Mixed Fractions

1. Problem: Calculate $4 \frac{2}{4} + 6 \frac{4}{8}$. Step 1: Convert mixed numbers to improper fractions. $4 \frac{2}{4} = 4 + \frac{2}{4} = \frac{16}{4} + \frac{2}{4} = \frac{18}{4}$. $6 \frac{4}{8} = 6 + \frac{4}{8} = \frac{48}{8} + \frac{4}{8} = \frac{52}{8}$. Step 2: Find common denominator for $\frac{18}{4}$ and $\frac{52}{8}$. LCM of 4 and 8 is 8. $\frac{18}{4} = \frac{18 \times 2}{4 \times 2} = \frac{36}{8}$. Step 3: Add fractions. $\frac{36}{8} + \frac{52}{8} = \frac{88}{8}$. Step 4: Simplify. $\frac{88}{8} = 11$. 2. Problem: Calculate $8 \frac{11}{14} - 5 \frac{3}{7}$. Step 1: Convert to improper fractions. $8 \frac{11}{14} = \frac{8 \times 14 + 11}{14} = \frac{112 + 11}{14} = \frac{123}{14}$. $5 \frac{3}{7} = \frac{5 \times 7 + 3}{7} = \frac{35 + 3}{7} = \frac{38}{7}$. Step 2: Find common denominator. LCM of 14 and 7 is 14. $\frac{38}{7} = \frac{38 \times 2}{7 \times 2} = \frac{76}{14}$. Step 3: Subtract. $\frac{123}{14} - \frac{76}{14} = \frac{47}{14}$. Step 4: Convert to mixed number. $\frac{47}{14} = 3 \frac{5}{14}$. 3. Problem: Calculate $6 \frac{2}{3} + 4 \frac{5}{15}$. Step 1: Convert to improper fractions. $6 \frac{2}{3} = \frac{6 \times 3 + 2}{3} = \frac{18 + 2}{3} = \frac{20}{3}$. $4 \frac{5}{15} = \frac{4 \times 15 + 5}{15} = \frac{60 + 5}{15} = \frac{65}{15}$. Step 2: Simplify $\frac{65}{15}$. Divide numerator and denominator by 5: $\frac{13}{3}$. Step 3: Add fractions with common denominator 3. $\frac{20}{3} + \frac{13}{3} = \frac{33}{3} = 11$. 4. Problem: Calculate $7 \frac{12}{16} - 5 \frac{1}{2}$. Step 1: Convert to improper fractions. $7 \frac{12}{16} = \frac{7 \times 16 + 12}{16} = \frac{112 + 12}{16} = \frac{124}{16}$. Simplify $\frac{124}{16}$ by dividing numerator and denominator by 4: $\frac{31}{4}$. $5 \frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}$. Step 2: Find common denominator. LCM of 4 and 2 is 4. $\frac{11}{2} = \frac{22}{4}$. Step 3: Subtract. $\frac{31}{4} - \frac{22}{4} = \frac{9}{4} = 2 \frac{1}{4}$. 5. Problem: Calculate $8 \frac{6}{7} - 7 \frac{2}{14}$. Step 1: Convert to improper fractions. $8 \frac{6}{7} = \frac{8 \times 7 + 6}{7} = \frac{56 + 6}{7} = \frac{62}{7}$. $7 \frac{2}{14} = \frac{7 \times 14 + 2}{14} = \frac{98 + 2}{14} = \frac{100}{14}$. Simplify $\frac{100}{14}$ by dividing numerator and denominator by 2: $\frac{50}{7}$. Step 2: Subtract. $\frac{62}{7} - \frac{50}{7} = \frac{12}{7} = 1 \frac{5}{7}$. 