X and Y, working together, can complete a work in 30 days, Y and Z, working together, can complete the same work in 40 days. and Z and X, working together, can complete the same work in 24 days. In how many days can X, Y and Z, working together, to complete the work?

Let's assume the total work is 120 units.

X and Y, working together, can complete a work in 30 days.

Efficiecny of X and Y toegther = $$\frac{120}{30}$$ = 4 units/day    Eq.(i)

Y and Z, working together, can complete the same work in 40 days.

Efficiecny of Y and Z toegther = $$\frac{120}{40}$$ = 3 units/day    Eq.(ii)

Z and X, working together, can complete the same work in 24 days.

Efficiecny of Z and X toegther = $$\frac{120}{24}$$ = 5 units/day     Eq.(iii)

By Eq.(i), Eq.(ii) and Eq.(iii), Efficiency of X, Y and Z together = $$\frac{4+3+5}{2}$$

= $$\frac{12}{2}$$

= 6 units/day

Time taken by X, Y and Z, working together, to complete the work = $$\frac{120}{6}$$

= 20 days