A, B and C can complete a work in $5$, $12$ and $30$ days respectively. In how many days can A, B and C together do $\frac{4}{7}$ of the same work?

Let's find the total work by taking LCM of $5,12,30$, which is $60$ units.Efficiency of A $=\frac{60}{5}=12$ units/dayEfficiency of B $=\frac{60}{12}=5$ units/dayEfficiency of C $=\frac{60}{30}=2$ units/dayTotal efficiency of A + B + C $=12+5+2=19$ units/day.We need to find the time taken to complete $\frac{4}{7}$ of the total work:
$$\text{Work to be done}=60\times \frac{4}{7}=\frac{240}{7}\text{ units}$$
$$\text{Time taken}=\frac{\text{Work}}{\text{Total Efficiency}}=\frac{\frac{240}{7}}{19}=\frac{240}{133}\text{ days}$$