1. **Problem Statement:** Calculate the product of mixed numbers by converting them to improper fractions and multiplying. 2. **Formula:** To multiply fractions, multiply the numerators and multiply the denominators: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Step-by-step for a):** - Convert mixed numbers to improper fractions: $$6 \frac{1}{4} = \frac{6 \times 4 + 1}{4} = \frac{25}{4}$$ $$\frac{3}{5}$$ (already a fraction) - Multiply: $$\frac{25}{4} \times \frac{3}{5} = \frac{25 \times 3}{4 \times 5} = \frac{75}{20}$$ - Simplify by dividing numerator and denominator by 5: $$\frac{\cancel{75}^{15}}{\cancel{20}^{4}} = \frac{15}{4}$$ 4. **Step-by-step for b):** - Convert mixed numbers: $$4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{14}{3}$$ $$1 \frac{2}{7} = \frac{1 \times 7 + 2}{7} = \frac{9}{7}$$ - Multiply: $$\frac{14}{3} \times \frac{9}{7} = \frac{14 \times 9}{3 \times 7} = \frac{126}{21}$$ - Simplify by dividing numerator and denominator by 21: $$\frac{\cancel{126}^{6}}{\cancel{21}^{1}} = 6$$ 5. **Step-by-step for c):** - Convert mixed numbers: $$5 \frac{4}{9} = \frac{5 \times 9 + 4}{9} = \frac{49}{9}$$ $$3 \frac{6}{7} = \frac{3 \times 7 + 6}{7} = \frac{27}{7}$$ - Multiply: $$\frac{49}{9} \times \frac{27}{7} = \frac{49 \times 27}{9 \times 7} = \frac{1323}{63}$$ - Simplify by dividing numerator and denominator by 63: $$\frac{\cancel{1323}^{21}}{\cancel{63}^{1}} = 21$$ 6. **Step-by-step for d):** - Convert mixed numbers: $$2 \frac{2}{5} = \frac{2 \times 5 + 2}{5} = \frac{12}{5}$$ $$5 \frac{5}{6} = \frac{5 \times 6 + 5}{6} = \frac{35}{6}$$ - Multiply: $$\frac{12}{5} \times \frac{35}{6} = \frac{12 \times 35}{5 \times 6} = \frac{420}{30}$$ - Simplify by dividing numerator and denominator by 30: $$\frac{\cancel{420}^{14}}{\cancel{30}^{1}} = 14$$ 7. **Step-by-step for e):** - Convert mixed numbers: $$1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{8}{5}$$ $$4 \frac{1}{8} = \frac{4 \times 8 + 1}{8} = \frac{33}{8}$$ - Multiply: $$\frac{8}{5} \times \frac{33}{8} = \frac{8 \times 33}{5 \times 8} = \frac{264}{40}$$ - Simplify by dividing numerator and denominator by 8: $$\frac{\cancel{264}^{33}}{\cancel{40}^{5}} = \frac{33}{5}$$ 8. **Step-by-step for f):** - Convert mixed numbers: $$3 \frac{3}{7} = \frac{3 \times 7 + 3}{7} = \frac{24}{7}$$ $$5 \frac{3}{6} = \frac{5 \times 6 + 3}{6} = \frac{33}{6}$$ - Multiply: $$\frac{24}{7} \times \frac{33}{6} = \frac{24 \times 33}{7 \times 6} = \frac{792}{42}$$ - Simplify by dividing numerator and denominator by 6: $$\frac{\cancel{792}^{132}}{\cancel{42}^{7}} = \frac{132}{7}$$ **Final answers:** - a) $\frac{15}{4}$ - b) $6$ - c) $21$ - d) $14$ - e) $\frac{33}{5}$ - f) $\frac{132}{7}$