To find the new weighted average when the ratio changes from 2:1 to 1:2, we can use the definition of the weighted average. The formula for the weighted average is:
\[ \text{Weighted Average} = \frac{(x_1 \cdot w_1) + (x_2 \cdot w_2)}{w_1 + w_2} \]
where \(x_1\) and \(x_2\) are the points on the number line, and \(w_1\) and \(w_2\) are the weights.
Step 1: Initial Setup
Given points:
- \(x_1 = 3\)
- \(x_2 = 12\)
Original ratio of weights is \(2:1\):
- \(w_1 = 2\)
- \(w_2 = 1\)
Step 2: Calculate Original Weighted Average
Calculate the original weighted average:
\[ \text{Weighted Average} = \frac{(3 \cdot 2) + (12 \cdot 1)}{2 + 1} = \frac{6 + 12}{3} = \frac{18}{3} = 6 \]
Step 3: Change the Ratio to 1:2
Now we change the ratio to \(1:2\):
- \(w_1 = 1\)
- \(w_2 = 2\)
Step 4: Calculate the New Weighted Average
Now we can calculate the new weighted average:
\[ \text{New Weighted Average} = \frac{(3 \cdot 1) + (12 \cdot 2)}{1 + 2} = \frac{3 + 24}{3} = \frac{27}{3} = 9 \]
Conclusion
The new weighted average when the ratio changes to \(1:2\) is:
\[ \boxed{9} \]
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