Sure, let's solve the problem step by step.
Step 1: Determine A's work rate
First, we need to find out how much work A can do in one day.
Given that A can complete [tex]\(\frac{3}{4}\)[/tex] of the work in 12 days, we calculate A's daily work rate by dividing the fraction of the work completed by the time it takes:
[tex]\[ \text{Daily Work Rate} = \frac{\frac{3}{4} \text{ of the work}}{12 \text{ days}} = \frac{3}{4} \div 12 \][/tex]
[tex]\[ \text{Daily Work Rate} = \frac{3}{4} \times \frac{1}{12} = \frac{3}{48} = \frac{1}{16} \][/tex]
So, A can complete [tex]\(\frac{1}{16}\)[/tex] of the work per day.
Step 2: Determine the time to finish [tex]\(\frac{1}{8}\)[/tex] of the work
Next, we need to find out how many days it will take for A to complete [tex]\(\frac{1}{8}\)[/tex] of the work. To do this, we divide the desired fraction of the work by A's daily work rate:
[tex]\[ \text{Days Needed} = \frac{\frac{1}{8}}{\text{Daily Work Rate}} = \frac{\frac{1}{8}}{\frac{1}{16}} \][/tex]
[tex]\[ \text{Days Needed} = \frac{1}{8} \times \frac{16}{1} = \frac{16}{8} = 2 \text{ days} \][/tex]
Therefore, it will take A 2 days to finish [tex]\(\frac{1}{8}\)[/tex] of the work.
So the answer is [tex]\( \boxed{2} \)[/tex].
