A takes twice as much time as B to finish a piece of work. Working together, they can complete the work in 2 hours. In how many hours can B alone complete the same work if he works by himself from the beginning?

Introduction / Context:
This is a classic Time and Work question involving two workers, A and B, whose times are in a simple ratio. We are told that A is slower, taking twice as much time as B to finish the entire work. When they work together, they complete the job in 2 hours. Using this information, we must determine the time B alone would need to finish the same work.

Given Data / Assumptions:

• A takes twice as much time as B to complete the work.
• Working together, A and B finish the work in 2 hours.
• Both work at constant rates throughout.
• Total work is considered as 1 unit.

Concept / Approach:
When time taken is known, efficiency (rate of work) is the reciprocal of time. If A takes twice as long as B, then A's efficiency is half of B's. We let B's time be x hours, so A's time is 2x hours, and then we convert these times to rates 1 / x and 1 / (2x). Their combined rate is the sum of individual rates. Since their combined time is 2 hours, the combined rate must be 1 / 2 of the work per hour. We set up an equation and solve for x, which gives B's time alone.

Step-by-Step Solution:
Let B alone take x hours to complete the work. Then A alone takes 2x hours (as given). Rate of B = 1 / x units per hour. Rate of A = 1 / (2x) units per hour. Combined rate of A and B = 1 / x + 1 / (2x). = (2 + 1) / (2x) = 3 / (2x) units per hour. They complete the work together in 2 hours, so combined rate = 1 / 2 unit per hour. Thus, 3 / (2x) = 1 / 2. Cross-multiplying: 3 * 2 = 2x * 1 ⇒ 6 = 2x. So, x = 3 hours. Therefore, B alone can complete the work in 3 hours.

Verification / Alternative check:
If B takes 3 hours alone, then A takes 2 * 3 = 6 hours alone. That means B's rate is 1 / 3 and A's rate is 1 / 6. Combined rate = 1 / 3 + 1 / 6 = 1 / 2. Time taken together = 1 / (1 / 2) = 2 hours, which exactly matches the given condition. This confirms the correctness of the result x = 3 hours.

Why Other Options Are Wrong:
If B took 6 hours alone (Option B), A would take 12 hours, giving rates 1 / 6 and 1 / 12; their combined rate would be 1 / 4, implying 4 hours together, not 2 hours. Similarly, with 5 hours (Option D), A would need 10 hours, giving combined time different from 2 hours. Option C (7 hours) is also inconsistent in the same way. Only 3 hours gives a combined time of exactly 2 hours, as required in the problem statement.

Common Pitfalls:
Some students mistakenly assume that since A takes twice as long as B, B's time must simply be 1 hour if the combined time is 2 hours, which is incorrect because work rates do not combine linearly in that way. Another common error is to forget that rates add, not times. Always convert times to rates before summing them, and then use the combined rate to find the total time.

Final Answer:
B alone can complete the work in 3 hours.