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In algebra, dividing by a fraction involves multiplying by its reciprocal. The reciprocal of a fraction is simply switching the numerator and the denominator. This is a crucial concept when simplifying divisions involving fractions.
For example, if you have a fraction \ \( \frac{b}{b-a} \ \), the reciprocal will be \ \( \frac{b-a}{b} \ \).
Now apply this concept to the given problem: instead of dividing by \ \( \frac{b}{b-a} \ \), you multiply by \ \( \frac{b-a}{b} \ \).
This transforms \ \( \frac{a}{a-b} \ \div \ \frac{b}{b-a} \ \) \ into \ \( \frac{a}{a-b} \ \times \ \frac{b-a}{b} \ \).
Switching to multiplication makes it easier to handle the fractions.

Simplifying algebraic expressions is all about transforming complex equations into simpler forms. This often involves recognizing opportunities to rewrite parts of the expression.
A useful technique is substituting equivalent values. In our example, notice that \ \( b-a \ \) can also be rewritten as \ \(-(a-b) \).
This gives us \ \( \frac{a}{a-b} \ \times \ \frac{-(a-b)}{b} \).
Simplification makes further steps easier. By simplifying first, we prepare the expression for easier factor cancellation later.

Cancelling common factors is essential to simplify the expressions thoroughly. The idea is to identify terms in the numerator and the denominator that are identical or can be made identical through equivalent transformations.
In the expression \ \frac{a}{a-b} \ \times \ \frac{-(a-b)}{b} \ the term \ \(a-b \) appears both in the numerator and the denominator.
This common factor can be cancelled out, simplifying the expression to \ \( \frac{a}{1} \ \times \ \frac{-1}{b} \).
After cancelling common factors, you can further simplify: \ \frac{a}{1} \ just becomes \ \a \, and multiplying \ \a \ by \ \frac{-1}{b} \ gives us \ \frac{-a}{b} \. By cancelling common factors, we've simplified the initial complex expression significantly.