Joint work time from individual rates: Worker A can complete the job in 4 days, while worker B can complete the same job in 12 days. If both A and B work together at their constant daily rates from start to finish, how many days will they take to complete the entire work?

Introduction / Context:
This problem reinforces the core Time-and-Work principle that individual work rates add when people work together. We convert each person’s completion time into a daily rate (jobs per day), add the rates, and take the reciprocal to get the total time.

Given Data / Assumptions:

• A alone finishes in 4 days ⇒ rate_A = 1/4 job per day.
• B alone finishes in 12 days ⇒ rate_B = 1/12 job per day.
• They work together continuously at constant rates.

Concept / Approach:
For one complete job, total time T equals 1 divided by the combined rate R. That is, T = 1 / (rate_A + rate_B). Always add rates, not times, when tasks are done simultaneously.

Step-by-Step Solution:
rate_A = 1/4rate_B = 1/12Combined rate R = 1/4 + 1/12 = 3/12 + 1/12 = 4/12 = 1/3 job/dayTime T = 1 / (1/3) = 3 days

Verification / Alternative check:
In 3 days, A would complete 3 * (1/4) = 3/4 of the job and B would complete 3 * (1/12) = 1/4 of the job. Total = 3/4 + 1/4 = 1 job, confirming correctness.

Why Other Options Are Wrong:

• 2 days: too short; at rate 1/3 per day, 2 days produce only 2/3 of the job.
• 4 days and 5 days: too long; they exceed the exact reciprocal of the combined rate.

Common Pitfalls:

• Adding the times 4 and 12 directly, instead of adding rates 1/4 and 1/12.
• Arithmetic slip in finding the common denominator and simplifying 1/4 + 1/12.

Final Answer:
3 days