Correct Answer: $22 \frac{2}{9}$ days

Solution : Given,
To do a certain task,
X takes 3 times as long as Y and Z together,
Z takes 4 times as long as Y and X together,
Three of them together can complete the task in 10 days.
We know, Total work = Time × Efficiency
Let the efficiency of X, Y, and Z be a, b, and c respectively and total work be W
According to the question,
W = 10(a + b + c) ....................(1)
X takes 3 times (Y + Z)
⇒ (b + c) = 3a
Putting this in (1)
⇒ W = 10(a + 3a) ⇒ W = 10(4a) ⇒ W = 40a ⇒ a = $\frac{\text{W}}{40}$
Z takes 4 times (X + Y)
⇒ (a + b) = 4c
Putting this in (1)
⇒ W = 10(4c + c) ⇒ W = 10(5c) ⇒ W = 50c ⇒ c = $\frac{\text{W}}{50}$
Now X and Z are working together and considering total work as 1 then,
⇒ X + Z = a + c
⇒ X + Z = $\frac{1}{40} + \frac{1}{50}$
⇒ X + Z = $\frac{5 + 4}{200}$
⇒ X + Z = $\frac{9}{200}$
⇒ Time taken by X and Z to complete a task = $\frac{200}{9}$ days = $22\frac29$ days
Hence, the correct answer is $22\frac29$ days.