Answer with explanation:
V=Volume of tank
It is given that, X and Y work at constant rates.
Let rate of doing work of X is x hour.
And, Rate of Doing work of Y is y hour.
Statement I----X and Y together fill order in 2⁄3 the time that X alone does.
![\frac{V}{x+y}=\frac{2V}{3x}\\\\x+y=\frac{3x}{2}\\\\y=\frac{x}{2}](https://tex.z-dn.net/?f=%5Cfrac%7BV%7D%7Bx%2By%7D%3D%5Cfrac%7B2V%7D%7B3x%7D%5C%5C%5C%5Cx%2By%3D%5Cfrac%7B3x%7D%7B2%7D%5C%5C%5C%5Cy%3D%5Cfrac%7Bx%7D%7B2%7D)
-------------------------------------------------(1)
Statement II----Y alone does it in twice the time as X alone does.
![\frac{V}{x}=\frac{V}{\frac{y}{2}}\\\\y=\frac{2}{x}](https://tex.z-dn.net/?f=%5Cfrac%7BV%7D%7Bx%7D%3D%5Cfrac%7BV%7D%7B%5Cfrac%7By%7D%7B2%7D%7D%5C%5C%5C%5Cy%3D%5Cfrac%7B2%7D%7Bx%7D)
Substituting the value of y in 1
![\rightarrow \frac{x}{2}=\frac{2}{x}\\\\\rightarrow x^2=4\\\\\rightarrow x=\pm 2\\\\\rightarrow x=2](https://tex.z-dn.net/?f=%5Crightarrow%20%5Cfrac%7Bx%7D%7B2%7D%3D%5Cfrac%7B2%7D%7Bx%7D%5C%5C%5C%5C%5Crightarrow%20x%5E2%3D4%5C%5C%5C%5C%5Crightarrow%20x%3D%5Cpm%202%5C%5C%5C%5C%5Crightarrow%20x%3D2)
→x≠ -2, because Rate of doing work can't be negative.
Substituting the value of x in 1, gives
![y=\frac{2}{2}\\\\y=1](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B2%7D%7B2%7D%5C%5C%5C%5Cy%3D1)
→Rate of doing work of X= 2 hour
→Rate of doing work of Y=1 hour