1. **State the problem:** We need to put the fractions $\frac{4}{9}$, $\frac{5}{12}$, and $\frac{2}{5}$ in ascending order. 2. **Formula and rules:** To compare fractions, convert them to a common denominator or convert to decimals. 3. **Find a common denominator:** The denominators are 9, 12, and 5. The least common denominator (LCD) is the least common multiple (LCM) of 9, 12, and 5. - Prime factors: 9 = $3^2$, 12 = $2^2 \times 3$, 5 = 5 - LCM = $2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$ 4. **Convert each fraction to have denominator 180:** - $\frac{4}{9} = \frac{4 \times 20}{9 \times 20} = \frac{80}{180}$ - $\frac{5}{12} = \frac{5 \times 15}{12 \times 15} = \frac{75}{180}$ - $\frac{2}{5} = \frac{2 \times 36}{5 \times 36} = \frac{72}{180}$ 5. **Compare numerators:** $72 < 75 < 80$ 6. **Write fractions in ascending order:** $$\frac{2}{5} < \frac{5}{12} < \frac{4}{9}$$ **Final answer:** $\frac{2}{5}$, $\frac{5}{12}$, $\frac{4}{9}$