### Math Notes

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#### Algebra Solutions

##### Topics || Problems

An airplane, flying with the wind, took 2 hours for a 1000-mile flight, and 2.5 hours for the return flight. Find the wind velocity and the speed of the airplane in still air.
Solution:

Let $$V_w$$ be the velocity of the wind, $$V_a$$ be the velocity of the airplane in still air.

Velocity $$V$$, $$V = \frac{d}{t}$$

Flying with the wind: $$V = V_w + V_a$$

$$\frac{1000}{2} = V_w + V_a$$

$$V_a = \frac{1000}{2} - V_w$$

Flying againts the wind: $$V = -V_w + V_a$$

$$\frac{1000}{2.5} = -V_w + V_a$$

$$V_a = V_w + \frac{1000}{2.5}$$

Equate $$V_a$$: $$\frac{1000}{2} - V_w =V_w + \frac{1000}{2.5}$$

$$V_w = 50 \text{mph}$$

$$V_a = 500-50$$

$$V_a = 450 \text{mph}$$

Answer: The velocity of the airplane is $$450 \text{mph}$$ and the the velocity of the wind is $$50 \text{mph}$$