A is twice as efficient as B, and together A and B take the same time to complete a job as C and D working together. If C alone can complete the work in 20 days and D alone can complete it in 30 days, in how many days can A alone complete the entire work?

Introduction / Context:
This time and work question tests your understanding of efficiency comparison between different workers and how to convert their relative efficiencies into actual time to complete a job. You are given information about workers C and D, and a relationship between the combined work of A and B and that of C and D, and you must determine how long A alone would take to finish the same work.

Given Data / Assumptions:
We are given that C alone can finish the job in 20 days and D alone can finish the job in 30 days. A is twice as efficient as B, which means A does work at twice the rate of B. A and B working together take the same time as C and D working together to finish the entire job. The total work is considered as one complete unit of work.

Concept / Approach:
The key concept is that efficiency is measured as work rate, which is equal to 1 / time. When two people work together, their combined rate is the sum of their individual rates. We first compute the combined rate of C and D. Then, using the fact that A and B together take the same time as C and D, we equate the combined rate of A and B to that of C and D. Using the relation A is twice as efficient as B, we solve for A’s rate and finally for A’s time to complete the work alone.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Then the rate of C is 1 / 20 work per day. Step 2: The rate of D is 1 / 30 work per day. Step 3: The combined rate of C and D is 1 / 20 + 1 / 30 = (3 + 2) / 60 = 5 / 60 = 1 / 12 work per day. Hence C and D together take 12 days. Step 4: We are told that A and B together take the same time as C and D, so the combined rate of A and B is also 1 / 12 work per day. Step 5: Let B’s rate be b work per day. Then A is twice as efficient as B, so A’s rate is 2b. Step 6: Combined rate of A and B is therefore 2b + b = 3b. We know this equals 1 / 12, so 3b = 1 / 12. Step 7: Solving for b gives b = 1 / 36 work per day. Step 8: The rate of A is 2b = 2 * (1 / 36) = 1 / 18 work per day. Step 9: Time taken by A alone is the reciprocal of his rate: time = 1 / (1 / 18) = 18 days.

Verification / Alternative check:
We can quickly verify this by computing A and B’s combined time. B takes 36 days alone, A takes 18 days alone. Together, their combined rate is 1 / 18 + 1 / 36 = 2 / 36 + 1 / 36 = 3 / 36 = 1 / 12. This matches the combined rate of C and D, which also gives 1 / 12 work per day. So the condition in the problem statement is satisfied and the answer is consistent.

Why Other Options Are Wrong:
12 days would mean A is far more efficient than implied by the relation with B and would not satisfy the equality with the combined work of C and D. 24 days and 30 days both imply slower rates for A and lead to a combined rate of A and B that is less than 1 / 12, so they would take longer than C and D. 36 days would make A even slower than B, which contradicts the statement that A is twice as efficient as B.

Common Pitfalls:
A common mistake is to confuse the phrase “twice as efficient” with “twice the time.” Efficiency and time are inversely related, so if someone is twice as efficient, they actually take half the time, not double. Another frequent error is to incorrectly add or subtract rates or to forget that when people work together, their rates add, but their times do not add directly. Also, students sometimes compute the combined time of C and D incorrectly, which affects the rest of the calculation.

Final Answer:
Therefore, A alone can complete the entire work in 18 days.