1. Simplify and evaluate each expression without a calculator. **a)** Simplify $$\dfrac{9}{5} \times \dfrac{-15}{21} \div \left(\dfrac{-45}{14}\right)$$ Step 1: Rewrite division as multiplication by reciprocal: $$\dfrac{9}{5} \times \dfrac{-15}{21} \times \dfrac{14}{-45}$$ Step 2: Simplify signs: negative times negative is positive. Step 3: Simplify numerators and denominators: $$\dfrac{9}{5} \times \dfrac{15}{21} \times \dfrac{14}{45}$$ Step 4: Factor and reduce: $$9 = 3 \times 3, \quad 15 = 3 \times 5, \quad 21 = 3 \times 7, \quad 14 = 2 \times 7, \quad 45 = 9 \times 5$$ Step 5: Cancel common factors: - Cancel 3 in numerator and denominator - Cancel 5 in numerator and denominator - Cancel 7 in numerator and denominator Resulting expression: $$\dfrac{3}{1} \times \dfrac{1}{1} \times \dfrac{2}{9} = \dfrac{3 \times 2}{9} = \dfrac{6}{9} = \dfrac{2}{3}$$ **Answer a):** $\dfrac{2}{3}$ **b)** Evaluate $$3 \left(-\dfrac{3}{4}\right) + 1$$ Step 1: Multiply: $$3 \times -\dfrac{3}{4} = -\dfrac{9}{4}$$ Step 2: Add 1 (which is $\dfrac{4}{4}$): $$-\dfrac{9}{4} + \dfrac{4}{4} = -\dfrac{5}{4}$$ **Answer b):** $-\dfrac{5}{4}$ **c)** Simplify $$\dfrac{3}{10} \times \dfrac{2}{5} - \left(-\dfrac{5}{8}\right)$$ Step 1: Multiply fractions: $$\dfrac{3}{10} \times \dfrac{2}{5} = \dfrac{6}{50} = \dfrac{3}{25}$$ Step 2: Subtract negative is addition: $$\dfrac{3}{25} + \dfrac{5}{8}$$ Step 3: Find common denominator $200$: $$\dfrac{3}{25} = \dfrac{24}{200}, \quad \dfrac{5}{8} = \dfrac{125}{200}$$ Step 4: Add: $$\dfrac{24}{200} + \dfrac{125}{200} = \dfrac{149}{200}$$ **Answer c):** $\dfrac{149}{200}$ **d)** Simplify $$\left(\dfrac{1.5}{-32}\right) \times \left(\dfrac{-8}{5}\right) \div \dfrac{21}{16}$$ Step 1: Convert decimals to fractions: $$1.5 = \dfrac{3}{2}$$ Step 2: Rewrite expression: $$\dfrac{3/2}{-32} \times \dfrac{-8}{5} \times \dfrac{16}{21}$$ Step 3: Simplify $$\dfrac{3/2}{-32} = \dfrac{3}{2} \times \dfrac{1}{-32} = -\dfrac{3}{64}$$ Step 4: Multiply all: $$-\dfrac{3}{64} \times -\dfrac{8}{5} \times \dfrac{16}{21}$$ Step 5: Negative times negative is positive: $$\dfrac{3}{64} \times \dfrac{8}{5} \times \dfrac{16}{21}$$ Step 6: Multiply numerators and denominators: $$\dfrac{3 \times 8 \times 16}{64 \times 5 \times 21} = \dfrac{384}{6720}$$ Step 7: Simplify fraction: Divide numerator and denominator by 96: $$\dfrac{4}{70} = \dfrac{2}{35}$$ **Answer d):** $\dfrac{2}{35}$ **e)** Evaluate $$\dfrac{1}{2} 3 \; \dfrac{3}{4} 3$$ Assuming multiplication: $$\dfrac{1}{2} \times 3 \times \dfrac{3}{4} \times 3$$ Step 1: Multiply numerators and denominators: $$\dfrac{1 \times 3 \times 3 \times 3}{2 \times 4} = \dfrac{27}{8}$$ **Answer e):** $\dfrac{27}{8}$ **f)** Simplify $$\dfrac{1}{2}\left(\dfrac{6}{7} - 8\right)$$ Step 1: