1. **State the problem:** Simplify the expression $$\frac{5}{9} \times \frac{6}{4} - \frac{4}{9} \div \frac{10}{3}$$. 2. **Recall the rules:** - Multiplication and division of fractions: multiply numerators and denominators. - Division by a fraction is the same as multiplying by its reciprocal. - Perform multiplication and division before subtraction. 3. **Calculate the multiplication part:** $$\frac{5}{9} \times \frac{6}{4} = \frac{5 \times 6}{9 \times 4} = \frac{30}{36}$$ Simplify by dividing numerator and denominator by 6: $$\frac{\cancel{30}^{5}}{\cancel{36}^{6}} = \frac{5}{6}$$ 4. **Calculate the division part:** $$\frac{4}{9} \div \frac{10}{3} = \frac{4}{9} \times \frac{3}{10} = \frac{4 \times 3}{9 \times 10} = \frac{12}{90}$$ Simplify by dividing numerator and denominator by 6: $$\frac{\cancel{12}^{2}}{\cancel{90}^{15}} = \frac{2}{15}$$ 5. **Subtract the two results:** $$\frac{5}{6} - \frac{2}{15}$$ Find common denominator, which is 30: $$\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}$$ $$\frac{2}{15} = \frac{2 \times 2}{15 \times 2} = \frac{4}{30}$$ 6. **Perform subtraction:** $$\frac{25}{30} - \frac{4}{30} = \frac{25 - 4}{30} = \frac{21}{30}$$ Simplify by dividing numerator and denominator by 3: $$\frac{\cancel{21}^{7}}{\cancel{30}^{10}} = \frac{7}{10}$$ **Final answer:** $$\frac{7}{10}$$