To find the new weighted average when the ratio changes from 3:1 to 4:1, let's start with the original information:
- The two points are 1 and 5.
- The ratio of their weights is 3:1.
Step 1: Calculate the original weighted average
The formula for the weighted average is given by:
\[ \text{Weighted Average} = \frac{w_1 \cdot x_1 + w_2 \cdot x_2}{w_1 + w_2} \]
where \( w_1 \) and \( w_2 \) are the weights, and \( x_1 \) and \( x_2 \) are the values at the points.
From the given ratio of 3:1, we can assign:
- \( w_1 = 3 \) (weight of 1)
- \( w_2 = 1 \) (weight of 5)
- \( x_1 = 1 \)
- \( x_2 = 5 \)
Now, substituting the values:
\[ \text{Weighted Average} = \frac{3 \cdot 1 + 1 \cdot 5}{3 + 1} = \frac{3 + 5}{4} = \frac{8}{4} = 2 \]
This confirms our original weighted average is correct.
Step 2: Calculate the new weighted average with a ratio of 4:1
Now we change the ratio to 4:1, which means:
- \( w_1 = 4 \) (weight of 1)
- \( w_2 = 1 \) (weight of 5)
Using the same formula for the weighted average, we find:
\[ \text{New Weighted Average} = \frac{4 \cdot 1 + 1 \cdot 5}{4 + 1} = \frac{4 + 5}{5} = \frac{9}{5} = 1.8 \]
Conclusion
The new weighted average when the ratio is 4:1 is 1.8.
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