$$B_{1}$$ and $$B_{2}$$ can do a piece of work together in 8 days, $$B_{2}$$ and $$B_{3}$$ can do the same work together in 16 days, while $$B_{1}$$ and $$B_{3}$$ can do it together in 32 days. What percentage of the total work can all the 3 people, working together, do in 4 days?

Let's assume the total work is 32 units.

$$B_{1}$$ and $$B_{2}$$ can do a piece of work together in 8 days.

Efficiency of $$B_{1}$$ and $$B_{2}$$ together = $$\frac{32}{8}$$ = 4 units/day    Eq.(i)

$$B_{2}$$ and $$B_{3}$$ can do the same work together in 16 days.

Efficiency of $$B_{2}$$ and $$B_{3}$$ together = $$\frac{32}{16}$$ = 2 units/day    Eq.(ii)

while $$B_{1}$$ and $$B_{3}$$ can do it together in 32 days.

Efficiency of $$B_{1}$$ and $$B_{3}$$ together = $$\frac{32}{32}$$ = 1 unit/day    Eq.(iii)

By  Eq.(i),  Eq.(ii) and  Eq.(iii), the efficiency of all the 3 people together = $$\frac{4+2+1}{2}$$ = $$\frac{7}{2}$$ = 3.5 units/day

total work can all the 3 people, working together, do in 4 days = $$3.5\times4$$ = 14 units

Percentage of the total work can all the 3 people, working together, do in 4 days = $$\frac{14}{32}\times\ 100$$

 = $$\frac{700}{16}$$

= 43.75%