A is twice as efficient a worker as B, and together they finish a piece of work in 16 days. In how many days will A alone finish the entire work?

Introduction / Context:
Here we are given a relationship between the efficiencies of two workers, A and B, and the time they take together to finish the work. The phrase that A is twice as good as B means that A works at twice the rate of B. Using this relationship, we can express their rates in terms of a variable, find the combined rate using the given joint time, and then determine A’s individual time to complete the job.

Given Data / Assumptions:
- A is twice as efficient as B.
- A and B together complete the work in 16 days.
- Total work is assumed to be 1 unit.
- Work rates are constant over time.

Concept / Approach:
Let the daily work rate of B be x units. Then the rate of A is 2x units per day. Together, their combined rate is 3x units per day. Since they finish the total work in 16 days, the combined rate is also 1 / 16 units per day. Equating 3x to 1 / 16 allows us to solve for x and then compute A’s individual time by taking the reciprocal of A’s rate, which is 2x.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Let the rate of B = x units per day. Step 3: Since A is twice as efficient, rate of A = 2x units per day. Step 4: Combined rate of A and B = 2x + x = 3x units per day. Step 5: They finish the work in 16 days, so 3x = 1 / 16. Step 6: Solve for x: x = 1 / (16 * 3) = 1 / 48. Step 7: Rate of A = 2x = 2 * (1 / 48) = 1 / 24 units per day. Step 8: Time taken by A alone = 1 / (1 / 24) = 24 days.

Verification / Alternative check:
If A alone takes 24 days, then B’s rate is half of A’s rate, so B alone would take 48 days. Combined rate should be 1 / 24 + 1 / 48 = 2 / 48 + 1 / 48 = 3 / 48 = 1 / 16, which matches the given joint time of 16 days. This confirms that A’s individual time is correctly computed as 24 days.

Why Other Options Are Wrong:
- 20, 21, 22 days: These values do not maintain the correct ratio between A and B’s times and fail to reproduce the combined time of 16 days.
- 18 days: This would imply an even higher efficiency for A, leading to a combined rate larger than 1 / 16 and thus a shorter joint time than given.

Common Pitfalls:
Learners sometimes confuse the ratio of efficiency with the ratio of time. If A is twice as efficient as B, then A’s time is half of B’s time, not double. Another typical mistake is forgetting to express the combined work rate in terms of the defined variable and equating it to 1 divided by the given total time.

Final Answer:
A alone will complete the work in 24 days.