A and B together can complete a piece of work in 9 days. In any given time interval, A does thrice as much work as B. In how many days will A alone finish the entire work if he works by himself from the beginning?

Introduction / Context:
This problem is based on relative efficiencies where one person does a multiple of the work done by another in the same time. We are told that A and B together complete the work in 9 days and that A does three times as much work as B in any given time. Using this relationship, we can find the individual rates of A and B and from that deduce how long A alone would take to complete the entire job.

Given Data / Assumptions:

• A and B together complete the work in 9 days.
• A does thrice the work of B in the same time (efficiency ratio A : B = 3 : 1).
• Both work at constant rates.
• Total work is treated as 1 unit.

Concept / Approach:
Let B's daily work rate be b units. Then A's rate is 3b units because A is three times as efficient as B. The combined rate of A and B is therefore 4b units per day. Since they can finish the work in 9 days together, we know that 4b * 9 = 1. From this, we find b and then determine A's rate as 3b. The time taken by A alone is simply 1 divided by his daily rate.

Step-by-Step Solution:
Let B's daily rate = b units per day. Then A's daily rate = 3b units per day (since A does thrice the work of B). Combined daily rate of A and B = 3b + b = 4b. They complete the work in 9 days, so total work W = 4b * 9 = 36b. Set W = 1 unit, so 36b = 1. Thus, b = 1 / 36 units per day. A's daily rate = 3b = 3 * (1 / 36) = 1 / 12 units per day. Time taken by A alone = total work / A's rate = 1 / (1 / 12) = 12 days.

Verification / Alternative check:
In 12 days, A alone would complete the whole work. B's time would be 1 / b = 36 days. As a cross-check, if A and B work together, their combined rate is 1 / 12 + 1 / 36 = (3 + 1) / 36 = 4 / 36 = 1 / 9. This means they together take 9 days to finish the work, which perfectly matches the given condition in the question.

Why Other Options Are Wrong:
Option A (4 days), Option B (6 days) and Option C (8 days) all correspond to incorrect values of A's rate that do not satisfy the condition that A and B together finish in 9 days while A does thrice the work of B. Only 12 days leads to a consistent set of rates (1 / 12 and 1 / 36) that satisfy all given relationships.

Common Pitfalls:
Students sometimes reverse the ratio and treat B as three times more efficient than A, which leads to incorrect answers. Another typical error is to try averaging times instead of converting to rates. Remember that time is inversely proportional to efficiency; if A is three times as efficient as B, his time alone should be one-third of B's time, not vice versa.

Final Answer:
A alone will finish the entire work in 12 days.