Question

Find the weighted average of the numbers −3 and 5 with three fifths of the weight on the first number and two fifths on the second number.

4.8
1.8
0.2
−1.8

Answer 1

To find the weighted average, we use the formula:

\[
\text{Weighted Average} = \frac{(w_1 \cdot x_1) + (w_2 \cdot x_2)}{w_1 + w_2}
\]

where \(w_1\) is the weight on the first number, \(x_1\) is the first number, \(w_2\) is the weight on the second number, and \(x_2\) is the second number.

In this case:
- \(x_1 = -3\)
- \(x_2 = 5\)
- \(w_1 = \frac{3}{5}\)
- \(w_2 = \frac{2}{5}\)

Now, substitute the values into the formula:

\[
\text{Weighted Average} = \frac{\left(\frac{3}{5} \cdot (-3)\right) + \left(\frac{2}{5} \cdot 5\right)}{\frac{3}{5} + \frac{2}{5}}
\]

Calculating the numerator:

\[
\frac{3}{5} \cdot (-3) = -\frac{9}{5}
\]

\[
\frac{2}{5} \cdot 5 = \frac{10}{5} = 2
\]

Thus, the numerator becomes:

\[
-\frac{9}{5} + 2 = -\frac{9}{5} + \frac{10}{5} = \frac{1}{5}
\]

Now, calculate the denominator:

\[
\frac{3}{5} + \frac{2}{5} = \frac{5}{5} = 1
\]

So, the weighted average is:

\[
\text{Weighted Average} = \frac{\frac{1}{5}}{1} = \frac{1}{5} = 0.2
\]

Therefore, the weighted average of the numbers -3 and 5 is **0.2**.