Find Total Work (LCM)

Take the LCM of the given days ($8$, $12$, and $6$) to assume a total work of $24$ units.

$\text{LCM}(8, 12, 6) = 24 \text{ units}$

Calculate Combined Efficiencies

Determine the daily work rates for combinations: $(A+B)$ does $24/8 = 3$ units/day, $(B+C)$ does $24/12 = 2$ units/day, and $(A+B+C)$ does $24/6 = 4$ units/day.

$E_{A+B} = 3, \quad E_{B+C} = 2, \quad E_{A+B+C} = 4$

Calculate Individual Efficiencies for A and C

Subtract $(B+C)$ efficiency from total efficiency to get A's rate: $4 - 2 = 2$ units/day. Subtract $(A+B)$ efficiency from total efficiency to get C's rate: $4 - 3 = 1$ unit/day.

$E_A = 4 - 2 = 2, \quad E_C = 4 - 3 = 1$

Calculate Time Taken by A and C

Combined efficiency of A and C is $2 + 1 = 3$ units/day. Total time is total work divided by combined efficiency: $24 / 3 = 8$ days.

$T_{A+C} = \frac{24}{2 + 1} = 8 \text{ days}$