1. **State the problem:** We have grades for laboratory, lecture, and recitation parts: 77, 83, and 86 respectively. The weights for each part are all 4. 2. **Find the weighted sum \( \sum w_i x_i \):** We multiply each grade by its weight and sum them: $$\sum w_i x_i = 4 \times 77 + 4 \times 83 + 4 \times 86$$ Calculate each product: $$4 \times 77 = 308$$ $$4 \times 83 = 332$$ $$4 \times 86 = 344$$ Add these results: $$308 + 332 + 344 = 984$$ 3. **Find the sum of weights \( \sum w_i \):** All weights are 4 and there are 3 items, $$\sum w_i = 4 + 4 + 4 = 12$$ 4. **Calculate the weighted mean \( \bar{X}_w \):** Weighted mean formula is $$\bar{X}_w = \frac{\sum w_i x_i}{\sum w_i} = \frac{984}{12}$$ Divide the sum: $$\frac{984}{12} = 82$$ Rounded to 3 decimal places: $$82.000$$ **Final answers:** 1. \( \sum w_i x_i = 984 \) 2. \( \sum w_i = 12 \) 3. \( \bar{X}_w = 82.000 \)