A and B can complete a task in $12$ days, B and C can complete the task in $16$ days, A and C can complete the task in $24$ days. In how many days will they together complete the task?

Let the total work be the LCM of $12,16,$ and $24$, which is $48$ units.
Now, calculate individual combined efficiencies:

* Efficiency of $(A+B)=\frac{48}{12}=4\text{ units/day}$
* Efficiency of $(B+C)=\frac{48}{16}=3\text{ units/day}$
* Efficiency of $(A+C)=\frac{48}{24}=2\text{ units/day}$

Adding all three equations:

$$2\times (\text{Efficiency of A + B + C})=4+3+2=9$$

$$\text{Efficiency of A + B + C}=\frac{9}{2}=4.5\text{ units/day}$$

Number of days required to complete the task together:

$$\text{Days}=\frac{\text{Total Work}}{\text{Combined Efficiency}}=\frac{48}{4.5}=\frac{96}{9}\approx 10.67\text{ days}$$