1. **Problem statement:** We are given a histogram showing the frequency density of paintings' ages in a gallery. We know 18 paintings are over 90 years old. We need to estimate how many paintings are over 70 years old. 2. **Understanding frequency density and area:** The number of paintings in an age interval is the area of the corresponding bar in the histogram, calculated as: $$\text{Number of paintings} = \text{frequency density} \times \text{class width}$$ 3. **Given data:** - Age intervals and frequency densities: - 0-20: 2 - 20-40: 1.5 - 40-60: 3.8 - 60-80: 2.6 - 80-100: 1 - 100-120: 0.2 - 120-140: 0 - Class width for each interval is 20 years. 4. **Calculate number of paintings over 90 years old:** - Over 90 means ages 90 to 140, which covers parts of 80-100, 100-120, and 120-140 intervals. - We know total paintings over 90 = 18. 5. **Calculate number of paintings in 80-100 interval:** - Frequency density = 1 - Class width = 20 - Number of paintings in 80-100 = $1 \times 20 = 20$ 6. **Calculate number of paintings in 100-120 interval:** - Frequency density = 0.2 - Class width = 20 - Number of paintings in 100-120 = $0.2 \times 20 = 4$ 7. **Calculate number of paintings in 120-140 interval:** - Frequency density = 0 - Number of paintings = $0 \times 20 = 0$ 8. **Calculate number of paintings over 90 in 80-100 interval:** - The 80-100 interval is 20 years wide, but over 90 means only 10 years (90-100). - Assuming uniform distribution, number over 90 in 80-100 = $20 \times \frac{10}{20} = 10$ 9. **Calculate number of paintings over 90 in 100-120 interval:** - Entire interval is over 90, so all 4 paintings count. 10. **Calculate number of paintings over 90 in 120-140 interval:** - All 0 paintings. 11. **Sum paintings over 90:** - $10 + 4 + 0 = 14$ 12. **Given in problem:** 18 paintings are over 90, but calculation gives 14, so actual total paintings must be scaled. 13. **Calculate scaling factor:** - Scaling factor = $\frac{18}{14} = 1.2857$ 14. **Calculate number of paintings over 70 years old:** - Over 70 means intervals 70-80, 80-100, 100-120, 120-140. - Calculate number in 70-80 interval: - Frequency density = 2.6 - Class width = 20 - Number in 60-80 = $2.6 \times 20 = 52$ - For 70-80 (10 years), number = $52 \times \frac{10}{20} = 26$ 15. **Calculate number in 80-100, 100-120, 120-140 intervals:** - 80-100: 20 - 100-120: 4 - 120-140: 0 16. **Sum number over 70 before scaling:** - $26 + 20 + 4 + 0 = 50$ 17. **Apply scaling factor:** - Estimated paintings over 70 = $50 \times 1.2857 = 64.29 \approx 64$ **Final answer:** Approximately 64 paintings are over 70 years old.