A and B can together complete a certain task in 18 hours. After working together for 6 hours, A leaves and B alone takes another 36 hours to complete the rest of the task. In how many hours would A alone complete the entire task if he worked by himself from the beginning?

Introduction / Context:
This time and work problem describes a situation where two people start a task together and then one person leaves, leaving the other to finish the remaining work. From the total time together and the solo finishing time, we deduce their individual work rates. Finally, we compute how long A alone would take to finish the entire task. This type of question strengthens understanding of how combined work and individual work relate.

Given Data / Assumptions:
- A and B together can complete the task in 18 hours.
- They actually work together for 6 hours and then A leaves.
- B alone takes an additional 36 hours to finish the remaining work.
- Work rates remain constant for both A and B.
- Total work is assumed to be 1 complete task.

Concept / Approach:
We convert all the times into rates. First we compute the combined rate of A and B from the 18 hour information. Then we calculate how much work is done in the first 6 hours together. The remaining work is done by B alone in 36 hours, so we find B's rate from that. The rate of A is then found by subtracting B's rate from the combined rate. Finally we invert A's rate to find the time A alone would take to complete the entire task.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Combined rate of A and B = 1/18 of the work per hour. Step 3: Work done together in the first 6 hours = 6 * (1/18) = 6/18 = 1/3 of the task. Step 4: Remaining work after 6 hours = 1 − 1/3 = 2/3 of the task. Step 5: B completes this remaining 2/3 of the work in 36 hours, so B's rate = (2/3) / 36 = 2/108 = 1/54 of the work per hour. Step 6: Combined rate was 1/18, so A's rate = combined rate − B's rate = 1/18 − 1/54. Step 7: With denominator 54, 1/18 = 3/54, so A's rate = 3/54 − 1/54 = 2/54 = 1/27 of the work per hour. Step 8: Time taken by A alone to complete the entire work = 1 / (1/27) = 27 hours.

Verification / Alternative check:
Check if the numbers are consistent. In 6 hours together, they do 1/3 of the work. B alone at 1/54 per hour in 36 hours does 36 * 1/54 = 2/3 of the work. Total completed work = 1/3 + 2/3 = 1, which is the whole task. Also, combined rate 1/18 equals 1/27 + 1/54 because 1/27 = 2/54 and 2/54 + 1/54 = 3/54 = 1/18. So all relationships are consistent.

Why Other Options Are Wrong:
Option 54: This would correspond to A working at 1/54 per hour, equal to B, which is not consistent with the combined time of 18 hours and the later solo work by B.
Option 45: Implies a slower A and would result in a combined rate that does not match 1/18 when rechecked with B's rate.
Option 21: Implies A is faster than computed, leading to combined work exceeding the given condition.

Common Pitfalls:
Some learners wrongly average hours or try to directly distribute the work without using rates. Another error is to miscalculate the remaining fraction of work after the first stage. Always use the structure: total work as 1, compute fractions completed in each phase, convert all times to rates, and then carefully solve for unknown rates and times.

Final Answer:
A alone would take 27 hours to complete the entire task.