Explanations:In work rate problems, the key concept is to understand the relationship between the time taken to complete a task and the rate at which the task is completed. The work rate is the fraction of the work done per unit time. If a person can complete a job in \( T \) days, their work rate is \( \frac{1}{T} \) of the job per day.
For this problem, we are given that A and B together can complete the work in 8 days. Therefore, their combined work rate is \( \frac{1}{8} \) of the work per day. We are also given that B alone can complete the work in 12 days, so B's work rate is \( \frac{1}{12} \) of the work per day.
To find A's work rate, we subtract B's work rate from the combined work rate:
\[
\text{A's work rate} = \text{Combined work rate} - \text{B's work rate}
\]
\[
\text{A's work rate} = \frac{1}{8} - \frac{1}{12}
\]
To subtract these fractions, we need a common denominator. The least common multiple of 8 and 12 is 24. Converting the fractions, we get:
\[
\frac{1}{8} = \frac{3}{24} \quad \text{and} \quad \frac{1}{12} = \frac{2}{24}
\]
\[
\text{A's work rate} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24}
\]
This means A's work rate is \( \frac{1}{24} \) of the work per day. To find the time A takes to complete the work alone, we take the reciprocal of A's work rate:
\[
\text{Time taken by A} = \frac{1}{\text{A's work rate}} = \frac{1}{\frac{1}{24}} = 24 \text{ days}
\]
Therefore, A alone can complete the work in 24 days.