Pairwise work times for three workers: A and B can complete a job in 12 days, B and C in 15 days, and C and A in 20 days. In how many days will A alone complete the work?

Introduction / Context:
When pairwise completion times are given, summing pair rates provides the total combined rate of all three workers. From that, each individual rate can be extracted by subtracting the appropriate pair rate. This method avoids solving simultaneous equations explicitly.

Given Data / Assumptions:

• (A + B) finish in 12 days ⇒ rate = 1/12.
• (B + C) finish in 15 days ⇒ rate = 1/15.
• (C + A) finish in 20 days ⇒ rate = 1/20.
• Job size assumed as 1 unit of work.

Concept / Approach:
Add the three pair rates: (A+B) + (B+C) + (C+A) = 2(A+B+C). Hence, total triple rate is half of that sum. Then, A’s rate = (A+B+C) − (B+C). Finally, the time for A alone is 1 divided by A’s rate.

Step-by-Step Solution:

Verification / Alternative check:
With A = 1/30, B + C = 1/15. Also A + B = 1/12, so B = 1/12 − 1/30 = 1/20; then C = 1/15 − B = 1/15 − 1/20 = 1/60. Checks with C + A = 1/60 + 1/30 = 1/20, consistent.

Why Other Options Are Wrong:
20 or 45 days contradict the derived individual rates; 60 days would halve A’s rate and mismatch pair totals.

Common Pitfalls:
Forgetting to divide the sum of pair rates by 2, or mixing up which pair to subtract to isolate A’s rate.

Final Answer:
30 days