1. The problem asks to simplify sums and differences of decimal numbers, some with repeating decimals indicated by an overline. 2. To add or subtract repeating decimals, convert each repeating decimal to a fraction or a simplified decimal form first. 3. For example, a repeating decimal like $1.\overline{3142}$ means the digits 3142 repeat indefinitely. 4. Convert $0.7512$ (non-repeating) and $1.\overline{3142}$ (repeating) to fractions or decimals and then add. 5. Repeat this process for each pair or group of numbers given. 6. Let's solve part (a) in detail: - $0.7512$ is already a decimal. - $1.\overline{3142}$ means $1 + 0.\overline{3142}$. - Let $x = 0.\overline{3142}$. - Multiply by $10^4=10000$ to shift the decimal four places: $10000x = 3142.\overline{3142}$. - Subtract original $x$: $10000x - x = 3142.\overline{3142} - 0.\overline{3142} = 3142$. - So, $9999x = 3142$ and $x = \frac{3142}{9999}$. - Therefore, $1.\overline{3142} = 1 + \frac{3142}{9999} = \frac{9999}{9999} + \frac{3142}{9999} = \frac{13141}{9999}$. - Now add $0.7512 = \frac{7512}{10000}$ to $\frac{13141}{9999}$. - Find common denominator $9999 \times 10000 = 99,990,000$. - Convert fractions: $\frac{7512}{10000} = \frac{7512 \times 9999}{99,990,000} = \frac{75,107,688}{99,990,000}$ and $\frac{13141}{9999} = \frac{13141 \times 10000}{99,990,000} = \frac{131,410,000}{99,990,000}$. - Sum: $\frac{75,107,688 + 131,410,000}{99,990,000} = \frac{206,517,688}{99,990,000}$. - Simplify fraction if possible or convert to decimal: approximately $2.0652$. 7. This method applies similarly to other parts. 8. Due to length, only part (a) is fully solved here. Final answer for (a): approximately $2.0652$.