Calculating weighted average

A simple average treats every value equally.
A weighted average is different: some values carry more weight because they represent a bigger share.
For example, a test worth $2$ grades should count twice as much as a quiz worth $1$.

The formula multiplies each value by its weight, adds those up, and divides by the total weight: $\frac{w_1 v_1 + w_2 v_2}{w_1 + w_2}$.
The weights are the sizes or importance of each group, and the values are what you are averaging.

Here is a concrete case.
One class of $30$ students averages $70$, and another of $20$ students averages $80$.
The overall average is $\frac{30(70) + 20(80)}{30 + 20} = \frac{2100 + 1600}{50} = \frac{3700}{50} = 74$.

Notice the result, $74$, is closer to $70$ than to $80$.
That is because the $70$ group is bigger, so it pulls the average toward itself.
A weighted average always lands between the values, nearer to the heavier weight.

A student scores $90$ on a test that counts as $2$ grades and $60$ on a quiz that counts as $1$ grade. What is the weighted average?
Multiply each score by its weight, add, and divide by the total weight: $\frac{2(90) + 1(60)}{2 + 1}$.
That is $\frac{180 + 60}{3} = \frac{240}{3} = 80$.

An investor puts $\$3000$ in Fund A, which returns $5\%$, and $\$2000$ in Fund B, which returns $10\%$. What is the overall percent return?
Weight each return by the amount invested: $\frac{3000(5\%) + 2000(10\%)}{3000 + 2000}$.
That is $\frac{150 + 200}{5000} = \frac{350}{5000} = 0.07 = 7\%$.

A store sold $40$ items at $\$5$ each and $10$ items at $\$15$ each. What was the average price per item?
Weight each price by how many sold: $\frac{40(5) + 10(15)}{40 + 10}$.
That is $\frac{200 + 150}{50} = \frac{350}{50} = 7$, so the average price is $\$7$.