fractions:

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

Assuming $a \neq 3$, we can cancel out $(a-3)$:

$\frac{3}{a-2}$.

Now, we multiply the result by $\frac{a}{a-3}$:

$\frac{3}{a-2} \cdot \frac{a}{a-3} = \frac{3a}{(a-2)(a-3)} = \frac{3a}{a^2 - 3a - 2a + 6} = \frac{3a}{a^2 - 5a + 6}$.

3. Final Answer

$\frac{3a}{a^2-5a+6}$