1. **Problem Statement:** Calculate the arithmetic mean, harmonic mean, geometric mean, and quadratic mean of the ages 10, 15, and 20. 2. **Formulas:** - Arithmetic Mean (AM): $$AM = \frac{x_1 + x_2 + x_3}{3}$$ - Harmonic Mean (HM): $$HM = \frac{3}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3}}$$ - Geometric Mean (GM): $$GM = \sqrt[3]{x_1 \times x_2 \times x_3}$$ - Quadratic Mean (QM) or Root Mean Square: $$QM = \sqrt{\frac{x_1^2 + x_2^2 + x_3^2}{3}}$$ 3. **Calculations:** - Given ages: $x_1=10$, $x_2=15$, $x_3=20$ - Arithmetic Mean: $$AM = \frac{10 + 15 + 20}{3} = \frac{45}{3} = 15$$ - Harmonic Mean: $$HM = \frac{3}{\frac{1}{10} + \frac{1}{15} + \frac{1}{20}} = \frac{3}{0.1 + 0.0667 + 0.05} = \frac{3}{0.2167} \approx 13.85$$ - Geometric Mean: $$GM = \sqrt[3]{10 \times 15 \times 20} = \sqrt[3]{3000} \approx 14.42$$ - Quadratic Mean: $$QM = \sqrt{\frac{10^2 + 15^2 + 20^2}{3}} = \sqrt{\frac{100 + 225 + 400}{3}} = \sqrt{\frac{725}{3}} = \sqrt{241.67} \approx 15.55$$ 4. **Comment on the relationship:** For positive numbers, the means satisfy the inequality: $$HM \leq GM \leq AM \leq QM$$ Here, approximately: $$13.85 \leq 14.42 \leq 15 \leq 15.55$$ This confirms the expected relationship between these means.