1. **State the problem:** We are given the system of equations: $$A + 4^2 + 10 = 30$$ $$B + 5^2 + 7 = 35$$ $$C + A^2 + B^2 = 25$$ and we need to find the value of $$\frac{A}{B} + B \times C$$. 2. **Simplify the first two equations:** For the first equation: $$A + 16 + 10 = 30$$ $$A + 26 = 30$$ $$A = 30 - 26$$ $$A = 4$$ For the second equation: $$B + 25 + 7 = 35$$ $$B + 32 = 35$$ $$B = 35 - 32$$ $$B = 3$$ 3. **Use values of A and B in the third equation:** $$C + A^2 + B^2 = 25$$ $$C + 4^2 + 3^2 = 25$$ $$C + 16 + 9 = 25$$ $$C + 25 = 25$$ $$C = 25 - 25$$ $$C = 0$$ 4. **Calculate the expression $$\frac{A}{B} + B \times C$$:** $$\frac{4}{3} + 3 \times 0 = \frac{4}{3} + 0 = \frac{4}{3}$$ 5. **Final answer:** $$\boxed{\frac{4}{3}}$$