Fraction Operations C8C93A

1. **State the problem:** We are given two expressions to evaluate and compare: $$\frac{5}{6} \times \frac{2}{1}$$ and $$\frac{4}{6} \div 2$$, and we want to understand how $$\frac{4}{6} \div 2$$ equals the mixed number $$1 \frac{2}{3}$$. 2. **Multiplication of fractions:** The formula for multiplying fractions is $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$. 3. **Calculate $$\frac{5}{6} \times \frac{2}{1}$$:** $$\frac{5}{6} \times \frac{2}{1} = \frac{5 \times 2}{6 \times 1} = \frac{10}{6}$$ 4. **Division of fractions:** Dividing by a whole number is the same as multiplying by its reciprocal: $$\frac{a}{b} \div c = \frac{a}{b} \times \frac{1}{c}$$ 5. **Calculate $$\frac{4}{6} \div 2$$:** $$\frac{4}{6} \div 2 = \frac{4}{6} \times \frac{1}{2} = \frac{4 \times 1}{6 \times 2} = \frac{4}{12}$$ 6. **Simplify $$\frac{4}{12}$$:** $$\frac{4}{12} = \frac{\cancel{4}^1}{\cancel{12}^3} = \frac{1}{3}$$ 7. **Check the mixed number $$1 \frac{2}{3}$$:** The mixed number $$1 \frac{2}{3}$$ equals $$\frac{5}{3}$$ as an improper fraction. 8. **Clarify the confusion:** The expression $$\frac{4}{6} \div 2$$ equals $$\frac{1}{3}$$, not $$1 \frac{2}{3}$$. Possibly, the intended expression was $$\frac{4}{6} \times 2$$ or $$\frac{4}{6} \div \frac{2}{3}$$. 9. **If dividing by $$\frac{2}{3}$$:** $$\frac{4}{6} \div \frac{2}{3} = \frac{4}{6} \times \frac{3}{2} = \frac{4 \times 3}{6 \times 2} = \frac{12}{12} = 1$$ 10. **If multiplying $$\frac{4}{6} \times 2$$:** $$\frac{4}{6} \times 2 = \frac{4 \times 2}{6} = \frac{8}{6} = 1 \frac{2}{6} = 1 \frac{1}{3}$$ 11. **Conclusion:** The original statement $$\frac{4}{6} \div 2 = 1 \frac{2}{3}$$ is incorrect. The correct value is $$\frac{1}{3}$$. **Final answers:** - $$\frac{5}{6} \times 2 = \frac{10}{6} = 1 \frac{2}{6} = 1 \frac{1}{3}$$ - $$\frac{4}{6} \div 2 = \frac{1}{3}$$