Time-and-Work basics, combined work rate: Worker A can finish a job in 12 days and worker B can finish the same job in 15 days. If A and B work together at their constant daily rates, how much time (in days) will they take to complete the entire work?

Introduction / Context:
This question checks your understanding of the fundamental “work = rate * time” idea used in Time and Work problems. When two workers collaborate, their rates add directly, and the combined time is computed from the sum of their individual daily work rates.

Given Data / Assumptions:

• A alone finishes in 12 days ⇒ rate_A = 1/12 job per day.
• B alone finishes in 15 days ⇒ rate_B = 1/15 job per day.
• Both work together at constant rates until the whole job is completed.

Concept / Approach:
The combined daily rate is the sum of individual rates. If R is the total daily rate, then time T to finish one full job is T = 1 / R. Always keep units consistent: here, all times are in days and rates are jobs per day.

Step-by-Step Solution:
rate_A = 1/12rate_B = 1/15Combined rate R = 1/12 + 1/15 = (5 + 4)/60 = 9/60 = 3/20 job/dayTime T = 1 / R = 1 / (3/20) = 20/3 days = 6 2/3 days

Verification / Alternative check:
In 6 2/3 days, the pair completes (3/20) * (20/3) = 1 job. Any deviation (for example, 6 days or 8 days) would not satisfy R * T = 1.

Why Other Options Are Wrong:

• 8 days: too long; at rate 3/20, that yields only 1.2 jobs? No—8*(3/20)=1.2 is more than one; 8 is inconsistent with the computed exact time.
• 10 1/15 days: not derived from rates; it overestimates time.
• 6 days: too short since 6*(3/20)=0.9 job.

Common Pitfalls:

• Adding times instead of rates. Times never add directly; rates do.
• Arithmetic slip when converting to a mixed fraction.

Final Answer:
6 2/3 days