Three Pairwise Times — Find All Three Together P and Q can complete a job in 6 days, Q and R in 10 days, and P and R in 5 days. If all three work together, how long will they take to finish the job?

Introduction / Context:
Using pairwise completion times, we can find the sum of individual rates. The total rate of all three equals half the sum of pairwise rates (because each person is counted in two pairs).

Given Data / Assumptions:

• (P + Q) rate = 1/6.
• (Q + R) rate = 1/10.
• (P + R) rate = 1/5.

Concept / Approach:
(P + Q + R) rate = ( (P + Q) + (Q + R) + (P + R) ) / 2. Then total time = 1 / (P + Q + R rate).

Step-by-Step Solution:
Sum of pair rates = 1/6 + 1/10 + 1/5 = 5/30 + 3/30 + 6/30 = 14/30 = 7/15.All-three rate = (7/15)/2 = 7/30 job/day.Time = 1 / (7/30) = 30/7 days ≈ 4.2857 days.

Verification / Alternative check:
Each pair's time is reasonable, and the all-together time is less than the fastest pair time (5 days), as expected.

Why Other Options Are Wrong:
42/7, 43/7, 44/7, and 45/7 do not equal the exact 30/7 derived from rates.

Common Pitfalls:
Forgetting to divide the sum of pairwise rates by 2 or adding days rather than rates.

Final Answer:
30/7 days