A is 30% more efficient than B. How much time will A and B, working together at their usual efficiencies, take to complete a job which A alone could complete in 23 days?

Introduction / Context:
This time and work problem involves comparing efficiencies of two workers and determining how long they will take to finish a job when working together. You are told that A is 30 percent more efficient than B, and that A alone could finish the job in 23 days. The objective is to calculate the combined time taken by A and B working together to complete the same job.

Given Data / Assumptions:
A alone can complete the job in 23 days. A is 30% more efficient than B. This means A’s efficiency is 1.3 times B’s efficiency. Both work at constant rates when working together. The total work is assumed to be one complete unit of work. There are no breaks or changes in efficiency over time.

Concept / Approach:
We use the relationship between efficiency and time: efficiency is equal to 1 / time taken. Since A’s time is known, we can derive A’s rate. Using the statement that A is 30% more efficient than B, we express B’s rate in terms of A’s rate. Their combined rate is then the sum of their individual rates. Finally, time taken together is the reciprocal of this combined rate. Working with ratios simplifies the calculations, especially when percentage comparisons are involved.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Since A alone finishes the job in 23 days, A’s rate is 1 / 23 work per day. Step 2: We are told that A is 30% more efficient than B. So efficiency(A) = 1.3 * efficiency(B). Step 3: Let B’s efficiency be x work per day. Then A’s efficiency is 1.3x work per day. Step 4: But we already know A’s efficiency is 1 / 23. Therefore, 1.3x = 1 / 23, giving x = 1 / (23 * 1.3). Step 5: Instead of using decimals, consider the ratio of efficiencies. A : B = 1.3 : 1 = 13 : 10. Step 6: Let us assume B’s rate is 10k and A’s rate is 13k for some constant k. Then A’s time alone is 1 / (13k) = 23 days, so 13k = 1 / 23 and the total work is 1 unit. Step 7: Combined rate of A and B = 13k + 10k = 23k. Step 8: Time taken together = total work / combined rate = (13k * 23) / (23k) = 13 days.

Verification / Alternative check:
We may verify by constructing concrete numbers. Since A’s time is 23 days, in 23 days A completes 1 job. Using the ratio 13 : 10, B is 10 / 13 as efficient as A, so B alone would take 23 * 13 / 10 = 29.9 days approximately. Combined rate using actual decimals would be 1 / 23 + 1 / 29.9 which is close to 1 / 13, confirming our exact ratio-based answer of 13 days for the joint work.

Why Other Options Are Wrong:
9 days and 11 days are too small and would imply a combined rate that is unrealistically high compared to A’s individual rate. 15 days and 16 days are too large and correspond to combined rates that are too close to A’s solo rate, ignoring the significant contribution of B. Only 13 days correctly reflects the extra work done by B while being consistent with A’s known efficiency and the 30% difference.

Common Pitfalls:
A common mistake is to treat 30% more efficient as simply adding or subtracting time directly instead of adjusting efficiency. Another pitfall is averaging the days instead of combining rates. Remember that when workers collaborate, always add their work rates, not their times. Misinterpreting percentage efficiency often leads to incorrect proportional relationships between A and B.

Final Answer:
Working together, A and B will complete the job in 13 days.