1. **State the problem:** We need to order the fractions $\frac{2}{3}$, $\frac{1}{2}$, and $\frac{4}{7}$ from least to greatest. 2. **Formula and rules:** To compare fractions, we find a common denominator or convert them to decimals. The fraction with the smaller value is less. 3. **Find a common denominator:** The denominators are 3, 2, and 7. The least common denominator (LCD) is the least common multiple of 3, 2, and 7. $$\text{LCM}(3,2,7) = 42$$ 4. **Convert each fraction to have denominator 42:** $$\frac{2}{3} = \frac{2 \times 14}{3 \times 14} = \frac{28}{42}$$ $$\frac{1}{2} = \frac{1 \times 21}{2 \times 21} = \frac{21}{42}$$ $$\frac{4}{7} = \frac{4 \times 6}{7 \times 6} = \frac{24}{42}$$ 5. **Compare the numerators:** $21 < 24 < 28$ 6. **Order the fractions from least to greatest:** $$\frac{1}{2} < \frac{4}{7} < \frac{2}{3}$$ **Final answer:** $\boxed{\frac{1}{2} < \frac{4}{7} < \frac{2}{3}}$