$$D_{1}$$ and $$D_{2}$$ can do a piece of work together in 16 days, $$D_{2}$$ and $$D_{3}$$ can do the same work together in 24 days, while $$D_{3}$$ and $$D_{1}$$ can do it together in 48 days. In how many days can all 3, working together, do $$\frac{5}{6}$$ of the work?

Let's assume the total work is 48 units.

$$D_{1}$$ and $$D_{2}$$ can do a piece of work together in 16 days.

Efficiency of $$D_{1}$$ and $$D_{2}$$ taken together = $$\frac{48}{16}$$ = 3 units/day

$$D_{2}$$ and $$D_{3}$$ can do the same work together in 24 days.

Efficiency of $$D_{2}$$ and $$D_{3}$$ taken together = $$\frac{48}{24}$$ = 2 units/day

while $$D_{3}$$ and $$D_{1}$$ can do it together in 48 days.

Efficiency of $$D_{3}$$ and $$D_{1}$$ taken together = $$\frac{48}{48}$$ = 1 unit/day

Efficiency of $$D_{1}$$, $$D_{2}$$ and $$D_{3}$$ taken together = $$\frac{3+2+1}{2}$$

= $$\frac{6}{2}$$

= 3 units/day

Time taken by all 3, working together, do $$\frac{5}{6}$$ of the work = $$\frac{\frac{5}{6}\ of\ 48}{3}$$

= $$\frac{\frac{5}{6}\ \times\ 48}{3}$$

= $$\frac{5\times8}{3}$$