A and B can do a piece of work together in 10 days, B and C can do the same work together in 15 days, while C and A can do the same work together in 20 days. In how many days can A, B and C do the same work, working together?

Let's assume the total work is 60 units.

A and B can do a piece of work together in 10 days.

Efficiency of A and B together = $$\frac{60}{10}$$ = 6 units/day    Eq.(i)

B and C can do the same work together in 15 days.

Efficiency of B and C together = $$\frac{60}{15}$$ = 4 units/day    Eq.(ii)

C and A can do the same work together in 20 days.

Efficiency of C and A together = $$\frac{60}{20}$$ = 3 units/day    Eq.(iii)

From  Eq.(i),  Eq.(ii) and  Eq.(iii), Efficiency of A, B and C together = $$\frac{6+4+3}{2}$$

= $$\frac{13}{2}$$

Time taken by A, B and C to do the same work, working together = $$\frac{60}{\frac{13}{2}}$$

= $$\frac{60\times\ 2}{13}$$

= $$\frac{120}{13}$$ days