To divide the fractions \( \frac{2x}{27} \) by \( \frac{4x+20}{9} \), we will first change the division of fractions into multiplication by the reciprocal.
The expression can be rewritten as:
\[ \frac{2x}{27} \div \frac{4x + 20}{9} = \frac{2x}{27} \times \frac{9}{4x + 20} \]
Now, we can simplify \( 4x + 20 \):
\[ 4x + 20 = 4(x + 5) \]
Next, substitute this back into the multiplication:
\[ \frac{2x}{27} \times \frac{9}{4(x + 5)} \]
Now multiply the numerators and the denominators:
\[ \frac{2x \cdot 9}{27 \cdot 4(x + 5)} = \frac{18x}{108(x + 5)} \]
Now, simplify the fraction \( \frac{18x}{108(x + 5)} \):
Both the numerator and denominator can be divided by 18:
\[ \frac{18x \div 18}{108 \div 18(x + 5)} = \frac{x}{6(x + 5)} \]
Thus, the simplified answer is:
\[ \frac{x}{6(x + 5)} \]