Intuition for weighted average. Why $\frac{w_1}{w_1 + w_2}x_1 + \frac{w_2}{w_1 + w_2}x_2 = \frac{\sum_i w_ix_i}{\sum_i w_i}$? I know $\dfrac{w_1}{w_1 + w_2}x_1 + \dfrac{w_2}{w_1 + w_2}x_2 = \dfrac{\sum_i w_ix_i}{\sum_i w_i}$, because $\sum_i w_i$ is common denominator. I'm not asking about this algebra.
It's intuitive that $\dfrac{w_i}{w_1 + w_2}$ weighs $x_i$.
Intuitively, why's $\dfrac{\sum w_ix_i}{\sum w_i}$  Weighted Average?  You are summing $w_ix_i$ and $w_i$ separately. Thus you lost information, because the weight for $x_i$ doesn't appear. When you sum $\sum w_ix_i$ and and $\sum w_i$, these end as totals. They inform nothing about weights! And you can't recover the weights for just these sums!
Can picture explain?


Here is an example from statistics.
The table shows the sales of sugar (in kilograms) during $10$ days:
$$\begin{array}{c|c|c}
\text{Sales of sugar (in kg)}, x & \text{Number of days}, f & \text{Percentage of days}, P(x)\\
\hline
0&1&0.1\\
1&3&0.3\\
2&4&0.4\\
3&2&0.2\\
\hline
&10&1
\end{array}$$
On $3$ days (or during $30\%$ of the $10$-day period) $1$ kilo of sugar was sold each day. Now we need to find the average sales during the $10$-day period.
Method 1. Convert the table data to raw data. Let's assume the following sales took place each day:
$$3,0,3,2,2,1,3,1,1,2$$
So, the average sale is:
$$\frac{\sum x}{n}=\frac{3+0+2+2+2+1+3+1+1+2}{10}=1.7$$
Method 2. Let's simplify the above expression:
$$\frac{\sum x}{n}=\frac{0+1+1+1+2+2+2+2+3+3}{10}=\\
\frac{0\cdot 1+1\cdot 3+2\cdot 4+3\cdot 2}{10}=\\
0\cdot \frac{1}{10}+1\cdot \frac{3}{10}+2\cdot \frac4{10}+3\cdot \frac{2}{10}=\\
0\cdot 0.1+1\cdot 0.3+2\cdot 0.4+3\cdot 0.2=1.7$$
So, the sales figures are elements (x) and the percentage of days (P(x)) are weights. The more percentage a partiular sales figure occurs, the more its impact on the average sales figure.
