A and B together can complete a work in 12 days; B and C together can complete it in 16 days. A works alone for 5 days; B works alone for 7 days; then C works alone and finishes the job in 13 days. In how many days can C alone complete the entire work?

Introduction / Context:
We are given pairwise joint times and a staged completion where each person works alone for specified days. Setting individual daily rates for A, B, C and using the data leads to a solvable linear system.

Given Data / Assumptions:

• a + b = 1/12 (job/day).
• b + c = 1/16 (job/day).
• 5 days of A, 7 days of B, and 13 days of C complete the job: 5a + 7b + 13c = 1.

Concept / Approach:
Solve the system for c. From the first two equations express a and c in terms of b, or use elimination. Once c is known, C’s solo time is 1/c.

Step-by-Step Solution:

Verification / Alternative check:
Check sums: a + b = 1/16 + 1/48 = 3/48 + 1/48 = 1/12; b + c = 1/48 + 1/24 = 1/48 + 2/48 = 1/16; and 5a + 7b + 13c = 5/16 + 7/48 + 13/24 = 1 (after common denominator), confirming consistency.

Why Other Options Are Wrong:
16/32/48/20 days do not match the solved value c = 1/24.

Common Pitfalls:
Trying to average the given times or mis-adding fractions; always set rate equations first.

Final Answer:
24 days