1. The problem asks to complete the multiplications by breaking down the multiplication of mixed numbers into sums of products of their whole and fractional parts. 2. The formula used is: $$\left(a + \frac{b}{c}\right) \times \left(d + \frac{e}{f}\right) = a \times d + a \times \frac{e}{f} + \frac{b}{c} \times d + \frac{b}{c} \times \frac{e}{f}$$ 3. Let's solve part (a): Given: $$3 \frac{1}{2} \times 3 \frac{8}{8}$$ Rewrite mixed numbers: $$3 \frac{1}{2} = 3 + \frac{1}{2}$$ $$3 \frac{8}{8} = 3 + \frac{8}{8} = 3 + 1 = 4$$ Multiply: $$\left(3 + \frac{1}{2}\right) \times 4 = 3 \times 4 + \frac{1}{2} \times 4$$ Calculate each term: $$3 \times 4 = 12$$ $$\frac{1}{2} \times 4 = 2$$ Sum: $$12 + 2 = 14$$ 4. Part (a) answer: $$14$$ 5. Part (b): Given: $$2 \frac{1}{4} \times 12 \frac{2}{12}$$ Rewrite mixed numbers: $$2 \frac{1}{4} = 2 + \frac{1}{4}$$ $$12 \frac{2}{12} = 12 + \frac{2}{12}$$ Multiply: $$\left(2 + \frac{1}{4}\right) \times \left(12 + \frac{2}{12}\right) = 2 \times 12 + 2 \times \frac{2}{12} + \frac{1}{4} \times 12 + \frac{1}{4} \times \frac{2}{12}$$ Calculate each term: $$2 \times 12 = 24$$ $$2 \times \frac{2}{12} = \frac{4}{12} = \frac{1}{3}$$ $$\frac{1}{4} \times 12 = 3$$ $$\frac{1}{4} \times \frac{2}{12} = \frac{2}{48} = \frac{1}{24}$$ Sum: $$24 + \frac{1}{3} + 3 + \frac{1}{24} = (24 + 3) + \left(\frac{1}{3} + \frac{1}{24}\right) = 27 + \frac{8}{24} + \frac{1}{24} = 27 + \frac{9}{24} = 27 + \frac{3}{8} = 27 \frac{3}{8}$$ 6. Part (b) answer: $$27 \frac{3}{8}$$ 7. Part (c): Given: $$4 \frac{9}{3} \times 9 \frac{2}{9}$$ Note: $$\frac{9}{3} = 3$$ so $$4 \frac{9}{3} = 4 + 3 = 7$$ $$9 \frac{2}{9} = 9 + \frac{2}{9}$$ Multiply: $$7 \times \left(9 + \frac{2}{9}\right) = 7 \times 9 + 7 \times \frac{2}{9} = 63 + \frac{14}{9} = 63 + 1 \frac{5}{9} = 64 \frac{5}{9}$$ 8. Part (c) answer: $$64 \frac{5}{9}$$ 9. Part (d): Given: $$8 \frac{3}{10} \times 8 \frac{10}{1}$$ Note: $$\frac{10}{1} = 10$$ so $$8 \frac{10}{1} = 8 + 10 = 18$$ $$8 \frac{3}{10} = 8 + \frac{3}{10}$$ Multiply: $$\left(8 + \frac{3}{10}\right) \times 18 = 8 \times 18 + \frac{3}{10} \times 18 = 144 + \frac{54}{10} = 144 + 5.4 = 149.4$$ 10. Part (d) answer: $$149.4$$