A and B can do a job in 18 days, B and C in 24 days, and A and C in 36 days. If all three work together, in how many days will the job be completed?

Introduction / Context:
Pairwise completion times can be used to extract the sum of individual rates and hence the time for all three working together. This is a standard system-of-rates technique.

Given Data / Assumptions:

• (A+B) = 18 days ⇒ r(A)+r(B) = 1/18.
• (B+C) = 24 days ⇒ r(B)+r(C) = 1/24.
• (C+A) = 36 days ⇒ r(C)+r(A) = 1/36.

Concept / Approach:
Add the three equations: 2[r(A)+r(B)+r(C)] = 1/18 + 1/24 + 1/36. Then divide by 2 to get r(A)+r(B)+r(C), the combined rate for all three.

Step-by-Step Solution:
Sum RHS with denominator 72: 1/18=4/72, 1/24=3/72, 1/36=2/72. Total = 9/72 = 1/8. Hence 2[r(A)+r(B)+r(C)] = 1/8 ⇒ r(A)+r(B)+r(C) = 1/16. Therefore, time together = 1 / (1/16) = 16 days.

Verification / Alternative check:
The pairwise rates are consistent; the derived combined rate 1/16 satisfies all constraints and is a common textbook result pattern.

Why Other Options Are Wrong:
12, 13, 26 days are incompatible with the exact rate sum; 10 is too small (too fast).

Common Pitfalls:
Forgetting to divide the summed pairwise rates by 2; arithmetic errors in fractional addition.

Final Answer:
16 days