To solve the weighted average problem in the image:

Given dataTo solve the weighted average problem in the image:

Given

• Numbers: $-3$, $-2$, $6$
• Corresponding weights:
  $w_{-3} = \frac{1}{3}, w_{-2} = \frac{1}{2}, w_6 = \frac{1}{6}$

The formula for the weighted average is:

$\text{Weighted Average} = \frac{\sum (x_i \cdot w_i)}{\sum w_i}$

Step 1: Calculate the weighted sum $\sum (x_i \cdot w_i)$

$\sum (x_i \cdot w_i) = (-3 \times \frac{1}{3}) + (-2 \times \frac{1}{2}) + (6 \times \frac{1}{6})$ $= -1 + (-1) + 1 = -1$

Step 2: Calculate the sum of the weights $\sum w_i$

$\sum w_i = \frac{1}{3} + \frac{1}{2} + \frac{1}{6}$

Finding a common denominator (which is 6): $\sum w_i = \frac{2}{6} + \frac{3}{6} + \frac{1}{6} = \frac{6}{6} = 1$

Step 3: Compute the weighted average

$\text{Weighted Average} = \frac{-1}{1} = -1$

Final Answer:

The weighted average is $-1$.

Would you like further clarification or more details?

Here are 5 related questions to consider:

1. How do you calculate the weighted average for more than three values?
2. What happens if the sum of the weights is not 1?
3. Can a weighted average be greater than any of the original numbers?
4. How would the result change if the weights were distributed differently?
5. In what scenarios is calculating a weighted average useful in real life?

Tip: Always ensure the sum of the weights equals 1 to simplify weighted average calculations.