1) Simplify each expression: 1.a) Simplify $3 \times \left(\frac{1}{2} - \frac{1}{3}\right)$ Step 1: Find common denominator inside parentheses: $\frac{1}{2} = \frac{3}{6}$, $\frac{1}{3} = \frac{2}{6}$ Step 2: Subtract: $\frac{3}{6} - \frac{2}{6} = \frac{1}{6}$ Step 3: Multiply by 3: $3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ Answer: $\frac{1}{2}$ 1.b) Simplify $\left(\frac{1}{2} \times \frac{1}{3}\right) \times \frac{1}{5}$ Step 1: Multiply first two fractions: $\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$ Step 2: Multiply by $\frac{1}{5}$: $\frac{1}{6} \times \frac{1}{5} = \frac{1}{30}$ Answer: $\frac{1}{30}$ 1.c) Simplify $\frac{\frac{3}{4} - \frac{1}{3}}{\frac{3}{4} \times \frac{1}{3}}$ Step 1: Find numerator: $\frac{3}{4} = \frac{9}{12}$, $\frac{1}{3} = \frac{4}{12}$, so $\frac{9}{12} - \frac{4}{12} = \frac{5}{12}$ Step 2: Find denominator: $\frac{3}{4} \times \frac{1}{3} = \frac{3}{12} = \frac{1}{4}$ Step 3: Divide numerator by denominator: $\frac{5}{12} \div \frac{1}{4} = \frac{5}{12} \times 4 = \frac{20}{12} = \frac{5}{3}$ Answer: $\frac{5}{3}$ 1.d) Simplify $\frac{2 \frac{1}{4} + 1 \frac{1}{3}}{\frac{1}{4} - \frac{1}{6}}$ Step 1: Convert mixed numbers to improper fractions: $2 \frac{1}{4} = \frac{9}{4}$ $1 \frac{1}{3} = \frac{4}{3}$ Step 2: Add numerator: $\frac{9}{4} + \frac{4}{3} = \frac{27}{12} + \frac{16}{12} = \frac{43}{12}$ Step 3: Subtract denominator: $\frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12}$ Step 4: Divide numerator by denominator: $\frac{43}{12} \div \frac{1}{12} = \frac{43}{12} \times 12 = 43$ Answer: $43$ 1.e) Simplify $3 \frac{3}{8} - 5 \frac{1}{2} + 4 \frac{1}{4}$ Step 1: Convert mixed numbers: $3 \frac{3}{8} = \frac{27}{8}$ $5 \frac{1}{2} = \frac{11}{2} = \frac{44}{8}$ $4 \frac{1}{4} = \frac{17}{4} = \frac{34}{8}$ Step 2: Calculate: $\frac{27}{8} - \frac{44}{8} + \frac{34}{8} = \frac{27 - 44 + 34}{8} = \frac{17}{8} = 2 \frac{1}{8}$ Answer: $2 \frac{1}{8}$ 1.f) Simplify $1 \frac{3}{10} \div \left(\frac{1}{5} + \frac{2}{3}\right)$ Step 1: Convert mixed number: $1 \frac{3}{10} = \frac{13}{10}$ Step 2: Add inside parentheses: $\frac{1}{5} + \frac{2}{3} = \frac{3}{15} + \frac{10}{15} = \frac{13}{15}$ Step 3: Divide: $\frac{13}{10} \div \frac{13}{15} = \frac{13}{10} \times \frac{15}{13} = \frac{15}{10} = \frac{3}{2} = 1 \frac{1}{2}$ Answer: $1 \frac{1}{2}$ 1.g) Simplify $6 - \left[1 \frac{1}{2} + 2 \frac{1}{2}\right]$ Step 1: Convert mixed numbers: $1 \frac{1}{2} = \frac{3}{2}$ $2 \frac{1}{2} = \frac{5}{2}$ Step 2: Add inside brackets: $\frac{3}{2} + \frac{5}{2} = \frac{8}{2} = 4$ Step 3: Subtract: $6 - 4 = 2$ Answer: $2$ 1.h) Simplify $3 \frac{1}{4} \times \left(5 \frac{1}{2} + 2 \frac{2}{3}\right)$ Step 1: Convert mixed numbers: $3 \frac{1}{4} = \frac{13}{4}$ $5 \frac{1}{2} = \frac{11}{2}$ $2 \frac{2}{3} = \frac{8}{3}$ Step 2: Add inside parentheses: $\frac{11}{2} + \frac{8}{3} = \frac{33}{6} + \frac{16}{6} = \frac{49}{6}$ Step 3: Multiply: $\frac{13}{4} \times \frac{49}{6} = \frac{637}{24} = 26 \frac{13}{24}$ Answer: $26 \frac{13}{24}$ 2) Connect fractions using "<": $\frac{5}{13} < \frac{4}{7} < \frac{2}{3}$ Explanation: Convert to decimals or cross multiply: $\frac{5}{13} \approx 0.3846$, $\frac{4}{7} \approx 0.5714$, $\frac{2}{3} \approx 0.6667$ 3) Naveen's library problem: i) Fraction donated: $\frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12}$ ii) If donated 150 books = $\frac{5}{12}$ of total, total books = $\frac{150 \times 12}{5} = 360$ iii) Fraction given to neighbour: $\frac{60}{360} = \frac{1}{6}$ Final answers: 1.a) $\frac{1}{2}$ 1.b) $\frac{1}{30}$ 1.c) $\frac{5}{3}$ 1.d) $43$ 1.e) $2 \frac{1}{8}$ 1.f) $1 \frac{1}{2}$ 1.g) $2$ 1.h) $26 \frac{13}{24}$ 2) $\frac{5}{13} < \frac{4}{7} < \frac{2}{3}$ 3.i) $\frac{5}{12}$ 3.ii) 360 books 3.iii) $\frac{1}{6}$