6. Problem: Calculate $7 \frac{3}{4} + 4 \frac{8}{20}$. Step 1: Convert to improper fractions. $7 \frac{3}{4} = \frac{7 \times 4 + 3}{4} = \frac{28 + 3}{4} = \frac{31}{4}$. $4 \frac{8}{20} = \frac{4 \times 20 + 8}{20} = \frac{80 + 8}{20} = \frac{88}{20}$. Simplify $\frac{88}{20}$ by dividing numerator and denominator by 4: $\frac{22}{5}$. Step 2: Find common denominator. LCM of 4 and 5 is 20. $\frac{31}{4} = \frac{31 \times 5}{4 \times 5} = \frac{155}{20}$. $\frac{22}{5} = \frac{22 \times 4}{5 \times 4} = \frac{88}{20}$. Step 3: Add. $\frac{155}{20} + \frac{88}{20} = \frac{243}{20} = 12 \frac{3}{20}$. 7. Problem: Calculate $8 \frac{2}{9} + 5 \frac{1}{3}$. Step 1: Convert to improper fractions. $8 \frac{2}{9} = \frac{8 \times 9 + 2}{9} = \frac{72 + 2}{9} = \frac{74}{9}$. $5 \frac{1}{3} = \frac{5 \times 3 + 1}{3} = \frac{15 + 1}{3} = \frac{16}{3}$. Step 2: Find common denominator. LCM of 9 and 3 is 9. $\frac{16}{3} = \frac{16 \times 3}{3 \times 3} = \frac{48}{9}$. Step 3: Add. $\frac{74}{9} + \frac{48}{9} = \frac{122}{9} = 13 \frac{5}{9}$. 8. Problem: Calculate $1 \frac{2}{9} + 3 \frac{12}{18}$. Step 1: Convert to improper fractions. $1 \frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{9 + 2}{9} = \frac{11}{9}$. $3 \frac{12}{18} = \frac{3 \times 18 + 12}{18} = \frac{54 + 12}{18} = \frac{66}{18}$. Simplify $\frac{66}{18}$ by dividing numerator and denominator by 6: $\frac{11}{3}$. Step 2: Find common denominator. LCM of 9 and 3 is 9. $\frac{11}{3} = \frac{11 \times 3}{3 \times 3} = \frac{33}{9}$. Step 3: Add. $\frac{11}{9} + \frac{33}{9} = \frac{44}{9} = 4 \frac{8}{9}$. 9. Problem: Calculate $5 \frac{3}{6} - 1 \frac{16}{18}$. Step 1: Convert to improper fractions. $5 \frac{3}{6} = \frac{5 \times 6 + 3}{6} = \frac{30 + 3}{6} = \frac{33}{6}$. Simplify $\frac{33}{6}$ by dividing numerator and denominator by 3: $\frac{11}{2}$. $1 \frac{16}{18} = \frac{1 \times 18 + 16}{18} = \frac{18 + 16}{18} = \frac{34}{18}$. Simplify $\frac{34}{18}$ by dividing numerator and denominator by 2: $\frac{17}{9}$. Step 2: Find common denominator. LCM of 2 and 9 is 18. $\frac{11}{2} = \frac{11 \times 9}{2 \times 9} = \frac{99}{18}$. $\frac{17}{9} = \frac{17 \times 2}{9 \times 2} = \frac{34}{18}$. Step 3: Subtract. $\frac{99}{18} - \frac{34}{18} = \frac{65}{18} = 3 \frac{11}{18}$. 10. Problem: Calculate $7 \frac{7}{12} - \frac{1}{3}$. Step 1: Convert to improper fraction. $7 \frac{7}{12} = \frac{7 \times 12 + 7}{12} = \frac{84 + 7}{12} = \frac{91}{12}$. Step 2: Find common denominator. LCM of 12 and 3 is 12. $\frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12}$. Step 3: Subtract. $\frac{91}{12} - \frac{4}{12} = \frac{87}{12}$. Step 4: Simplify. Divide numerator and denominator by 3: $\frac{29}{4} = 7 \frac{1}{4}$. Final answers: 1. 11 2. $3 \frac{5}{14}$ 3. 11 4. $2 \frac{1}{4}$ 5. $1 \frac{5}{7}$ 6. $12 \frac{3}{20}$ 7. $13 \frac{5}{9}$ 8. $4 \frac{8}{9}$ 9. $3 \frac{11}{18}$ 10. $7 \frac{1}{4}$