Answer
$40$ miles per hour.
Work Step by Step
First we determine the Least Common Denominator (LCD) to clear fractions.
The LCD is $(450+x)(450-x)$.
Multiply both sides by the LCD.
$\Rightarrow (450+x)(450-x)\cdot \left ( \frac{980}{450+x}\right )=(450+x)(450-x)\cdot \left ( \frac{820}{450-x}\right )$
Simplify.
$\Rightarrow 980(450-x)=820(450+x)$
Use the distributive property.
$\Rightarrow 441000-980x=369000+820x$
Add $-369000+980x$ to both sides.
$\Rightarrow 441000-980x-369000+980x=369000+820x-369000+980x$
Simplify.
$\Rightarrow 72000=1800x$
Divide both sides by $1800$.
$\Rightarrow \frac{72000}{1800}=\frac{1800x}{1800}$
Simplify.
$\Rightarrow 40=x$
Hence, the average rate of the wind is $40$ miles per hour.
Note: Check if the solution is correct. The equation is defined for all real values of $x$ except the zeros of the denominators, which are $-450$ and $450$. As the solution $x=40$ is not a zero of a denominator, it is correct.
