To find the weighted average of the three points on the number line, we can use the formula for the weighted average:
\[ \text{Weighted Average} = \frac{(x_1 \cdot w_1) + (x_2 \cdot w_2) + (x_3 \cdot w_3)}{w_1 + w_2 + w_3} \]
Where:
- \( x_1, x_2, x_3 \) are the points (2, 4, and 8),
- \( w_1, w_2, w_3 \) are the respective weights (10%, 30%, 60%).
First, convert the weights from percentages to decimals:
- \( w_1 = 0.10 \)
- \( w_2 = 0.30 \)
- \( w_3 = 0.60 \)
Next, plug in the values into the weighted average formula:
\[ \text{Weighted Average} = \frac{(2 \cdot 0.10) + (4 \cdot 0.30) + (8 \cdot 0.60)}{0.10 + 0.30 + 0.60} \]
Calculating the numerator:
\[ 2 \cdot 0.10 = 0.2 \] \[ 4 \cdot 0.30 = 1.2 \] \[ 8 \cdot 0.60 = 4.8 \]
Adding these products together:
\[ 0.2 + 1.2 + 4.8 = 6.2 \]
Now, calculate the denominator:
\[ 0.10 + 0.30 + 0.60 = 1.00 \]
Now divide the total by the sum of weights:
\[ \text{Weighted Average} = \frac{6.2}{1.00} = 6.2 \]
Thus, the weighted average is 6.2.
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