1. **Problem statement:** We need to find the sum of three natural numbers given that the sum of their squares is 6525. The second number is $\frac{2}{3}$ of the first and also $\frac{1}{2}$ of the third. 2. **Define variables:** Let the first number be $x$. 3. **Express other numbers:** - Second number: $y = \frac{2}{3}x$ - Third number: $z = 2y = 2 \times \frac{2}{3}x = \frac{4}{3}x$ 4. **Sum of squares equation:** $$x^2 + y^2 + z^2 = 6525$$ Substitute $y$ and $z$: $$x^2 + \left(\frac{2}{3}x\right)^2 + \left(\frac{4}{3}x\right)^2 = 6525$$ 5. **Simplify:** $$x^2 + \frac{4}{9}x^2 + \frac{16}{9}x^2 = 6525$$ Combine terms: $$x^2 + \frac{4}{9}x^2 + \frac{16}{9}x^2 = x^2 + \frac{20}{9}x^2 = \frac{9}{9}x^2 + \frac{20}{9}x^2 = \frac{29}{9}x^2$$ 6. **Solve for $x^2$:** $$\frac{29}{9}x^2 = 6525 \implies x^2 = \frac{6525 \times 9}{29} = \frac{58725}{29} = 2025$$ 7. **Find $x$:** $$x = \sqrt{2025} = 45$$ 8. **Find $y$ and $z$:** $$y = \frac{2}{3} \times 45 = 30$$ $$z = \frac{4}{3} \times 45 = 60$$ 9. **Find the sum:** $$x + y + z = 45 + 30 + 60 = 135$$ **Final answer:** 135 (Option C)