1. **Problem Statement:** Evaluate the given addition and subtraction problems involving mixed numbers and fractions. 2. **Formula and Rules:** - Convert mixed numbers to improper fractions. - Use the rule: $a + (-b) = a - b$. - Find common denominators to add or subtract fractions. - Simplify the result. 3. **Step-by-step Solutions:** **Q1 d.** $-3 \frac{1}{6} + (-4 \frac{2}{3})$ - Convert to improper fractions: $-\frac{19}{6} + (-\frac{14}{3})$ - Find common denominator 6: $-\frac{19}{6} + (-\frac{28}{6})$ - Add: $-\frac{19}{6} - \frac{28}{6} = -\frac{47}{6} = -7 \frac{5}{6}$ **Q1 e.** $-\frac{8}{5} - (-2 \frac{1}{2}) - \frac{1}{2}$ - Convert mixed number: $-\frac{8}{5} - (-\frac{5}{2}) - \frac{1}{2}$ - Simplify signs: $-\frac{8}{5} + \frac{5}{2} - \frac{1}{2}$ - Combine $\frac{5}{2} - \frac{1}{2} = \frac{4}{2} = 2$ - Now: $-\frac{8}{5} + 2$ - Convert 2 to $\frac{10}{5}$: $-\frac{8}{5} + \frac{10}{5} = \frac{2}{5}$ **Q2 a.** $3 \frac{4}{7} + 1 \frac{2}{5} - (-\frac{3}{7})$ - Convert: $\frac{25}{7} + \frac{7}{5} + \frac{3}{7}$ - Combine fractions with denominator 7: $\frac{25}{7} + \frac{3}{7} = \frac{28}{7} = 4$ - Now: $4 + \frac{7}{5} = \frac{20}{5} + \frac{7}{5} = \frac{27}{5} = 5 \frac{2}{5}$ **Q2 b.** $\frac{2}{3} - (-3 \frac{3}{20}) + (-\frac{2}{5})$ - Convert: $\frac{2}{3} + \frac{63}{20} - \frac{2}{5}$ - Find common denominator 60: $\frac{40}{60} + \frac{189}{60} - \frac{24}{60} = \frac{205}{60} = 3 \frac{25}{60} = 3 \frac{5}{12}$ **Q2 c.** $-6 \frac{4}{9} - 3 \frac{3}{4} - 3 \frac{5}{9}$ - Convert: $-\frac{58}{9} - \frac{15}{4} - \frac{32}{9}$ - Combine fractions with denominator 9: $-\frac{58}{9} - \frac{32}{9} = -\frac{90}{9} = -10$ - Now: $-10 - \frac{15}{4} = -\frac{40}{4} - \frac{15}{4} = -\frac{55}{4} = -13 \frac{3}{4}$ **Q2 d.** $(-\frac{1}{2} + \frac{1}{3}) + (\frac{1}{4} + (-\frac{1}{3})) + (-\frac{1}{20})$ - Calculate inside parentheses: $-\frac{1}{2} + \frac{1}{3} = -\frac{3}{6} + \frac{2}{6} = -\frac{1}{6}$ $\frac{1}{4} - \frac{1}{3} = \frac{3}{12} - \frac{4}{12} = -\frac{1}{12}$ - Sum all: $-\frac{1}{6} + (-\frac{1}{12}) + (-\frac{1}{20})$ - Common denominator 60: $-\frac{10}{60} - \frac{5}{60} - \frac{3}{60} = -\frac{18}{60} = -\frac{3}{10}$ **Q3 a.** $\frac{7}{8} - (-2 \frac{3}{4}) + (-\frac{1}{3})$ - Convert: $\frac{7}{8} + \frac{11}{4} - \frac{1}{3}$ - Convert $\frac{11}{4}$ to $\frac{22}{8}$ - Sum: $\frac{7}{8} + \frac{22}{8} = \frac{29}{8}$ - Now subtract $\frac{1}{3}$ - Common denominator 24: $\frac{87}{24} - \frac{8}{24} = \frac{79}{24} = 3 \frac{7}{24}$ **Q3 b.** $\{-\frac{1}{2} + \frac{1}{3} + \frac{1}{6}\} + (-\frac{2}{5})$ - Inside braces: $-\frac{1}{2} + \frac{1}{3} + \frac{1}{6} = -\frac{3}{6} + \frac{2}{6} + \frac{1}{6} = 0$ - Now add $-\frac{2}{5} = -\frac{2}{5}$ **Q3 c.