X and Y together can do a work in $2\frac{2}{5}$ day days. Y and Z together can do the same working 3 days, and X and Z together can do the same work in 4 days. The time taken by X, Y and Z together to do the same work is:

2 day

X + Y = \( {2 }_{5 }^{2 } \) days,

Y + Z = 3 days,

Z + X = 4 days, 

⇒ Adding all the efficiencies,

⇒ 2(X + Y + Z)= 12,

⇒ X + Y + Z = 6,

⇒ Time taken by X + Y + Z to complete the work = \(\frac{12}{2}\) = 6 days,  ..(\(\frac{Work}{Efficiency}\) = Time)