Sum of rates method, two workers: A can finish a job in 14 days, while B can finish it in 21 days. If both A and B work together at their constant rates, in how many days will they complete the entire work?

Introduction / Context:
When two workers collaborate, their individual work rates add up. The total time to complete one whole job equals 1 divided by the combined rate. This problem reinforces that principle with clean numbers.

Given Data / Assumptions:

• A alone: 14 days ⇒ rate_A = 1/14 job/day.
• B alone: 21 days ⇒ rate_B = 1/21 job/day.
• They work together from start to finish.

Concept / Approach:
Combined rate R = rate_A + rate_B. Time T = 1/R. Keep fractions precise to avoid rounding mistakes; convert to decimal at the end if needed.

Step-by-Step Solution:
rate_A = 1/14rate_B = 1/21R = 1/14 + 1/21 = (3 + 2)/42 = 5/42 job/dayT = 1 / (5/42) = 42/5 = 8.4 days

Verification / Alternative check:
Check work done in 8.4 days: 8.4 * (5/42) = 8.4 * 0.119047… = 1 job (exact since 42/5 * 5/42 = 1).

Why Other Options Are Wrong:

• 10.5: equals A’s half-time notionally, not a valid combined time.
• 8 and 9: rounded guesses not consistent with exact 42/5.

Common Pitfalls:

• Adding times instead of rates.
• Arithmetic error in LCM of 14 and 21 (LCM is 42).

Final Answer:
8.4