Fraction Multiplication Division 884D73

1. Problem: Multiply $\frac{3}{4}$ by $\frac{6}{7}$. Formula: Multiply numerators and denominators: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ Calculation: $$\frac{3}{4} \times \frac{6}{7} = \frac{3 \times 6}{4 \times 7} = \frac{18}{28}$$ Simplify by dividing numerator and denominator by 2: $$\frac{\cancel{18}^{9}}{\cancel{28}^{14}} = \frac{9}{14}$$ Answer: $\frac{9}{14}$ 2. Problem: Multiply $1 \frac{1}{3}$ by $8 \frac{4}{8}$. Convert mixed numbers to improper fractions: $$1 \frac{1}{3} = \frac{4}{3}, \quad 8 \frac{4}{8} = 8 \frac{1}{2} = \frac{17}{2}$$ Multiply: $$\frac{4}{3} \times \frac{17}{2} = \frac{4 \times 17}{3 \times 2} = \frac{68}{6}$$ Simplify by dividing numerator and denominator by 2: $$\frac{\cancel{68}^{34}}{\cancel{6}^{3}} = \frac{34}{3} = 11 \frac{1}{3}$$ Answer: $11 \frac{1}{3}$ 3. Problem: Multiply $\frac{5}{6}$ by $9$. Rewrite $9$ as $\frac{9}{1}$: $$\frac{5}{6} \times \frac{9}{1} = \frac{5 \times 9}{6 \times 1} = \frac{45}{6}$$ Simplify by dividing numerator and denominator by 3: $$\frac{\cancel{45}^{15}}{\cancel{6}^{2}} = \frac{15}{2} = 7 \frac{1}{2}$$ Answer: $7 \frac{1}{2}$ 4. Problem: Multiply $3 \frac{1}{2}$ by $\frac{5}{9}$. Convert mixed number: $$3 \frac{1}{2} = \frac{7}{2}$$ Multiply: $$\frac{7}{2} \times \frac{5}{9} = \frac{7 \times 5}{2 \times 9} = \frac{35}{18} = 1 \frac{17}{18}$$ Answer: $1 \frac{17}{18}$ 5. Problem: Divide $10 \frac{1}{4}$ by $\frac{3}{5}$. Convert mixed number: $$10 \frac{1}{4} = \frac{41}{4}$$ Division of fractions: multiply by reciprocal: $$\frac{41}{4} \div \frac{3}{5} = \frac{41}{4} \times \frac{5}{3} = \frac{41 \times 5}{4 \times 3} = \frac{205}{12} = 17 \frac{1}{12}$$ Answer: $17 \frac{1}{12}$ 6. Problem: Divide $\frac{6}{9}$ by $\frac{4}{10}$. Simplify $\frac{6}{9}$: $$\frac{\cancel{6}^{2}}{\cancel{9}^{3}} = \frac{2}{3}$$ Divide by multiplying by reciprocal: $$\frac{2}{3} \div \frac{4}{10} = \frac{2}{3} \times \frac{10}{4} = \frac{2 \times 10}{3 \times 4} = \frac{20}{12}$$ Simplify by dividing numerator and denominator by 4: $$\frac{\cancel{20}^{5}}{\cancel{12}^{3}} = \frac{5}{3}$$ Answer: $\frac{5}{3}$ 7. Problem: Divide $8 \frac{1}{18}$ by $12$. Convert mixed number: $$8 \frac{1}{18} = \frac{145}{18}$$ Rewrite $12$ as $\frac{12}{1}$: $$\frac{145}{18} \div \frac{12}{1} = \frac{145}{18} \times \frac{1}{12} = \frac{145}{216}$$ This fraction cannot be simplified further. Answer: $\frac{145}{216}$ 8. Problem: Divide $\frac{6}{5}$ by $\frac{8}{10}$. Rewrite division as multiplication by reciprocal: $$\frac{6}{5} \times \frac{10}{8} = \frac{6 \times 10}{5 \times 8} = \frac{60}{40}$$ Simplify by dividing numerator and denominator by 20: $$\frac{\cancel{60}^{3}}{\cancel{40}^{2}} = \frac{3}{2}$$ Answer: $\frac{3}{2}$ 9. Problem: Multiply $3 \frac{3}{5}$ by $\frac{5}{12}$. Convert mixed number: $$3 \frac{3}{5} = \frac{18}{5}$$ Multiply: $$\frac{18}{5} \times \frac{5}{12} = \frac{18 \times 5}{5 \times 12}$$ Cancel $5$ in numerator and denominator: $$\frac{18 \times \cancel{5}}{\cancel{5} \times 12} = \frac{18}{12}$$ Simplify by dividing numerator and denominator by 6: $$\frac{\cancel{18}^{3}}{\cancel{12}^{2}} = \frac{3}{2}$$ Answer: $\frac{3}{2}$ 10. Problem: Divide $\frac{9}{10}$ by $4$. Rewrite $4$ as $\frac{4}{1}$: $$\frac{9}{10} \div \frac{4}{1} = \frac{9}{10} \times \frac{1}{4} = \frac{9}{40}$$ Answer: $\frac{9}{40}$ 11. Problem: Divide $8 \frac{3}{12}$ by $\frac{6}{10}$. Convert mixed number: $$8 \frac{3}{12} = 8 \frac{1}{4} = \frac{33}{4}$$ Divide by multiplying by reciprocal: $$\frac{33}{4} \times \frac{10}{6} = \frac{33 \times 10}{4 \times 6} = \frac{330}{24}$$ Simplify by dividing numerator and denominator by 6: $$\frac{\cancel{330}^{55}}{\cancel{24}^{4}} = \frac{55}{4} = 13 \frac{3}{4}$$ Answer: $13 \frac{3}{4}$ 12. Problem: Multiply $\frac{1}{2}$ by $\frac{4}{9}$. Multiply: $$\frac{1}{2} \times \frac{4}{9} = \frac{1 \times 4}{2 \times 9} = \frac{4}{18}$$ Simplify by dividing numerator and denominator by 2: $$\frac{\cancel{4}^{2}}{\cancel{18}^{9}} = \frac{2}{9}$$ Answer: $\frac{2}{9}$ 13. Problem: Multiply $\frac{8}{12}$ by $4$. Simplify $\frac{8}{12}$: $$\frac{\cancel{8}^{2}}{\cancel{12}^{3}} = \frac{2}{3}$$ Rewrite $4$ as $\frac{4}{1}$: $$\frac{2}{3} \times \frac{4}{1} = \frac{8}{3} = 2 \frac{2}{3}$$ Answer: $2 \frac{2}{3}$ 14. Problem: Multiply $\frac{6}{8}$ by $3 \frac{3}{9}$. Simplify $\frac{6}{8}$: $$\frac{\cancel{6}^{3}}{\cancel{8}^{4}} = \frac{3}{4}$$ Convert mixed number: $$3 \frac{3}{9} = 3 \frac{1}{3} = \frac{10}{3}$$ Multiply: $$\frac{3}{4} \times \frac{10}{3} = \frac{3 \times 10}{4 \times 3}$$ Cancel $3$: $$\frac{\cancel{3} \times 10}{4 \times \cancel{3}} = \frac{10}{4}$$ Simplify by dividing numerator and denominator by 2: $$\frac{\cancel{10}^{5}}{\cancel{4}^{2}} = \frac{5}{2} = 2 \frac{1}{2}$$ Answer: $2 \frac{1}{2}$ 15. Problem: Multiply $9 \frac{10}{11}$ by $\frac{3}{9}$. Convert mixed number: $$9 \frac{10}{11} = \frac{109}{11}$$ Simplify $\frac{3}{9}$: $$\frac{\cancel{3}^{1}}{\cancel{9}^{3}} = \frac{1}{3}$$ Multiply: $$\frac{109}{11} \times \frac{1}{3} = \frac{109}{33} = 3 \frac{10}{33}$$ Answer: $3 \frac{10}{33}$ 16. Problem: Multiply $\frac{14}{18}$ by $\frac{13}{10}$. Simplify $\frac{14}{18}$: $$\frac{\cancel{14}^{7}}{\cancel{18}^{9}} = \frac{7}{9}$$ Multiply: $$\frac{7}{9} \times \frac{13}{10} = \frac{7 \times 13}{9 \times 10} = \frac{91}{90} = 1 \frac{1}{90}$$ Answer: $1 \frac{1}{90}$ 17. Problem: Divide $\frac{6}{13}$ by $3 \frac{9}{10}$. Convert mixed number: $$3 \frac{9}{10} = \frac{39}{10}$$ Divide by multiplying by reciprocal: $$\frac{6}{13} \times \frac{10}{39} = \frac{6 \times 10}{13 \times 39} = \frac{60}{507}$$ Simplify by dividing numerator and denominator by 3: $$\frac{\cancel{60}^{20}}{\cancel{507}^{169}} = \frac{20}{169}$$ Answer: $\frac{20}{169}$ 18. Problem: Divide $18 \frac{1}{2}$ by $5$. Convert mixed number: $$18 \frac{1}{2} = \frac{37}{2}$$ Rewrite $5$ as $\frac{5}{1}$: $$\frac{37}{2} \div \frac{5}{1} = \frac{37}{2} \times \frac{1}{5} = \frac{37}{10} = 3 \frac{7}{10}$$ Answer: $3 \frac{7}{10}$ 19. Problem: Multiply $\frac{16}{17}$ by $\frac{18}{19}$. Multiply: $$\frac{16}{17} \times \frac{18}{19} = \frac{16 \times 18}{17 \times 19} = \frac{288}{323}$$ This fraction cannot be simplified further. Answer: $\frac{288}{323}$ 20. Problem: Divide $14 \frac{3}{5}$ by $\frac{5}{6}$. Convert mixed number: $$14 \frac{3}{5} = \frac{73}{5}$$ Divide by multiplying by reciprocal: $$\frac{73}{5} \times \frac{6}{5} = \frac{73 \times 6}{5 \times 5} = \frac{438}{25} = 17 \frac{13}{25}$$ Answer: $17 \frac{13}{25}$