1. **Problem statement:** We have 80 items on sale with discounts up to 60. We want to find the upper and lower quartiles and the interquartile range (IQR) of the discounts from the cumulative frequency graph. 2. **Understanding quartiles:** - The lower quartile (Q1) is the value below which 25% of the data lies. - The upper quartile (Q3) is the value below which 75% of the data lies. - The interquartile range (IQR) is $Q3 - Q1$. 3. **Using the cumulative frequency graph:** - Total items = 80. - Find the discount values at cumulative frequencies 20 (25% of 80) and 60 (75% of 80). 4. **Finding Q1 (25th percentile):** - At cumulative frequency 20, the discount lies between 10 (12 items) and 20 (25 items). - Interpolate linearly: $$Q1 = 10 + \frac{20 - 12}{25 - 12} \times (20 - 10) = 10 + \frac{8}{13} \times 10 \approx 10 + 6.15 = 16.15$$ 5. **Finding Q3 (75th percentile):** - At cumulative frequency 60, the discount lies between 30 (40 items) and 40 (70 items). - Interpolate linearly: $$Q3 = 30 + \frac{60 - 40}{70 - 40} \times (40 - 30) = 30 + \frac{20}{30} \times 10 = 30 + \frac{2}{3} \times 10 = 30 + 6.67 = 36.67$$ 6. **Calculate IQR:** $$IQR = Q3 - Q1 = 36.67 - 16.15 = 20.52$$ **Final answers:** - Lower quartile (Q1) = 16.15 - Upper quartile (Q3) = 36.67 - Interquartile range (IQR) = 20.52