1. **State the problem:** Hana thinks of a fraction equivalent to $\frac{5}{9}$ where the numerator is greater than 18 and the denominator is less than 40. 2. **Recall the rule for equivalent fractions:** Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if $a \times d = b \times c$. 3. **Set up the equivalent fraction:** Let the fraction Hana thinks of be $\frac{x}{y}$ such that $\frac{x}{y} = \frac{5}{9}$. 4. **Express the relationship:** From equivalence, $x = 5k$ and $y = 9k$ for some integer $k$. 5. **Apply the conditions:** - Numerator $x > 18$ implies $5k > 18$ so $k > \frac{18}{5} = 3.6$. - Denominator $y < 40$ implies $9k < 40$ so $k < \frac{40}{9} \approx 4.44$. 6. **Find integer $k$ satisfying both:** $k$ must be an integer greater than 3.6 and less than 4.44, so $k = 4$. 7. **Calculate the fraction:** - Numerator $x = 5 \times 4 = 20$. - Denominator $y = 9 \times 4 = 36$. 8. **Final answer:** Hana is thinking of the fraction $\frac{20}{36}$, which is equivalent to $\frac{5}{9}$, with numerator greater than 18 and denominator less than 40.