1. **State the problem:** We are given volumes of spheres A and B, and the ratio of the radii of spheres B and C. We need to find the ratio of the surface areas of spheres A and C. 2. **Recall formulas:** - Volume of a sphere: $$V = \frac{4}{3} \pi r^3$$ - Surface area of a sphere: $$S = 4 \pi r^2$$ 3. **Given data:** - Volume of A: $$V_A = 125$$ cm³ - Volume of B: $$V_B = 27$$ cm³ - Radius ratio $$r_B : r_C = 1 : 2$$ 4. **Find radii of A and B:** From volume formula, $$V = \frac{4}{3} \pi r^3 \implies r = \sqrt[3]{\frac{3V}{4\pi}}$$ Since $$\frac{V_A}{V_B} = \frac{125}{27} = \left(\frac{r_A}{r_B}\right)^3$$, $$\frac{r_A}{r_B} = \sqrt[3]{\frac{125}{27}} = \frac{5}{3}$$ 5. **Express radius of C in terms of radius of B:** Given $$r_B : r_C = 1 : 2$$, so $$r_C = 2 r_B$$ 6. **Find ratio of surface areas:** Surface area ratio $$S_A : S_C = r_A^2 : r_C^2 = \left(\frac{r_A}{r_C}\right)^2$$ Calculate $$\frac{r_A}{r_C}$$: $$\frac{r_A}{r_C} = \frac{r_A}{r_B} \times \frac{r_B}{r_C} = \frac{5}{3} \times \frac{1}{2} = \frac{5}{6}$$ Therefore, $$S_A : S_C = \left(\frac{5}{6}\right)^2 = \frac{25}{36}$$ **Final answer:** The ratio of the surface areas of sphere A to sphere C is $$25 : 36$$.