** $2 \frac{3}{4} - 3 \frac{11}{12} - \frac{1}{5} - \frac{2}{7}$ - Convert: $\frac{11}{4} - \frac{47}{12} - \frac{1}{5} - \frac{2}{7}$ - Common denominator 420: $\frac{1155}{420} - \frac{1645}{420} - \frac{84}{420} - \frac{120}{420} = \frac{1155 - 1645 - 84 - 120}{420} = \frac{-694}{420} = -1 \frac{274}{420} = -1 \frac{137}{210}$ **Q3 d.** $-\frac{3}{4} + (-\frac{1}{2}) = -\frac{3}{4} - \frac{1}{2} = -\frac{3}{4} - \frac{2}{4} = -\frac{5}{4} = -1 \frac{1}{4}$ **Q3 e.** $-1 - (-\frac{5}{6}) + (-\frac{6}{5}) = -1 + \frac{5}{6} - \frac{6}{5}$ - Common denominator 30: $-\frac{30}{30} + \frac{25}{30} - \frac{36}{30} = -\frac{41}{30} = -1 \frac{11}{30}$ **Q3 f.** $(-\frac{7}{11})^2 - \frac{7}{11} = \frac{49}{121} - \frac{7}{11}$ - Convert $\frac{7}{11}$ to $\frac{77}{121}$ - Subtract: $\frac{49}{121} - \frac{77}{121} = -\frac{28}{121}$ **Q3 g.** $\frac{2}{3} - (\frac{1}{4} + \frac{1}{8})^2$ - Sum inside parentheses: $\frac{1}{4} + \frac{1}{8} = \frac{2}{8} + \frac{1}{8} = \frac{3}{8}$ - Square: $\left(\frac{3}{8}\right)^2 = \frac{9}{64}$ - Subtract: $\frac{2}{3} - \frac{9}{64}$ - Common denominator 192: $\frac{128}{192} - \frac{27}{192} = \frac{101}{192}$ **Q1 j.** $-\frac{1}{2} + (-\frac{3}{4}) = -\frac{1}{2} - \frac{3}{4} = -\frac{2}{4} - \frac{3}{4} = -\frac{5}{4} = -1 \frac{1}{4}$ **Q1 k.** $3 \frac{1}{8} + (-\frac{1}{4}) = \frac{25}{8} - \frac{2}{8} = \frac{23}{8} = 2 \frac{7}{8}$ **Q1 l.** $5 \frac{1}{5} - 4 \frac{1}{2} = \frac{26}{5} - \frac{9}{2}$ - Common denominator 10: $\frac{52}{10} - \frac{45}{10} = \frac{7}{10}$ **Q1 o.** $6 \frac{1}{5} - (-\frac{3}{4}) + (-4 \frac{1}{10})$ - Convert: $\frac{31}{5} + \frac{3}{4} - \frac{41}{10}$ - Common denominator 20: $\frac{124}{20} + \frac{15}{20} - \frac{82}{20} = \frac{57}{20} = 2 \frac{17}{20}$ **Q1 p.** $4 \frac{2}{7} + (-6 \frac{1}{3}) - (-\frac{4}{21})$ - Convert: $\frac{30}{7} - \frac{19}{3} + \frac{4}{21}$ - Common denominator 21: $\frac{90}{21} - \frac{133}{21} + \frac{4}{21} = -\frac{39}{21} = -1 \frac{18}{21} = -1 \frac{6}{7}$ **Q1 q.** $-4 + (-3 \frac{1}{8}) + (-\frac{4}{3})$ - Convert: $-4 - \frac{25}{8} - \frac{4}{3}$ - Common denominator 24: $-\frac{96}{24} - \frac{75}{24} - \frac{32}{24} = -\frac{203}{24} = -8 \frac{11}{24}$ 4. **Final answers:** - Q1 d: $-7 \frac{5}{6}$ - Q1 e: $\frac{2}{5}$ - Q2 a: $5 \frac{2}{5}$ - Q2 b: $3 \frac{5}{12}$ - Q2 c: $-13 \frac{3}{4}$ - Q2 d: $-\frac{3}{10}$ - Q3 a: $3 \frac{7}{24}$ - Q3 b: $-\frac{2}{5}$ - Q3 c: $-1 \frac{137}{210}$ - Q3 d: $-1 \frac{1}{4}$ - Q3 e: $-1 \frac{11}{30}$ - Q3 f: $-\frac{28}{121}$ - Q3 g: $\frac{101}{192}$ - Q1 j: $-1 \frac{1}{4}$ - Q1 k: $2 \frac{7}{8}$ - Q1 l: $\frac{7}{10}$ - Q1 o: $2 \frac{17}{20}$ - Q1 p: $-1 \frac{6}{7}$ - Q1 q: $-8 \frac{11}{24}$ These steps show conversion, common denominators, addition/subtraction, and simplification clearly for learning.