1. Calculate the following sums and differences of rational numbers. 1.a) $\frac{3}{10} + \frac{1}{5}$ - Convert $\frac{1}{5}$ to $\frac{2}{10}$ to have a common denominator. - Add: $\frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2}$. 1.b) $2 \frac{1}{3} + (-1 \frac{1}{4})$ - Convert mixed numbers to improper fractions: $2 \frac{1}{3} = \frac{7}{3}$, $-1 \frac{1}{4} = -\frac{5}{4}$. - Find common denominator 12: $\frac{7}{3} = \frac{28}{12}$, $-\frac{5}{4} = -\frac{15}{12}$. - Add: $\frac{28}{12} - \frac{15}{12} = \frac{13}{12} = 1 \frac{1}{12}$. 1.c) $\frac{5}{12} - \frac{5}{12} = 0$. 1.d) $-2 \frac{1}{2} - (-3 \frac{1}{3})$ - Convert: $-2 \frac{1}{2} = -\frac{5}{2}$, $-3 \frac{1}{3} = -\frac{10}{3}$. - Subtracting a negative is addition: $-\frac{5}{2} + \frac{10}{3}$. - Common denominator 6: $-\frac{15}{6} + \frac{20}{6} = \frac{5}{6}$. 1.e) $-\frac{5}{6} + \frac{1}{3}$ - Convert $\frac{1}{3} = \frac{2}{6}$. - Add: $-\frac{5}{6} + \frac{2}{6} = -\frac{3}{6} = -\frac{1}{2}$. 1.f) $\frac{3}{8} - (-\frac{1}{4})$ - Subtracting a negative is addition: $\frac{3}{8} + \frac{1}{4}$. - Convert $\frac{1}{4} = \frac{2}{8}$. - Add: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$. 2. Calculate the following sums and differences. 2.a) $\frac{2}{3} - \frac{3}{4}$ - Common denominator 12: $\frac{8}{12} - \frac{9}{12} = -\frac{1}{12}$. 2.b) $-\frac{2}{9} + (-\frac{1}{3})$ - Convert $-\frac{1}{3} = -\frac{3}{9}$. - Add: $-\frac{2}{9} - \frac{3}{9} = -\frac{5}{9}$. 2.c) $\frac{1}{4} + (-\frac{3}{5})$ - Common denominator 20: $\frac{5}{20} - \frac{12}{20} = -\frac{7}{20}$. 2.d) $-\frac{3}{4} - (-\frac{5}{8})$ - Subtracting a negative is addition: $-\frac{3}{4} + \frac{5}{8}$. - Convert $-\frac{3}{4} = -\frac{6}{8}$. - Add: $-\frac{6}{8} + \frac{5}{8} = -\frac{1}{8}$. 2.e) $1 \frac{1}{2} - 2 \frac{1}{4}$ - Convert: $1 \frac{1}{2} = \frac{3}{2}$, $2 \frac{1}{4} = \frac{9}{4}$. - Common denominator 4: $\frac{6}{4} - \frac{9}{4} = -\frac{3}{4}$. 2.f) $1 \frac{2}{5} + (-1 \frac{3}{4})$ - Convert: $1 \frac{2}{5} = \frac{7}{5}$, $-1 \frac{3}{4} = -\frac{7}{4}$. - Common denominator 20: $\frac{28}{20} - \frac{35}{20} = -\frac{7}{20}$. 3. Calculate the following products and quotients. 3.a) $\frac{4}{5} \div \frac{5}{6}$ - Division of fractions: multiply by reciprocal. - $\frac{4}{5} \times \frac{6}{5} = \frac{24}{25}$. 3.b) $3 \frac{1}{3} \times 1 \frac{3}{4}$ - Convert: $3 \frac{1}{3} = \frac{10}{3}$, $1 \frac{3}{4} = \frac{7}{4}$. - Multiply: $\frac{10}{3} \times \frac{7}{4} = \frac{70}{12} = 5 \frac{10}{12} = 5 \frac{5}{6}$. 3.c) $\frac{1}{8} \times (-\frac{2}{5}) = -\frac{2}{40} = -\frac{1}{20}$. 3.d) $-\frac{9}{10} \div (-\frac{4}{5})$ - Division: multiply by reciprocal. - $-\frac{9}{10} \times -\frac{5}{4} = \frac{45}{40} = 1 \frac{1}{8}$. 3.e) $-\frac{3}{8} \times 5 \frac{1}{3}$ - Convert $5 \frac{1}{3} = \frac{16}{3}$. - Multiply: $-\frac{3}{8} \times \frac{16}{3} = -\frac{48}{24} = -2$. 3.f) $\frac{1}{10} \div (-\frac{3}{8})$ - Multiply by reciprocal: $\frac{1}{10} \times -\frac{8}{3} = -\frac{8}{30} = -\frac{4}{15}$. 4. Calculate the following products and quotients. 4.a) $-\frac{3}{4} \times -\frac{1}{9} = \frac{3}{36} = \frac{1}{12}$. 4.b) $1 \frac{1}{3} \div 1 \frac{1}{4}$ - Convert: $1 \frac{1}{3} = \frac{4}{3}$, $1 \frac{1}{4} = \frac{5}{4}$. - Divide: $\frac{4}{3} \times \frac{4}{5} = \frac{16}{15} = 1 \frac{1}{15}$. 4.c) $-\frac{3}{8} \div \frac{7}{10}$ - Multiply by reciprocal: $-\frac{3}{8} \times \frac{10}{7} = -\frac{30}{56} = -\frac{15}{28}$. 4.d) $-2 \frac{1}{8} \div 1 \frac{1}{4}$ - Convert: $-2 \frac{1}{8} = -\frac{17}{8}$, $1 \frac{1}{4} = \frac{5}{4}$. - Divide: $-\frac{17}{8} \times \frac{4}{5} = -\frac{68}{40} = -\frac{17}{10} = -1 \frac{7}{10}$. 4.e) $\frac{7}{9} \times -\frac{6}{11} = -\frac{42}{99} = -\frac{14}{33}$. 4.f) $-1 \frac{1}{2} \div -2 \frac{1}{2}$ - Convert: $-1 \frac{1}{2} = -\frac{3}{2}$, $-2 \frac{1}{2} = -\frac{5}{2}$. - Divide: $-\frac{3}{2} \times -\frac{2}{5} = \frac{6}{10} = \frac{3}{5}$. Final answers summarized: 1.a) $\frac{1}{2}$ 1.b) $1 \frac{1}{12}$ 1.c) $0$ 1.d) $\frac{5}{6}$ 1.e) $-\frac{1}{2}$ 1.f) $\frac{5}{8}$ 2.a) $-\frac{1}{12}$ 2.b) $-\frac{5}{9}$ 2.c) $-\frac{7}{20}$ 2.d) $-\frac{1}{8}$ 2.e) $-\frac{3}{4}$ 2.f) $-\frac{7}{20}$ 3.a) $\frac{24}{25}$ 3.b) $5 \frac{5}{6}$ 3.c) $-\frac{1}{20}$ 3.d) $1 \frac{1}{8}$ 3.e) $-2$ 3.f) $-\frac{4}{15}$ 4.a) $\frac{1}{12}$ 4.b) $1 \frac{1}{15}$ 4.c) $-\frac{15}{28}$ 4.d) $-1 \frac{7}{10}$ 4.e) $-\frac{14}{33}$ 4.f) $\frac{3}{5}$