Flying against the wind, an airplane travels 5100 kilometers in 6 hours. Flying with the wind, the same plane travels 3870 kilometers in 3 hours. What is the rate of the plane in still air and what is the rate of the wind?

If an object moves at a speed [tex]v_p[/tex] with respect to another object at a speed [tex]v_w[/tex], the relative speeds between them can be [tex]v_p+v_w[/tex] or [tex]v_p-v_w[/tex] depending if they are collaborative or one against the other.

The speed of an object who travels a distance x in a time t is

[tex]\displaystyle v=\frac{x}{t}[/tex]

We know that an airplane travels 5100 kilometers in 6 hours when flying against the wind, and travels 3870 kilometers in 3 hours when flying with the wind. Let's call [tex]v_p[/tex] and [tex]v_w[/tex] the speeds of the plane in still air and the wind, respectively. The first travel is performed by the plane with a wind whose speed subtracts from its own, so the relative speed is

[tex]v_p-v_w=\displaystyle \frac{5100}{6}[/tex]

[tex]v_p-v_w=850[/tex]

The second travel is performed with the wind pushing in the same direction, so

[tex]\displaystyle v_p+v_w=\frac{3870}{3}[/tex]

[tex]v_p+v_w=1290[/tex]

Adding both equations, we have

[tex]v_p-v_w+v_p+v_w=850+1290[/tex]

Simplifying and solving

[tex]2v_p=850+1290[/tex]

[tex]v_p=1070[/tex]

Replacing into the second equation

[tex]1070+v_w=1290[/tex]

[tex]v_w=1290-1070=220[/tex]

The rate of the plane in still air is 1070 km/h and what is the rate of the wind is 220 Km/h