Two points are on a number line at 1 and 5. The original weighted average is 2, with a ratio of 3:1 .(1 point) When the ratio changes to 4:1 , the new weighted average becomes

Answer 1

To find the new weighted average when the ratio changes from 3:1 to 4:1, we can use the concept of weighted averages.

Given:

• Two points: \( x_1 = 1 \) and \( x_2 = 5 \)
• Original ratio \( 3:1 \)

Step 1: Calculate the original weighted average

For a ratio of \( 3:1 \):

• The weights for \( x_1 \) and \( x_2 \) are \( 3 \) and \( 1 \), respectively.
• The weighted average \( W \) is calculated as:

\[ W = \frac{(3 \cdot 1) + (1 \cdot 5)}{3 + 1} \] \[ W = \frac{3 + 5}{4} = \frac{8}{4} = 2 \]

This confirms the original weighted average is indeed \( 2 \).

Step 2: Calculate the new weighted average for the ratio \( 4:1 \)

Now we shift the ratio to \( 4:1 \):

• The weights for \( x_1 \) and \( x_2 \) will now be \( 4 \) and \( 1 \), respectively.

Using the weighted average formula again:

\[ W' = \frac{(4 \cdot 1) + (1 \cdot 5)}{4 + 1} \] \[ W' = \frac{4 + 5}{5} = \frac{9}{5} = 1.8 \]

Final Answer: The new weighted average becomes \( 1.8 \).