a) \( -6 \frac{2}{5} \times 3 \div 2 \frac{2}{4} \) c) \( -4 \frac{1}{5} \times\left(-\frac{15}{6}\right) \div 3 \frac{2}{3} \) \( -2 \frac{3}{7}+1 \frac{8}{5}-\left(-1 \frac{3}{5}\right) \) (e) \( 3 \frac{2}{11} \times\left(-\frac{1}{12}+4 \frac{1}{4}\right) \) (i) \( 4 \frac{4}{5} \times 5 \frac{7}{9} \div 6 \frac{8}{9} \) (k) \( 5 \frac{3}{7} \div 2 \frac{10}{14} \)

Sure, let's evaluate each of the given expressions step by step. We'll convert mixed numbers to improper fractions, perform the necessary operations, and then convert the results back to mixed numbers where appropriate. --- ### **a) \( -6 \frac{2}{5} \times 3 \div 2 \frac{2}{4} \)** 1. **Convert Mixed Numbers to Improper Fractions:** - \( -6 \frac{2}{5} = -\left(\frac{6 \times 5 + 2}{5}\right) = -\frac{32}{5} \) - \( 2 \frac{2}{4} = 2 + \frac{2}{4} = 2 + \frac{1}{2} = \frac{5}{2} \) 2. **Rewrite the Expression:** \[ -\frac{32}{5} \times \frac{3}{1} \div \frac{5}{2} \] 3. **Perform Division by a Fraction (Multiply by Its Reciprocal):** \[ -\frac{32}{5} \times \frac{3}{1} \times \frac{2}{5} = -\frac{192}{25} \] 4. **Convert to Mixed Number:** \[ -\frac{192}{25} = -7 \frac{17}{25} \] **Final Answer:** \[ -7 \frac{17}{25} \] --- ### **c) \( -4 \frac{1}{5} \times \left(-\frac{15}{6}\right) \div 3 \frac{2}{3} \)** 1. **Convert Mixed Numbers to Improper Fractions:** - \( -4 \frac{1}{5} = -\left(\frac{4 \times 5 + 1}{5}\right) = -\frac{21}{5} \) - \( 3 \frac{2}{3} = \frac{11}{3} \) 2. **Rewrite the Expression:** \[ -\frac{21}{5} \times \left(-\frac{15}{6}\right) \div \frac{11}{3} \] 3. **Multiply the First Two Fractions:** \[ -\frac{21}{5} \times -\frac{15}{6} = \frac{315}{30} = \frac{21}{2} \] 4. **Divide by the Third Fraction (Multiply by Its Reciprocal):** \[ \frac{21}{2} \times \frac{3}{11} = \frac{63}{22} \] 5. **Convert to Mixed Number:** \[ \frac{63}{22} = 2 \frac{19}{22} \] **Final Answer:** \[ 2 \frac{19}{22} \] --- ### **\( -2 \frac{3}{7} + 1 \frac{8}{5} - \left(-1 \frac{3}{5}\right) \)** 1. **Convert Mixed Numbers to Improper Fractions:** - \( -2 \frac{3}{7} = -\left(\frac{2 \times 7 + 3}{7}\right) = -\frac{17}{7} \) - \( 1 \frac{8}{5} = \frac{13}{5} \) - \( -\left(-1 \frac{3}{5}\right) = +1 \frac{3}{5} = \frac{8}{5} \) 2. **Rewrite the Expression:** \[ -\frac{17}{7} + \frac{13}{5} + \frac{8}{5} \] 3. **Combine Like Terms:** \[ -\frac{17}{7} + \left(\frac{13}{5} + \frac{8}{5}\right) = -\frac{17}{7} + \frac{21}{5} \] 4. **Find a Common Denominator (35):** \[ -\frac{85}{35} + \frac{147}{35} = \frac{62}{35} \] 5. **Convert to Mixed Number:** \[ \frac{62}{35} = 1 \frac{27}{35} \] **Final Answer:** \[ 1 \frac{27}{35} \] --- ### **e) \( 3 \frac{2}{11} \times \left(-\frac{1}{12} + 4 \frac{1}{4}\right) \)** 1. **Simplify Inside the Parentheses:** - \( 4 \frac{1}{4} = \frac{17}{4} \) - \( -\frac{1}{12} + \frac{17}{4} = -\frac{1}{12} + \frac{51}{12} = \frac{50}{12} = \frac{25}{6} \) 2. **Convert Mixed Number to Improper Fraction:** - \( 3 \frac{2}{11} = \frac{35}{11} \) 3. **Multiply the Fractions:** \[ \frac{35}{11} \times \frac{25}{6} = \frac{875}{66} \] 4. **Convert to Mixed Number:** \[ \frac{875}{66} = 13 \frac{17}{66} \] **Final Answer:** \[ 13 \frac{17}{66} \] --- ### **i) \( 4 \frac{4}{5} \times 5 \frac{7}{9} \div 6 \frac{8}{9} \)** 1. **Convert Mixed Numbers to Improper Fractions:** - \( 4 \frac{4}{5} = \frac{24}{5} \) - \( 5 \frac{7}{9} = \frac{52}{9} \) - \( 6 \frac{8}{9} = \frac{62}{9} \) 2. **Rewrite the Expression:** \[ \frac{24}{5} \times \frac{52}{9} \div \frac{62}{9} \] 3. **Convert Division to Multiplication by Reciprocal:** \[ \frac{24}{5} \times \frac{52}{9} \times \frac{9}{62} \] 4. **Simplify by Canceling Common Factors:** - The 9 in the numerator and denominator cancels out. - \( \frac{24}{5} \times \frac{52}{62} \) 5. **Simplify \( \frac{52}{62} \) by Dividing Numerator and Denominator by 2:** \[ \frac{26}{31} \] 6. **Multiply the Fractions:** \[ \frac{24}{5} \times \frac{26}{31} = \frac{624}{155} \] 7. **Convert to Mixed Number:** \[ \frac{624}{155} = 4 \frac{4}{155} \] **Final Answer:** \[ 4 \frac{4}{155} \] --- ### **k) \( 5 \frac{3}{7} \div 2 \frac{10}{14} \)** 1. **Simplify the Divisor:** - \( 2 \frac{10}{14} = 2 \frac{5}{7} \) (since \( \frac{10}{14} = \frac{5}{7} \)) 2. **Convert Mixed Numbers to Improper Fractions:** - \( 5 \frac{3}{7} = \frac{38}{7} \) - \( 2 \frac{5}{7} = \frac{19}{7} \) 3. **Perform the Division:** \[ \frac{38}{7} \div \frac{19}{7} = \frac{38}{7} \times \frac{7}{19} = \frac{38 \times 7}{7 \times 19} = \frac{38}{19} = 2 \] **Final Answer:** \[ 2 \] --- Let me know if you need further clarification on any of these problems!