A and B can do a work in 8 days, B and C can do the same work in 12 days. A, B and C together can finish it in 6 days. A and C together will do it in

### Concept & Formula / Logic To find the work rate of a specific pair $(A+C)$ when given the combined rate $(A+B+C)$ and the rates of other pairs, you can isolate the individual rates by subtracting the given pairs from the total combined rate. ### Step-by-Step Solution 1. **Identify the given 1-day work rates:** (A + B)'s 1-day work = $\frac{1}{8}$ (B + C)'s 1-day work = $\frac{1}{12}$ (A + B + C)'s 1-day work = $\frac{1}{6}$ 2. **Find A's 1-day work:** Subtract (B + C)'s rate from (A + B + C)'s rate. A = (A + B + C) - (B + C) $A = \frac{1}{6} - \frac{1}{12} = \frac{2 - 1}{12} = \frac{1}{12}$ 3. **Find C's 1-day work:** Subtract (A + B)'s rate from (A + B + C)'s rate. C = (A + B + C) - (A + B) $C = \frac{1}{6} - \frac{1}{8} = \frac{4 - 3}{24} = \frac{1}{24}$ 4. **Calculate (A + C)'s combined 1-day work:** (A + C)'s 1-day work = A + C $A + C = \frac{1}{12} + \frac{1}{24} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}$ Since their combined 1-day work is $\frac{1}{8}$, A and C together will finish the work in 8 days. ### Exam Strategy & Shortcut Use the LCM (Total Work) method. Let total work = LCM of 8, 12, and 6, which is 24 units. Efficiency of $(A+B) = \frac{24}{8} = 3$ units/day. Efficiency of $(B+C) = \frac{24}{12} = 2$ units/day. Efficiency of $(A+B+C) = \frac{24}{6} = 4$ units/day. Efficiency of A = $(A+B+C) - (B+C) = 4 - 2 = 2$ units/day. Efficiency of C = $(A+B+C) - (A+B) = 4 - 3 = 1$ unit/day. Combined efficiency of $(A+C) = 2 + 1 = 3$ units/day. Time taken by $(A+C) = \frac{24}{3} = 8$ days. ### Common Pitfall A frequent error is trying to add the rates of $(A+B)$ and $(B+C)$ to find $(A+C)$, but this gives $A+2B+C$. Subtracting variables strategically is required. ### Final Answer Therefore, the correct answer is **8 days**.