A + B finish a work in 8 days; B + C finish it in 12 days; A + B + C together finish it in 6 days. In how many days will A + C together finish the work?

Problem restatement
From joint times for three pairs and the trio, deduce individual rates and obtain the time for (A + C).

Given data

• a + b = 1/8
• b + c = 1/12
• a + b + c = 1/6

Concept/Approach
Add the first two equations to relate to the third, solve for b, then get a + c.

Step-by-step calculation
(a + b) + (b + c) = 1/8 + 1/12 = 5/24 Left side = a + 2b + c = (a + b + c) + b = 1/6 + b 1/6 + b = 5/24 → b = 5/24 − 1/6 = 5/24 − 4/24 = 1/24 a + c = (a + b + c) − b = 1/6 − 1/24 = 3/24 = 1/8 Time for (A + C) = 1 ÷ (1/8) = 8 days

Verification
With b = 1/24, we get a = 1/8 − b = 1/8 − 1/24 = 1/12; c = 1/6 − (a + b) = 1/6 − (1/12 + 1/24) = 1/24. Then a + c = 1/12 + 1/24 = 1/8 (consistent).

Common pitfalls

• Trying to find each of a, b, c first; unnecessary if only a + c is needed.

Final Answer
8 days