1. **Problem statement:** We want to find how many kilograms of pure gold must be added to 150 kg of 800-fineness gold to obtain gold of 925 fineness. 2. **Understanding fineness:** Fineness indicates the purity of gold in parts per thousand. For example, 800-fineness gold means 800 parts gold per 1000 parts alloy. 3. **Given:** - Initial mass of gold alloy: $m_1 = 150$ kg - Initial fineness: $f_1 = 800$ - Desired fineness: $f_2 = 925$ - Pure gold fineness: $f_p = 1000$ 4. **Let $x$ be the mass of pure gold to add.** 5. **Total gold content before adding:** $$\text{Gold in initial alloy} = m_1 \times \frac{f_1}{1000} = 150 \times \frac{800}{1000} = 120 \text{ kg}$$ 6. **Total gold content after adding $x$ kg of pure gold:** $$120 + x$$ 7. **Total mass after adding:** $$150 + x$$ 8. **Final fineness equation:** $$\frac{120 + x}{150 + x} = \frac{925}{1000}$$ 9. **Solve for $x$:** $$120 + x = 0.925(150 + x)$$ $$120 + x = 138.75 + 0.925x$$ $$x - 0.925x = 138.75 - 120$$ $$0.075x = 18.75$$ $$x = \frac{18.75}{0.075} = 250$$ 10. **Answer:** You must add **250 kilograms** of pure gold to the 150 kg of 800-fineness gold to obtain 925-fineness gold.