1. **State the problem:** Convert the decimal number 0.315 into a simplified fraction. 2. **Write the decimal as a fraction:** Since 0.315 has three decimal places, it can be written as $ \frac{315}{1000} $. 3. **Simplify the fraction:** Find the greatest common divisor (GCD) of 315 and 1000. 4. The prime factors of 315 are 3, 3, 5, and 7 (since $315 = 3^2 \times 5 \times 7$). 5. The prime factors of 1000 are 2, 2, 2, 5, 5, 5 (since $1000 = 2^3 \times 5^3$). 6. The common factor is 5. 7. Divide numerator and denominator by 5: $$\frac{\cancel{315}^{{\div 5}}}{\cancel{1000}^{{\div 5}}} = \frac{63}{200}$$ 8. Check if 63 and 200 have any common factors: - 63 factors: 3, 3, 7 - 200 factors: 2, 2, 2, 5, 5 No common factors remain. 9. **Final answer:** $0.315 = \frac{63}{200}$.