ANSWER
40 meters per minute
EXPLANATION
We have to find the rate of the wind - in other words, the speed of the wind.
We know that the duck can fly 2400 m in 10 minutes with the wind, so the speed of the duck when it is flying with the wind is,
[tex]s_{with\text{ }the\text{ }wind}=\frac{2400m}{10min}=240m/min[/tex]
However, when the duck is flying against the wind, it can only fly 2/3 of this distance in the same time, which is,
[tex]2400m\cdot\frac{2}{3}=1600m[/tex]
So, the speed of the duck against the wind is,
[tex]s_{against\text{ }the\text{ }wind}=\frac{1600m}{10min}=160m/min[/tex]
The speed of the duck with the wind is the sum of the actual speed of the duck and the speed of the wind,
[tex]s_{with\text{ }the\text{ }wind}=s_{duck}+s_{wind}[/tex]
While the speed of the duck against the wind is the actual speed of the duck minus the speed of the wind,
[tex]s_{against\text{ }the\text{ }wind}=s_{duck}-s_{wind}[/tex]
If we subtract the equation of the speed against the wind from the equation of the speed with the wind, we have,
[tex]\begin{gathered} s_{with\text{ }the\text{ }wind}-s_{against\text{ }the\text{ }wind}=s_{duck}-s_{duck}+s_{wind}-(-s_{wind}) \\ \\ s_{with\text{ }the\text{ }wind}-s_{against\text{ }the\text{ }wind}=0+s_{wind}+s_{wind} \\ \\ s_{with\text{ }the\text{ }wind}-s_{against\text{ }the\text{ }wind}=2s_{wind} \end{gathered}[/tex]
Solving for the speed of the wind and replacing the values,
[tex]s_{wind}=\frac{s_{with\text{ }the\text{ }wind}-s_{against\text{ }the\text{ }wind}}{2}=\frac{240m/min-160m/min}{2}=\frac{80m/min}{2}=40m/min[/tex]
Hence, the speed (or rate) of the wind is 40 meters per minute.
