To solve the expression $\frac{5}{9} \div 1 \frac{2}{3}$, follow these steps:

Step 1: Convert the mixed number to an improper fraction.

The mixed number $1 \frac{2}{3}$ can be converted as follows: $1 \frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3}$

Step 2: Rewrite the division as multiplication by the reciprocal.

Dividing by a fraction is the same as multiplying by its reciprocal. So the expression becomes: $\frac{5}{9} \div \frac{5}{3} = \frac{5}{9} \times \frac{3}{5}$

Step 3: Multiply the fractions.

Multiply the numerators together and the denominators together: $\frac{5 \times 3}{9 \times 5} = \frac{15}{45}$

Step 4: Simplify the fraction.

$\frac{15}{45} = \frac{1}{3}$

So, $\frac{5}{9} \div 1 \frac{2}{3} = \frac{1}{3}$.

Would you like more details or further explanations?

Here are 5 related questions to consider:

1. How do you convert improper fractions back to mixed numbers?
2. What is the general rule for dividing fractions?
3. How do you simplify fractions effectively?
4. What happens if both fractions have the same numerator and denominator?
5. Can you use this method to solve more complex division problems?

Tip: When dividing by a fraction, always remember to multiply by the reciprocal!