Comparing speeds to infer joint time A is three times as efficient as B and thus completes the job 30 days sooner than B. In how many days will A and B finish the job working together from start to finish?

Introduction / Context:
The problem gives a relative efficiency (A is 3x B) and a difference in their individual completion times (A finishes 30 days sooner). From this, we can find individual times and then the joint time.

Given Data / Assumptions:

• A is three times as efficient as B.
• A needs 30 fewer days than B to complete the job alone.

Concept / Approach:
If B alone takes t days, then A alone takes t/3 days (because A is 3x as fast). The difference t − t/3 = 30 allows solving for t. Then combine rates.

Step-by-Step Solution:

Verification / Alternative check:
In 11.25 days, A contributes 11.25/15 = 0.75 of the job, B contributes 11.25/45 = 0.25, totaling 1 job.

Why Other Options Are Wrong:
15 days is A-alone time, not joint. 45/5 (=9) and 12 days do not match the combined rate.

Common Pitfalls:
Equating “3 times as efficient” with “30 days less” without forming equations, or adding times instead of rates.

Final Answer:
45/4 days