1. Reduce each numerical fraction to lowest terms: (a) $\frac{13}{26} = \frac{13 \div 13}{26 \div 13} = \frac{1}{2}$ (b) $\frac{9}{12} = \frac{9 \div 3}{12 \div 3} = \frac{3}{4}$ (c) $\frac{18}{30} = \frac{18 \div 6}{30 \div 6} = \frac{3}{5}$ (d) $\frac{24}{72} = \frac{24 \div 24}{72 \div 24} = \frac{1}{3}$ (e) $\frac{36}{27} = \frac{36 \div 9}{27 \div 9} = \frac{4}{3}$ 2. Smartphone ownership fractions: (a) 2011: $\frac{35}{100} = \frac{7}{20}$ after dividing numerator and denominator by 5. 2013: $\frac{56}{100} = \frac{14}{25}$ after dividing numerator and denominator by 4. (b) Increase factor = $\frac{\frac{14}{25}}{\frac{7}{20}} = \frac{14}{25} \times \frac{20}{7} = \frac{14 \times 20}{25 \times 7} = \frac{280}{175} = \frac{16}{10} = \frac{8}{5} = 1 \frac{3}{5}$ 3. Reduce algebraic fractions: (a) $\frac{6x}{9} = \frac{6 \div 3 \times x}{9 \div 3} = \frac{2x}{3}$ (b) $\frac{x}{2x^2} = \frac{x}{2 \times x \times x} = \frac{1}{2x}$ (cancel one $x$) (c) $\frac{b}{abc} = \frac{b}{a \times b \times c} = \frac{1}{ac}$ (cancel $b$) (d) $\frac{4x}{6x^2 y} = \frac{4 \times x}{6 \times x \times x \times y} = \frac{2}{3xy}$ (cancel $2$ and one $x$) (e) $\frac{15a^2 b}{20ab^2} = \frac{15 \times a \times a \times b}{20 \times a \times b \times b} = \frac{3a}{4b}$ (cancel 5, $a$, and $b$) 4. Factorise and reduce: (a) $\frac{2p}{4q + 6r} = \frac{2p}{2(2q + 3r)} = \frac{p}{2q + 3r}$ (b) $\frac{x}{x^2 - 4x} = \frac{x}{x(x - 4)} = \frac{1}{x - 4}$ (cancel $x$) (c) $\frac{3ab}{6a^2 + 3a} = \frac{3ab}{3a(2a + 1)} = \frac{b}{2a + 1}$ (cancel $3a$) (d) $\frac{14d}{21d - 7de} = \frac{14d}{7d(3 - e)} = \frac{2}{3 - e}$ (cancel $7d$) (e) $\frac{x + 2}{x^2 - 4} = \frac{x + 2}{(x + 2)(x - 2)} = \frac{1}{x - 2}$ (cancel $x + 2$) 5. Simplify and explain: - $\frac{x - 1}{2x - 2} = \frac{x - 1}{2(x - 1)} = \frac{1}{2}$ (simplifiable by canceling $x - 1$) - $\frac{x - 2}{x + 2}$ cannot be simplified because numerator and denominator share no common factors. - $\frac{5t}{10t - s}$ cannot be simplified because numerator and denominator share no common factors. Final answers are shown in each step.