Mixed Fraction Division 0A03A8

1. **State the problem:** Calculate $2 \frac{4}{9} \div 4 \frac{2}{3} - \frac{3}{7}$. 2. **Convert mixed fractions to improper fractions:** $$2 \frac{4}{9} = \frac{2 \times 9 + 4}{9} = \frac{18 + 4}{9} = \frac{22}{9}$$ $$4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}$$ 3. **Rewrite the expression:** $$\frac{22}{9} \div \frac{14}{3} - \frac{3}{7}$$ 4. **Division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{22}{9} \div \frac{14}{3} = \frac{22}{9} \times \frac{3}{14}$$ 5. **Multiply the fractions:** $$\frac{22}{9} \times \frac{3}{14} = \frac{22 \times 3}{9 \times 14} = \frac{66}{126}$$ 6. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{66}^{22 \times 3}}{\cancel{126}^{42 \times 3}} = \frac{22}{42}$$ Further simplify by dividing numerator and denominator by 2: $$\frac{\cancel{22}^{11 \times 2}}{\cancel{42}^{21 \times 2}} = \frac{11}{21}$$ 7. **Now subtract $\frac{3}{7}$:** Find common denominator for $\frac{11}{21}$ and $\frac{3}{7}$, which is 21. $$\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}$$ 8. **Perform the subtraction:** $$\frac{11}{21} - \frac{9}{21} = \frac{11 - 9}{21} = \frac{2}{21}$$ **Final answer:** $$\boxed{\frac{2}{21}}$$