Convert 8 to fraction: $$8 = \dfrac{56}{7}$$ Step 2: Subtract inside parentheses: $$\dfrac{6}{7} - \dfrac{56}{7} = -\dfrac{50}{7}$$ Step 3: Multiply by $\dfrac{1}{2}$: $$\dfrac{1}{2} \times -\dfrac{50}{7} = -\dfrac{50}{14} = -\dfrac{25}{7}$$ **Answer f):** $-\dfrac{25}{7}$ **g)** Evaluate $$2 \left(\dfrac{5}{3} \times 2\right)$$ Step 1: Multiply inside parentheses: $$\dfrac{5}{3} \times 2 = \dfrac{10}{3}$$ Step 2: Multiply by 2: $$2 \times \dfrac{10}{3} = \dfrac{20}{3}$$ **Answer g):** $\dfrac{20}{3}$ **h)** Simplify $$\left[\dfrac{5}{12} \div 15 - \dfrac{5}{3} \times \dfrac{3}{8} \div 10 \right]$$ Step 1: Rewrite division as multiplication by reciprocal: $$\dfrac{5}{12} \times \dfrac{1}{15} - \dfrac{5}{3} \times \dfrac{3}{8} \times \dfrac{1}{10}$$ Step 2: Multiply: $$\dfrac{5}{180} - \dfrac{15}{240}$$ Step 3: Simplify fractions: $$\dfrac{1}{36} - \dfrac{1}{16}$$ Step 4: Find common denominator 144: $$\dfrac{4}{144} - \dfrac{9}{144} = -\dfrac{5}{144}$$ **Answer h):** $-\dfrac{5}{144}$ **i)** Evaluate $$0.08 \times 1.2 \times 0.5 - 1.2$$ Step 1: Multiply: $$0.08 \times 1.2 = 0.096$$ Step 2: Multiply by 0.5: $$0.096 \times 0.5 = 0.048$$ Step 3: Subtract 1.2: $$0.048 - 1.2 = -1.152$$ **Answer i):** $-1.152$ **j)** Evaluate $$3.58 - \dfrac{14.5}{4.2}(3.7 - 5.8)$$ Step 1: Subtract inside parentheses: $$3.7 - 5.8 = -2.1$$ Step 2: Divide: $$\dfrac{14.5}{4.2} \approx 3.45238$$ Step 3: Multiply: $$3.45238 \times -2.1 = -7.25$$ Step 4: Subtract: $$3.58 - (-7.25) = 3.58 + 7.25 = 10.83$$ **Answer j):** $10.83$ **k)** Evaluate $$-4.8 + 12 \div \left[-\dfrac{50.5}{12.5}\right]$$ Step 1: Divide inside brackets: $$-\dfrac{50.5}{12.5} = -4.04$$ Step 2: Divide 12 by -4.04: $$12 \div -4.04 = -2.97$$ Step 3: Add: $$-4.8 + (-2.97) = -7.77$$ **Answer k):** $-7.77$ **l)** Evaluate $$0.8 \times (0.375 - 1.75) - 1.5$$ Step 1: Subtract inside parentheses: $$0.375 - 1.75 = -1.375$$ Step 2: Multiply: $$0.8 \times -1.375 = -1.1$$ Step 3: Subtract: $$-1.1 - 1.5 = -2.6$$ **Answer l):** $-2.6$ **m)** Evaluate $$(-3) - \left(-\dfrac{3}{4}\right)$$ Step 1: Subtracting a negative is addition: $$-3 + \dfrac{3}{4} = -\dfrac{12}{4} + \dfrac{3}{4} = -\dfrac{9}{4}$$ **Answer m):** $-\dfrac{9}{4}$ 2. Indicate mistakes in students' work. **a)** The steps and numbers are unclear and inconsistent; the student seems to confuse operations and values, leading to incorrect final answers. **b)** The student incorrectly changes subtraction to addition and misapplies multiplication steps, resulting in an invalid final equality. **c)** The student incorrectly combines terms and fractions, miscalculates products, and arrives at an incorrect final value. **Summary:** Each student made errors in arithmetic operations, fraction handling, or sign management.