1. **Problem statement:** Simoné surveyed learners' favorite colors and recorded the number of learners for each color. We need to represent this data in a pie chart and calculate probabilities for certain colors. 2. **Data summary:** - Red: 14 - Blue: 18 - Green: 11 - Orange: 6 - Yellow: 8 - Purple: 4 - Black: 3 3. **Total learners:** $$14 + 18 + 11 + 6 + 8 + 4 + 3 = 64$$ 4. **Pie chart representation:** Each sector's angle in the pie chart is proportional to the fraction of learners who chose that color. Formula for angle: $$\text{Angle} = \frac{\text{Number of learners for color}}{\text{Total learners}} \times 360^\circ$$ Calculate each angle: - Red: $$\frac{14}{64} \times 360 = 78.75^\circ$$ - Blue: $$\frac{18}{64} \times 360 = 101.25^\circ$$ - Green: $$\frac{11}{64} \times 360 = 61.875^\circ$$ - Orange: $$\frac{6}{64} \times 360 = 33.75^\circ$$ - Yellow: $$\frac{8}{64} \times 360 = 45^\circ$$ - Purple: $$\frac{4}{64} \times 360 = 22.5^\circ$$ - Black: $$\frac{3}{64} \times 360 = 16.875^\circ$$ 5. **Probability calculations:** - Probability of Yellow: $$P(\text{Yellow}) = \frac{8}{64} = \frac{1}{8} = 0.125$$ - Probability of Orange or Purple: $$P(\text{Orange or Purple}) = \frac{6 + 4}{64} = \frac{10}{64} = \frac{5}{32} \approx 0.15625$$ 6. **Plotting points M(5, 2) and N(-2, 4):** - Point M is at coordinates (5, 2). - Point N is at coordinates (-2, 4). 7. **Reflection of M in the y-axis:** Reflection across the y-axis changes the x-coordinate sign: $$M'(x', y') = (-x, y)$$ So, $$M' = (-5, 2)$$ 8. **Translation of N 5 places right and 6 places down:** Translation changes coordinates by adding/subtracting values: $$N''(x'', y'') = (x + 5, y - 6)$$ So, $$N'' = (-2 + 5, 4 - 6) = (3, -2)$$ **Final answers:** - Pie chart angles as above. - $P(\text{Yellow}) = 0.125$ - $P(\text{Orange or Purple}) \approx 0.15625$ - $M' = (-5, 2)$ - $N'' = (3, -2)$