1. **State the problem:** We need to order the fractions $\frac{1}{2}$, $\frac{5}{11}$, and $\frac{2}{5}$ from least to greatest. 2. **Formula and rules:** To compare fractions, we can find a common denominator or convert them to decimals. 3. **Find a common denominator:** The denominators are 2, 11, and 5. The least common denominator (LCD) is the least common multiple of these numbers. 4. Calculate the LCD: $$\text{LCM}(2, 11, 5) = 2 \times 5 \times 11 = 110$$ 5. Convert each fraction to have denominator 110: $$\frac{1}{2} = \frac{1 \times 55}{2 \times 55} = \frac{55}{110}$$ $$\frac{5}{11} = \frac{5 \times 10}{11 \times 10} = \frac{50}{110}$$ $$\frac{2}{5} = \frac{2 \times 22}{5 \times 22} = \frac{44}{110}$$ 6. **Compare numerators:** $44 < 50 < 55$ 7. **Order from least to greatest:** $$\frac{2}{5} < \frac{5}{11} < \frac{1}{2}$$ **Final answer:** $\frac{2}{5}$, $\frac{5}{11}$, $\frac{1}{2}$