A can finish one-third (1/3) of a job in 5 days, while B can finish two-fifths (2/5) of the job in 10 days. In how many days can A and B together complete the entire job?

Introduction / Context: This problem gives partial completion benchmarks. Convert each to a full-job time or directly to daily rates, then add to get the joint rate and invert for total time.

Given Data / Assumptions:

• A completes 1/3 in 5 days ⇒ daily rate r(A) = (1/3)/5 = 1/15.
• B completes 2/5 in 10 days ⇒ daily rate r(B) = (2/5)/10 = 1/25.

Concept / Approach: Combined daily rate r = r(A) + r(B) = 1/15 + 1/25. Total time = 1 / r.

Step-by-Step Solution: 1/15 + 1/25 = (5 + 3)/75 = 8/75. Total time = 1 / (8/75) = 75/8 = 9.375 days = 9 3/8 days.

Verification / Alternative check: Decimal verification: 9.375 * (8/75) = 1, so the calculation is exact.

Why Other Options Are Wrong: 7 3/4, 8 4/5, 10 days do not match the sum-of-rates inversion; 9 days is close but not exact.

Common Pitfalls: Treating partial-job times as direct times instead of converting to rates; missing fraction simplifications.

Final Answer: 9 3/8 days