$$S_{1}$$ and $$S_{2}$$ can do a piece of work together in 18 days, $$S_{2}$$ and $$S_{3}$$ can do the same work together in 27 days, while $$S_{3}$$ and $$S_{1}$$ can do it together in 54 days. In how many days can all 3, working together, do 50% of the work?

LCM of 18, 27 and 54 is 54.

Let's assume the total work is 54 units.

$$S_{1}$$ and $$S_{2}$$ can do a piece of work together in 18 days.

Efficiency of $$S_{1}$$ and $$S_{2}$$ together = $$\frac{54}{18}\ =\ 3\ $$ units/day    Eq.(i)

$$S_{2}$$ and $$S_{3}$$ can do the same work together in 27 days.

Efficiency of $$S_{2}$$ and $$S_{3}$$ together = $$\frac{54}{27}\ =\ 2\ $$ units/day    Eq.(ii)
while $$S_{3}$$ and $$S_{1}$$ can do it together in 54 days.

Efficiency of $$S_{3}$$ and $$S_{1}$$ together = $$\frac{54}{54}\ =\ 1\ $$ unit/day    Eq.(iii)

From Eq.(i), Eq.(ii) and Eq.(iii), total efficiency of all three = $$\frac{3+2+1}{2}$$

= $$\frac{6}{2}$$

= 3 units/day

Number of days taken by all 3, working together to do 50% of the work = 50% of $$\frac{54}{3}$$

= 50% of 18

= $$\frac{1}{2}\times\ 18$$

= 9 days