1. **Problem Statement:** Given that A and B together can complete a work in 12 days, and B alone can complete the same work in 30 days, find how many days A alone will take to finish the work. 2. **Formula and Important Rules:** The work done per day by a person or group is the reciprocal of the number of days they take to complete the work. If $T$ is the number of days to complete the work, then work done per day $= \frac{1}{T}$. When two people work together, their combined work per day is the sum of their individual work per day: $$\text{Work per day of } A + B = \text{Work per day of } A + \text{Work per day of } B$$ 3. **Given:** - $(A + B)$'s 1 day's work $= \frac{1}{12}$ - $B$'s 1 day's work $= \frac{1}{30}$ 4. **Find:** $A$'s 1 day's work and then the number of days $A$ alone takes to finish the work. 5. **Calculation:** $$\text{Work per day of } A = \text{Work per day of } (A + B) - \text{Work per day of } B = \frac{1}{12} - \frac{1}{30}$$ Find a common denominator for $\frac{1}{12}$ and $\frac{1}{30}$, which is 60: $$\frac{1}{12} = \frac{5}{60}, \quad \frac{1}{30} = \frac{2}{60}$$ So, $$\text{Work per day of } A = \frac{5}{60} - \frac{2}{60} = \frac{3}{60} = \frac{1}{20}$$ 6. **Interpretation:** Since $A$ does $\frac{1}{20}$ of the work in one day, $A$ alone will take 20 days to complete the work. **Final answer:** $$\boxed{20 \text{ days}}$$