1. **Stating the problem:** We need to solve the fraction and ratio calculations given: - Calculate $\frac{13}{2} \times \frac{5}{3}$ - Calculate $\frac{15 \times 5}{2 \times 3}$ - Calculate $\frac{7 \times 5}{6}$ - Given a total amount 189000 split in ratio 3:2:4, find the amounts for each part. - Simplify the fraction $\frac{22}{5} \times \frac{5}{2}$ and express as a mixed number. 2. **Formula and rules:** - Multiplying fractions: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ - Simplify fractions by canceling common factors. - For ratios, total parts = sum of ratio numbers; each part's amount = total amount $\times$ (part ratio / total parts). - Convert improper fractions to mixed numbers by dividing numerator by denominator. 3. **Calculations:** **(a) Calculate $\frac{13}{2} \times \frac{5}{3}$:** $$\frac{13}{2} \times \frac{5}{3} = \frac{13 \times 5}{2 \times 3} = \frac{65}{6}$$ This is an improper fraction. Convert to mixed number: $$65 \div 6 = 10 \text{ remainder } 5 \Rightarrow 10 \frac{5}{6}$$ **(b) Calculate $\frac{15 \times 5}{2 \times 3}$:** $$\frac{15 \times 5}{2 \times 3} = \frac{75}{6}$$ Simplify numerator and denominator by dividing both by 3: $$\frac{75 \div 3}{6 \div 3} = \frac{25}{2}$$ Convert to mixed number: $$25 \div 2 = 12 \text{ remainder } 1 \Rightarrow 12 \frac{1}{2}$$ **(c) Calculate $\frac{7 \times 5}{6}$:** $$\frac{7 \times 5}{6} = \frac{35}{6}$$ Convert to mixed number: $$35 \div 6 = 5 \text{ remainder } 5 \Rightarrow 5 \frac{5}{6}$$ **(d) Split 189000 in ratio 3:2:4:** Total parts = $3 + 2 + 4 = 9$ Amount for part 1: $$189000 \times \frac{3}{9} = 189000 \times \frac{1}{3} = 63000$$ Amount for part 2: $$189000 \times \frac{2}{9} = 42000$$ Amount for part 3: $$189000 \times \frac{4}{9} = 84000$$ **(e) Simplify $\frac{22}{5} \times \frac{5}{2}$:** Multiply numerators and denominators: $$\frac{22 \times 5}{5 \times 2} = \frac{110}{10}$$ Simplify by dividing numerator and denominator by 10: $$\frac{110 \div 10}{10 \div 10} = \frac{11}{1} = 11$$ 4. **Final answers:** - $\frac{13}{2} \times \frac{5}{3} = 10 \frac{5}{6}$ - $\frac{15 \times 5}{2 \times 3} = 12 \frac{1}{2}$ - $\frac{7 \times 5}{6} = 5 \frac{5}{6}$ - Amounts in ratio 3:2:4 for 189000 are 63000, 42000, and 84000 respectively. - $\frac{22}{5} \times \frac{5}{2} = 11$