Work allocation with partial progress: A and B together can complete a job in 24 days. B alone completes one-third of the job in 12 days. How long will A alone take to complete the remaining two-thirds of the job?

Introduction / Context:
This problem blends combined work with a partial-completion scenario. We are told B's performance for one-third of the job and the joint time for the full job. From these, we compute A’s rate and then the time A takes to finish the remaining portion alone.

Given Data / Assumptions:

• Together time (A + B) = 24 days ⇒ joint rate = 1/24 job/day.
• B does 1/3 of the job in 12 days ⇒ B’s rate = (1/3) / 12 = 1/36 job/day.
• Uniform work rates; total work = 1 job.

Concept / Approach:
Compute A’s rate as (joint rate − B’s rate). The remaining work after B’s one-third is two-thirds. Time for A to finish that amount is (remaining work) / (A’s rate).

Step-by-Step Solution:
Joint rate = 1/24; B’s rate = 1/36. A’s rate = 1/24 − 1/36 = (3 − 2)/72 = 1/72 job/day. Remaining work after B’s 1/3 = 2/3. Time for A = (2/3) / (1/72) = (2/3)*72 = 48 days.

Verification / Alternative check:
A alone for the whole job would take 72 days (since rate 1/72). Doing only two-thirds thus requires 48 days, which matches the computation.

Why Other Options Are Wrong:
36 or 24 days underestimate A’s slower solo rate; 72 days corresponds to the full job by A, not just the remainder.

Common Pitfalls:
Treating 12 days as B's whole-job time (it is only for one-third). Always convert to per-day rates before combining or subtracting.

Final Answer:
48 days