To find the new weighted average when the ratio changes to 2:1, we can first compute the initial weighted average with the given values and then apply the new ratio.
Given points:
- Point A at 2
- Point B at 6
Original ratio is 1:1, so the weighted average calculation is as follows:
\[ \text{Weighted Average} = \frac{w_1 \cdot x_1 + w_2 \cdot x_2}{w_1 + w_2} \]
Where:
- \( w_1 = 1 \) (weight for point A)
- \( w_2 = 1 \) (weight for point B)
- \( x_1 = 2 \) (value of point A)
- \( x_2 = 6 \) (value of point B)
Calculating the weighted average with a 1:1 ratio:
\[ \text{Weighted Average} = \frac{(1 \cdot 2) + (1 \cdot 6)}{1 + 1} = \frac{2 + 6}{2} = \frac{8}{2} = 4 \]
Now, for the new ratio of 2:1:
This means point A will have a weight of 2 and point B will have a weight of 1:
\[ w_1 = 2, \quad w_2 = 1 \]
Now, we calculate the new weighted average:
\[ \text{Weighted Average} = \frac{(2 \cdot 2) + (1 \cdot 6)}{2 + 1} = \frac{4 + 6}{3} = \frac{10}{3} \approx 3.33 \]
Therefore, the new weighted average is:
3.33 (or 3.3 with a repeating decimal).
So the answer is 3.3¯¯¯ (3 point repeating).
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