1. **Problem statement:** Find equivalent fractions and simplify fractions using the properties given: Property 1: $-\frac{a}{b} = \frac{-a}{b}$ for $b > 0$. Property 2: Equivalent fractions can be found by multiplying numerator and denominator by the same nonzero number. 2. **Find equivalent fractions for each given fraction:** (a) $\frac{21}{13}$ - Multiply numerator and denominator by $-1$ to get an equivalent fraction with a negative numerator: $$\frac{21}{13} = \frac{21 \times (-1)}{13 \times (-1)} = \frac{-21}{-13}$$ (b) $\frac{12}{-25}$ - Use property 1 to write with positive denominator: $$\frac{12}{-25} = \frac{-12}{25}$$ (c) $\frac{18}{-48}$ - Use property 1: $$\frac{18}{-48} = \frac{-18}{48}$$ (d) $\frac{-42}{-24}$ - Both numerator and denominator negative, multiply numerator and denominator by $-1$: $$\frac{-42}{-24} = \frac{-42 \times (-1)}{-24 \times (-1)} = \frac{42}{24}$$ 3. **Simplify the fractions:** (a) $\frac{12}{-24}$ - Use property 1: $$\frac{12}{-24} = \frac{-12}{24}$$ - Simplify numerator and denominator by dividing both by 12: $$\frac{\cancel{-12}^{-1}}{\cancel{24}^2} = \frac{-1}{2}$$ (b) $\frac{-39}{75}$ - Simplify numerator and denominator by dividing both by 3: $$\frac{\cancel{-39}^{-13}}{\cancel{75}^{25}} = \frac{-13}{25}$$ (c) $\frac{132}{-264}$ - Use property 1: $$\frac{132}{-264} = \frac{-132}{264}$$ - Simplify numerator and denominator by dividing both by 132: $$\frac{\cancel{-132}^{-1}}{\cancel{264}^2} = \frac{-1}{2}$$ 4. **Write fractions with positive denominators:** (a) $\frac{1}{-2} = \frac{-1}{2}$ (b) $\frac{-3}{-5} = \frac{3}{5}$ (c) $\frac{2}{-7} = \frac{-2}{7}$ 5. **Express minutes as fractions of an hour (60 minutes):** (a) 15 minutes: $$\frac{15}{60} = \frac{\cancel{15}^1}{\cancel{60}^4} = \frac{1}{4}$$ (b) 20 minutes: $$\frac{20}{60} = \frac{\cancel{20}^1}{\cancel{60}^3} = \frac{1}{3}$$ (c) 45 minutes: $$\frac{45}{60} = \frac{\cancel{45}^{3}}{\cancel{60}^{4}} = \frac{3}{4}$$ (d) 50 minutes: $$\frac{50}{60} = \frac{\cancel{50}^{5}}{\cancel{60}^{6}} = \frac{5}{6}$$ 6. **Express weights as fractions of tạ (100 kg) and tấn (1000 kg):** (a) 20 kg: - As fraction of tạ: $$\frac{20}{100} = \frac{1}{5}$$ - As fraction of tấn: $$\frac{20}{1000} = \frac{1}{50}$$ (b) 55 kg: - As fraction of tạ: $$\frac{55}{100} = \frac{11}{20}$$ - As fraction of tấn: $$\frac{55}{1000} = \frac{11}{200}$$ (c) 87 kg: - As fraction of tạ: $$\frac{87}{100}$$ (cannot simplify further) - As fraction of tấn: $$\frac{87}{1000}$$ (cannot simplify further) (d) 91 kg: - As fraction of tạ: $$\frac{91}{100}$$ (cannot simplify further) - As fraction of tấn: $$\frac{91}{1000}$$ (cannot simplify further) 7. **Express shaded parts as fractions with positive denominators:** (a) Circle with 8 sectors, 3 shaded: $$\frac{3}{8}$$ (b) Rectangle with 12 circles, 9 shaded: $$\frac{9}{12} = \frac{3}{4}$$ (c) 6x6 grid (36 squares), 20 shaded: $$\frac{20}{36} = \frac{5}{9}$$ (d) 6x6 grid, 13 shaded: $$\frac{13}{36}$$ (cannot simplify further) **Final answers:** 1a) $\frac{-21}{-13}$ 1b) $\frac{-12}{25}$ 1c) $\frac{-18}{48}$ 1d) $\frac{42}{24}$ 2a) $\frac{-1}{2}$ 2b) $\frac{-13}{25}$ 2c) $\frac{-1}{2}$ 3a) $\frac{-1}{2}$ 3b) $\frac{3}{5}$ 3c) $\frac{-2}{7}$ 4a) $\frac{1}{4}$ 4b) $\frac{1}{3}$ 4c) $\frac{3}{4}$ 4d) $\frac{5}{6}$ 5a) $\frac{1}{5}$ tạ, $\frac{1}{50}$ tấn 5b) $\frac{11}{20}$ tạ, $\frac{11}{200}$ tấn 5c) $\frac{87}{100}$ tạ, $\frac{87}{1000}$ tấn 5d) $\frac{91}{100}$ tạ, $\frac{91}{1000}$ tấn 6a) $\frac{3}{8}$ 6b) $\frac{3}{4}$ 6c) $\frac{5}{9}$ 6d) $\frac{13}{36}$