1. **State the problem:** Find the reciprocal of $$\frac{1}{\frac{a}{b}} + \frac{1}{\frac{c}{ab}}$$. 2. **Rewrite the expression:** The expression inside the reciprocal is $$\frac{1}{\frac{a}{b}} + \frac{1}{\frac{c}{ab}}$$. 3. **Simplify each term:** - $$\frac{1}{\frac{a}{b}} = \frac{b}{a}$$ because dividing by a fraction is the same as multiplying by its reciprocal. - $$\frac{1}{\frac{c}{ab}} = \frac{ab}{c}$$ for the same reason. So the expression becomes $$\frac{b}{a} + \frac{ab}{c}$$. 4. **Find a common denominator to add the fractions:** - The denominators are $$a$$ and $$c$$. - The common denominator is $$ac$$. Rewrite each fraction: $$\frac{b}{a} = \frac{b \cdot c}{a \cdot c} = \frac{bc}{ac}$$ $$\frac{ab}{c} = \frac{ab \cdot a}{c \cdot a} = \frac{a^2 b}{ac}$$ 5. **Add the fractions:** $$\frac{bc}{ac} + \frac{a^2 b}{ac} = \frac{bc + a^2 b}{ac}$$ 6. **Factor the numerator:** $$bc + a^2 b = b(c + a^2)$$ So the sum is $$\frac{b(c + a^2)}{ac}$$. 7. **The original problem asks for the reciprocal of this sum:** $$\text{Reciprocal} = \frac{1}{\frac{b(c + a^2)}{ac}} = \frac{ac}{b(c + a^2)}$$. 8. **Final answer:** $$\boxed{\frac{ac}{b(c + a^2)}}$$