To do a certain task X would take 3 times as long as Y and Z together; and Z would takes 4 times as long as Y and X together. Three of them together can complete the task in 10 days. How much time is taken by X and Z to complete the task?

$22\frac{2}{9}$days

X + Y + Z = 10 days,

Efficiency α \(\frac{1}{Time}\)

Therefore,      X : Y + Z,    and    Z : Y + X

Efficiency       1 :  3                    1 :   4

Making addition of Efficiency equal, by multiplying (X : Y + Z) with 5 and (Z : Y + X) with 4. we get,

 ⇒                    X : Y + Z    and    Z : Y + X

 Efficiency         5 :  15                 4 :  16

Simplifying, we get X = 5, Y = 11, Z = 4,

⇒ Total work = (5 + 11 + 4) x 10 days = 200 units    ..(Efficiency × Days = Total work)

⇒ Time taken by X and Z to complete the task together = \(\frac{200}{5 +4}\) = \(\frac{200}{9}\) =\( {200 }_{ 9}^{ 2} \) days.