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, 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**.