1. **Stating the problem:** We need to perform addition and subtraction of mixed numbers and fractions for the given expressions. 2. **Formula and rules:** - To add or subtract mixed numbers, first convert them to improper fractions. - Find a common denominator to add or subtract fractions. - Simplify the result and convert back to mixed numbers if needed. --- ### Addition problems: **a) $12 \frac{3}{8} + 6 \frac{5}{6}$** 1. Convert to improper fractions: $$12 \frac{3}{8} = \frac{12 \times 8 + 3}{8} = \frac{99}{8}$$ $$6 \frac{5}{6} = \frac{6 \times 6 + 5}{6} = \frac{41}{6}$$ 2. Find common denominator: $\mathrm{lcm}(8,6) = 24$ 3. Convert fractions: $$\frac{99}{8} = \frac{99 \times 3}{24} = \frac{297}{24}$$ $$\frac{41}{6} = \frac{41 \times 4}{24} = \frac{164}{24}$$ 4. Add: $$\frac{297}{24} + \frac{164}{24} = \frac{461}{24}$$ 5. Convert back to mixed number: $$461 \div 24 = 19 \text{ remainder } 5 \Rightarrow 19 \frac{5}{24}$$ **b) $8 \frac{5}{9} + 6 \frac{6}{9}$** 1. Convert to improper fractions: $$8 \frac{5}{9} = \frac{8 \times 9 + 5}{9} = \frac{77}{9}$$ $$6 \frac{6}{9} = \frac{6 \times 9 + 6}{9} = \frac{60}{9}$$ 2. Same denominator 9, add: $$\frac{77}{9} + \frac{60}{9} = \frac{137}{9}$$ 3. Convert to mixed number: $$137 \div 9 = 15 \text{ remainder } 2 \Rightarrow 15 \frac{2}{9}$$ **c) $1 \frac{3}{4} + 2 \frac{1}{6}$** 1. Convert to improper fractions: $$1 \frac{3}{4} = \frac{7}{4}$$ $$2 \frac{1}{6} = \frac{13}{6}$$ 2. Find common denominator: $\mathrm{lcm}(4,6) = 12$ 3. Convert fractions: $$\frac{7}{4} = \frac{21}{12}$$ $$\frac{13}{6} = \frac{26}{12}$$ 4. Add: $$\frac{21}{12} + \frac{26}{12} = \frac{47}{12}$$ 5. Convert to mixed number: $$47 \div 12 = 3 \text{ remainder } 11 \Rightarrow 3 \frac{11}{12}$$ **d) $4 \frac{3}{14} + 3 \frac{5}{8}$** 1. Convert to improper fractions: $$4 \frac{3}{14} = \frac{59}{14}$$ $$3 \frac{5}{8} = \frac{29}{8}$$ 2. Find common denominator: $\mathrm{lcm}(14,8) = 56$ 3. Convert fractions: $$\frac{59}{14} = \frac{236}{56}$$ $$\frac{29}{8} = \frac{203}{56}$$ 4. Add: $$\frac{236}{56} + \frac{203}{56} = \frac{439}{56}$$ 5. Convert to mixed number: $$439 \div 56 = 7 \text{ remainder } 47 \Rightarrow 7 \frac{47}{56}$$ --- ### Subtraction problems: **a) $12 - 6 \frac{5}{6}$** 1. Convert $6 \frac{5}{6}$ to improper fraction: $$6 \frac{5}{6} = \frac{41}{6}$$ 2. Convert 12 to fraction with denominator 6: $$12 = \frac{72}{6}$$ 3. Subtract: $$\frac{72}{6} - \frac{41}{6} = \frac{31}{6}$$ 4. Convert to mixed number: $$31 \div 6 = 5 \text{ remainder } 1 \Rightarrow 5 \frac{1}{6}$$ **b) $2 \frac{5}{7} - \frac{6}{7}$** 1. Convert $2 \frac{5}{7}$ to improper fraction: $$2 \frac{5}{7} = \frac{19}{7}$$ 2. Subtract: $$\frac{19}{7} - \frac{6}{7} = \frac{13}{7}$$ 3. Convert to mixed number: $$13 \div 7 = 1 \text{ remainder } 6 \Rightarrow 1 \frac{6}{7}$$ **c) $2 \frac{2}{5} - 1 \frac{1}{10}$** 1. Convert to improper fractions: $$2 \frac{2}{5} = \frac{12}{5}$$ $$1 \frac{1}{10} = \frac{11}{10}$$ 2. Find common denominator: $\mathrm{lcm}(5,10) = 10$ 3. Convert fractions: $$\frac{12}{5} = \frac{24}{10}$$ 4. Subtract: $$\frac{24}{10} - \frac{11}{10} = \frac{13}{10}$$ 5. Convert to mixed number: $$13 \div 10 = 1 \text{ remainder } 3 \Rightarrow 1 \frac{3}{10}$$ **d) $4 \frac{5}{12} - 3 \frac{1}{8}$** 1. Convert to improper fractions: $$4 \frac{5}{12} = \frac{53}{12}$$ $$3 \frac{1}{8} = \frac{25}{8}$$ 2. Find common denominator: $\mathrm{lcm}(12,8) = 24$ 3. Convert fractions: $$\frac{53}{12} = \frac{106}{24}$$ $$\frac{25}{8} = \frac{75}{24}$$ 4. Subtract: $$\frac{106}{24} - \frac{75}{24} = \frac{31}{24}$$ 5. Convert to mixed number: $$31 \div 24 = 1 \text{ remainder } 7 \Rightarrow 1 \frac{7}{24}$$ --- **Final answers:** **Addition:** - a) $19 \frac{5}{24}$ - b) $15 \frac{2}{9}$ - c) $3 \frac{11}{12}$ - d) $7 \frac{47}{56}$ **Subtraction:** - a) $5 \frac{1}{6}$ - b) $1 \frac{6}{7}$ - c) $1 \frac{3}{10}$ - d) $1 \frac{7}{24}$