$A$ can complete $\frac{2}{3}$ of a work in $8$ days. $B$ can complete $\frac{3}{5}$ of the same work in $12$ days and $C$ can complete $\frac{4}{9}$ of the same work in $8$ days. If $A$ and $B$ worked together for $5$ days, then find how much time (in days) will $C$ alone take to complete the remaining work?

First, find the total time taken by each individual to complete the entire work:Time taken by $A=8\times \frac{3}{2}=12\text{ days}$ Time taken by $B=12\times \frac{5}{3}=20\text{ days}$ Time taken by $C=8\times \frac{9}{4}=18\text{ days}$ Let the total work be the LCM of $12,20,$ and $18$, which is $180\text{ units}$.Now find their efficiencies:Efficiency of $A=\frac{180}{12}=15\text{ units/day}$ Efficiency of $B=\frac{180}{20}=9\text{ units/day}$ Efficiency of $C=\frac{180}{18}=10\text{ units/day}$$A$ and $B$ worked together for $5$ days:
$$\text{Work done by }A+B=(15+9)\times 5=24\times 5=120\text{ units}$$
Remaining work:
$$\text{Remaining Work}=180-120=60\text{ units}$$
Time taken by $C$ alone to complete the remaining work:
$$\text{Time}=\frac{60}{10}=6\text{ days}$$