The problem is to evaluate the expression $\frac{4}{9} \div \frac{2}{5} \div \frac{20}{21}$.

1. Problem Description

The problem is to evaluate the expression $\frac{4}{9} \div \frac{2}{5} \div \frac{20}{21}$.

2. Solution Steps

To divide by a fraction, we multiply by its reciprocal. Therefore, the given expression can be rewritten as:

$\frac{4}{9} \div \frac{2}{5} \div \frac{20}{21} = \frac{4}{9} \times \frac{5}{2} \times \frac{21}{20}$

We can simplify the expression by canceling out common factors between numerators and denominators:

$\frac{4}{9} \times \frac{5}{2} \times \frac{21}{20} = \frac{2 \times 2}{3 \times 3} \times \frac{5}{2} \times \frac{3 \times 7}{4 \times 5} = \frac{2 \times 2 \times 5 \times 3 \times 7}{3 \times 3 \times 2 \times 4 \times 5}$

$= \frac{2 \times 2 \times 5 \times 3 \times 7}{3 \times 3 \times 2 \times 2 \times 2 \times 5}$

$= \frac{2}{2} \times \frac{2}{2} \times \frac{5}{5} \times \frac{3}{3} \times \frac{7}{3 \times 2} = \frac{7}{3 \times 2} = \frac{7}{6}$

3. Final Answer

$\frac{7}{6}$