Isolating a third worker’s rate from total: Workers A, B, and C together complete a job in 2 hours. A alone would take 6 hours and B alone would take 5 hours. How long would C alone take to finish the job if working at a constant rate?

Introduction / Context:
When three workers together finish a job in a given time, we can subtract known individual rates from the total joint rate to discover the unknown third worker’s rate. This is a standard pipes-and-cisterns/time-and-work technique based on linear additivity of rates.

Given Data / Assumptions:

• Total time (A+B+C) = 2 h ⇒ combined rate = 1/2 job per hour.
• A alone: 6 h ⇒ rate_A = 1/6 job per hour.
• B alone: 5 h ⇒ rate_B = 1/5 job per hour.

Concept / Approach:
Let rate_C be C’s solo rate. Since rates add: rate_A + rate_B + rate_C = 1/2. Hence, rate_C = 1/2 − 1/6 − 1/5. Once rate_C is known, C’s time alone is the reciprocal, 1 / rate_C.

Step-by-Step Solution:
rate_A = 1/6, rate_B = 1/5, total rate = 1/2rate_C = 1/2 − 1/6 − 1/5Use denominator 30: 1/2 = 15/30, 1/6 = 5/30, 1/5 = 6/30rate_C = 15/30 − 5/30 − 6/30 = 4/30 = 2/15 job per hourTime_C = 1 / (2/15) = 15/2 hours = 7 1/2 h

Verification / Alternative check:
Re-sum rates: 1/6 + 1/5 + 2/15 = 5/30 + 6/30 + 4/30 = 15/30 = 1/2 job per hour → total time 2 h, consistent.

Why Other Options Are Wrong:

• 5 1/2 h and 4 1/2 h: imply rates greater than 2/15, which would overshoot the total when summed with A and B.
• 9 h: implies a rate lower than 2/15 and fails to reach the total combined rate of 1/2.

Common Pitfalls:

• Subtracting times instead of rates.
• Arithmetic mistakes in handling fractions with different denominators.

Final Answer:
7 1/2 h