Question

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 \( 1.8 \).