To find the weighted average of the numbers [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], with the weight [tex]\(\frac{2}{3}\)[/tex] on the first number and [tex]\(\frac{1}{3}\)[/tex] on the second number, follow these steps:
1. Identify the numbers and their corresponding weights:
- First number: [tex]\(-1\)[/tex]
- Weight on the first number: [tex]\(\frac{2}{3}\)[/tex]
- Second number: [tex]\(1\)[/tex]
- Weight on the second number: [tex]\(\frac{1}{3}\)[/tex]
2. Multiply each number by its corresponding weight:
- [tex]\(-1 \times \frac{2}{3} = -\frac{2}{3}\)[/tex]
- [tex]\(1 \times \frac{1}{3} = \frac{1}{3}\)[/tex]
3. Add these weighted values together:
- Weighted sum: [tex]\(-\frac{2}{3} + \frac{1}{3}\)[/tex]
4. Find a common denominator and perform the addition:
- Both fractions already have the common denominator [tex]\(\frac{3}{3}\)[/tex]:
[tex]\(-\frac{2}{3} + \frac{1}{3} = \frac{-2 + 1}{3} = \frac{-1}{3}\)[/tex]
5. Convert the fraction to a decimal (if necessary):
- [tex]\(\frac{-1}{3} \approx -0.3333\)[/tex]
Therefore, the weighted average of the given numbers, with their specified weights, is approximately [tex]\(-0.3333\)[/tex].
Given these calculations, the closest value among the provided options is:
[tex]\[
\boxed{-0.3}
\][/tex]
