3. Determine the value of the following without the use or a. \( \frac{1}{2} \times \frac{1}{2} \) b. \( \frac{1}{2} \times \frac{1}{3} \) c. \( \frac{1}{2} \times \frac{2}{3} \) d. \( \frac{3}{2} \times \frac{2}{3} \) e. \( \frac{2}{5} \times \frac{15}{8} \) f. \( \frac{3}{7} \times \frac{7}{12} \) g. \( \frac{24}{27} \times \frac{12}{8} \) 1. \( \frac{0}{5} \times \frac{3}{4} \) 1. \( \frac{2}{3} \times \frac{3}{5} \times \frac{9}{11} \) k. \( \frac{2}{4}-\frac{2}{5} \times \frac{15}{3} \) 1. \( \frac{3}{2}\left(\frac{4}{5}+\frac{2}{3}\right) \) m. \( \frac{2}{9} \times \frac{5}{7}=\frac{10}{12} \) n. \( \frac{2}{3} \times \frac{a}{b} \)

Sure! Let's evaluate each of the given expressions step by step. ### a. \( \frac{1}{2} \times \frac{1}{2} \) Multiply the numerators and denominators: \[ \frac{1 \times 1}{2 \times 2} = \frac{1}{4} \] **Answer:** \( \frac{1}{4} \) --- ### b. \( \frac{1}{2} \times \frac{1}{3} \) \[ \frac{1 \times 1}{2 \times 3} = \frac{1}{6} \] **Answer:** \( \frac{1}{6} \) --- ### c. \( \frac{1}{2} \times \frac{2}{3} \) \[ \frac{1 \times 2}{2 \times 3} = \frac{2}{6} = \frac{1}{3} \] **Answer:** \( \frac{1}{3} \) --- ### d. \( \frac{3}{2} \times \frac{2}{3} \) \[ \frac{3 \times 2}{2 \times 3} = \frac{6}{6} = 1 \] **Answer:** \( 1 \) --- ### e. \( \frac{2}{5} \times \frac{15}{8} \) Simplify before multiplying: \[ \frac{2}{5} \times \frac{15}{8} = \frac{2 \times 15}{5 \times 8} = \frac{30}{40} = \frac{3}{4} \] **Answer:** \( \frac{3}{4} \) --- ### f. \( \frac{3}{7} \times \frac{7}{12} \) Cancel out the common factor (7): \[ \frac{3}{7} \times \frac{7}{12} = \frac{3 \times \cancel{7}}{\cancel{7} \times 12} = \frac{3}{12} = \frac{1}{4} \] **Answer:** \( \frac{1}{4} \) --- ### g. \( \frac{24}{27} \times \frac{12}{8} \) Simplify the fractions first: \[ \frac{24}{27} = \frac{8}{9} \quad \text{and} \quad \frac{12}{8} = \frac{3}{2} \] Now multiply: \[ \frac{8}{9} \times \frac{3}{2} = \frac{8 \times 3}{9 \times 2} = \frac{24}{18} = \frac{4}{3} \quad \text{or} \quad 1 \frac{1}{3} \] **Answer:** \( \frac{4}{3} \) or \( 1 \frac{1}{3} \) --- ### h. \( \frac{0}{5} \times \frac{3}{4} \) Any number multiplied by zero is zero: \[ \frac{0}{5} \times \frac{3}{4} = 0 \times \frac{3}{4} = 0 \] **Answer:** \( 0 \) --- ### i. \( \frac{2}{3} \times \frac{3}{5} \times \frac{9}{11} \) Multiply the numerators and denominators: \[ \frac{2 \times 3 \times 9}{3 \times 5 \times 11} = \frac{54}{165} \] Simplify by dividing numerator and denominator by 3: \[ \frac{54 \div 3}{165 \div 3} = \frac{18}{55} \] **Answer:** \( \frac{18}{55} \) --- ### j. \( \frac{2}{4} - \frac{2}{5} \times \frac{15}{3} \) Follow the order of operations (PEMDAS/BODMAS): 1. **Multiplication first:** \[ \frac{2}{5} \times \frac{15}{3} = \frac{2 \times 15}{5 \times 3} = \frac{30}{15} = 2 \] 2. **Then subtraction:** \[ \frac{2}{4} - 2 = \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2} \] **Answer:** \( -\frac{3}{2} \) or \( -1.5 \) --- ### k. \( \frac{3}{2}\left(\frac{4}{5} + \frac{2}{3}\right) \) 1. **Add the fractions inside the parentheses:** Find a common denominator (15): \[ \frac{4}{5} = \frac{12}{15} \quad \text{and} \quad \frac{2}{3} = \frac{10}{15} \] \[ \frac{12}{15} + \frac{10}{15} = \frac{22}{15} \] 2. **Multiply by \( \frac{3}{2} \):** \[ \frac{3}{2} \times \frac{22}{15} = \frac{66}{30} = \frac{11}{5} \quad \text{or} \quad 2.2 \] **Answer:** \( \frac{11}{5} \) or \( 2.2 \) --- ### m. \( \frac{2}{9} \times \frac{5}{7} = \frac{10}{12} \) Let's evaluate and check the given equality. **Calculation:** \[ \frac{2}{9} \times \frac{5}{7} = \frac{10}{63} \] **Given Equality:** \[ \frac{10}{63} \neq \frac{10}{12} \] It appears there's an error in the provided equality. **Correct Answer:** \( \frac{10}{63} \) --- ### n. \( \frac{2}{3} \times \frac{a}{b} \) Multiply the numerators and denominators: \[ \frac{2 \times a}{3 \times b} = \frac{2a}{3b} \] **Answer:** \( \frac{2a}{3b} \) --- If you have any further questions or need additional explanations, feel free to ask!