Step 1: Understanding the Question:
The question requires us to calculate the exact number of days person A would take to complete a task entirely alone.
We are given two major conditions.
First, A is three times as efficient as B, and together they can finish the work in exactly 3 days.
Second, if they were to work individually, B would take 8 days more than A to complete the same work.
Step 2: Key Formula or Approach:
Efficiency is defined as the amount of work done per unit of time and is inversely proportional to the time taken.
If the efficiency ratio of A to B is $3:1$, the time taken ratio of A to B will be $1:3$.
Total work can be calculated as the product of total combined efficiency and the total time taken together.
Step 3: Detailed Explanation:
Let us assume the efficiency of worker B is $x$ units of work per day.
Since A is thrice as efficient as B, the efficiency of worker A will be $3x$ units per day.
When A and B work together, their combined efficiency is the sum of their individual efficiencies.
Combined efficiency = $3x + x = 4x$ units per day.
We are given that they finish the whole work together in exactly 3 days.
Total work can be found by multiplying their combined efficiency by the number of days they worked together.
Total work = $4x \text{ units/day} \times 3 \text{ days} = 12x$ units.
Now, we must find the time taken by A alone to finish this total work.
Time taken by A alone = $\frac{\text{Total Work}}{\text{Efficiency of A}}$.
Time taken by A = $\frac{12x}{3x} = 4$ days.
To verify our answer, we can check the secondary condition given in the problem.
Time taken by B alone = $\frac{12x}{x} = 12$ days.
The difference in their times is $12 - 4 = 8$ days.
This matches the problem's statement that B takes 8 days more than A perfectly.
Thus, A takes 4 days to finish the work alone.
Step 4: Final Answer:
The number of days taken by A to finish the whole work alone is 4.