1. **Rational numbers equalities and simplifications:** - Given equalities like $\frac{-8}{14} = \frac{-4}{3.5} = \frac{16}{-2.1}$,... we check if the fractions simplify to the same value. - For example, simplify $\frac{-8}{14} = \frac{-4}{3.5}$. Note $3.5 = \frac{7}{2}$, so $\frac{-4}{3.5} = \frac{-4}{7/2} = -4 \times \frac{2}{7} = \frac{-8}{7}$ which is not equal to $\frac{-8}{14} = \frac{-4}{7}$. 2. **Simplifying given fractions:** - $a = \frac{18}{27} = \frac{2}{3}$ after dividing numerator and denominator by 9. - $b = \frac{-49}{91} = \frac{-7}{13}$ dividing numerator and denominator by 7. - $c = \frac{196}{252} = \frac{14}{18} = \frac{7}{9}$. - $d = \frac{-999}{444}$. Dividing numerator and denominator by 3, $= \frac{-333}{148}$ (no further simplification). - $e = \frac{225}{-45} = \frac{-225}{45} = -5$. - $f = \frac{-5120}{-768} = \frac{5120}{768} = \frac{640}{96} = \frac{80}{12} = \frac{20}{3}$. 3. **Simplify $\frac{-12}{18}$:** - Dividing numerator and denominator by 6, $= \frac{-2}{3}$. 4. **Pairwise fractions:** - $\frac{7}{8}$ and $\frac{17}{5}$ are distinct fractions. - $\frac{-5}{4}$ and $\frac{-8}{25}$ distinct. - $\frac{11}{36}$ and $\frac{-13}{12}$ distinct. - Triplets such as $\frac{5}{6}, \frac{1}{10}, \frac{4}{15}$ are separate. 5. **Solving for a in equations:** - $\frac{a}{12} = \frac{2}{3} \implies a = 12 \times \frac{2}{3} = 8$. - $\frac{15}{a} = \frac{3}{5} \implies 15 \times 5 = 3a \implies a = \frac{75}{3} = 25$. - $\frac{a}{2} = \frac{21}{-6} = -\frac{7}{2} \implies a = 2 \times -\frac{7}{2} = -7$. - $\frac{5}{a} = -\frac{1}{4} \implies 5 \times 4 = -1 \times a \implies a = -20$. 6. **Solving equations:** - a) $\frac{3+x}{5-x} = \frac{1}{3}$ 1. Cross multiply: $3(3 + x) = 1(5 - x)$ 2. $9 + 3x = 5 - x$ 3. $3x + x = 5 - 9$ 4. $4x = -4 \implies x = -1$ - b) $\frac{1-2x}{x+2} = \frac{3}{4}$ 1. Cross multiply: $4(1 - 2x) = 3(x + 2)$ 2. $4 - 8x = 3x + 6$ 3. $-8x - 3x = 6 - 4$ 4. $-11x = 2 \implies x = -\frac{2}{11}$ 7. **Solving systems:** - a) $\frac{x}{5} = \frac{y}{8}$ and $x+2y=30$ 1. From first: $\frac{x}{5} = \frac{y}{8} \implies 8x = 5y \implies y = \frac{8}{5}x$ 2. Substitute into second: $x + 2 \cdot \frac{8}{5}x = 30$ 3. $x + \frac{16}{5}x = 30$ 4. $\frac{5}{5}x + \frac{16}{5}x = 30 \implies \frac{21}{5}x = 30$ 5. $x = \frac{30 \times 5}{21} = \frac{150}{21} = \frac{50}{7}$ 6. $y = \frac{8}{5} \times \frac{50}{7} = \frac{400}{35} = \frac{80}{7}$ - b) $\frac{x}{y}=\frac{2}{7}$ and $x + y = 9$ 1. $x = \frac{2}{7}y$ 2. Substitute: $\frac{2}{7}y + y = 9$ 3. $\frac{2}{7}y + \frac{7}{7}y = 9 \implies \frac{9}{7}y = 9$ 4. $y = 9 \times \frac{7}{9} = 7$ 5. $x = \frac{2}{7} \times 7 = 2$ - c) $\frac{x}{y} = -\frac{5}{3}$ and $x - y = 8$ 1. $x = -\frac{5}{3}y$ 2. Substitute: $-\frac{5}{3}y - y = 8$ 3. $-\frac{5}{3}y - \frac{3}{3}y = 8 \implies -\frac{8}{3}y = 8$ 4. $y = \frac{8 \times 3}{-8} = -3$ 5. $x = -\frac{5}{3} \times (-3) = 5$ 8. **Pairs fractions simplification check:** - $\frac{10}{11}$ and $\frac{31}{33} = \frac{31}{33}$ - $\frac{-3}{16}$ and $\frac{-7}{20}$ - $\frac{11}{13}$ and $\frac{110}{130} = \frac{11}{13}$ (simplify second by dividing numerator and denominator by 10) - $\frac{-78}{13}$ and $\frac{5}{3}$ 9. **ABCD is a square with $AB = 4$ cm:** - So all sides are length $4$ cm since a square has equal sides. **Final answers summarized:** - Simplifications: $a=\frac{2}{3}$; $b=\frac{-7}{13}$; $c=\frac{7}{9}$; $d=\frac{-333}{148}$; $e=-5$; $f=\frac{20}{3}$. - $\frac{-12}{18} = \frac{-2}{3}$. - Solved value of $a$ in different fractions given. - Solutions for $x$ in equations: a) $x=-1$, b) $x=-\frac{2}{11}$. - Solutions for systems: a) $x=\frac{50}{7}, y=\frac{80}{7}$; b) $x=2, y=7$; c) $x=5, y=-3$. - Square side length is $4$ cm.