1. **State the problem:** We need to put the fractions $\frac{1}{10}$, $\frac{2}{3}$, and $\frac{5}{11}$ in order from least to greatest. 2. **Formula and rules:** To compare fractions, we can find a common denominator or convert them to decimals. The fraction with the smaller value is the lesser fraction. 3. **Find a common denominator:** The denominators are 10, 3, and 11. The least common denominator (LCD) is the least common multiple of these numbers. - Prime factors: 10 = 2 \times 5, 3 = 3, 11 = 11 - LCD = 2 \times 3 \times 5 \times 11 = 330 4. **Convert each fraction to have denominator 330:** $$\frac{1}{10} = \frac{1 \times 33}{10 \times 33} = \frac{33}{330}$$ $$\frac{2}{3} = \frac{2 \times 110}{3 \times 110} = \frac{220}{330}$$ $$\frac{5}{11} = \frac{5 \times 30}{11 \times 30} = \frac{150}{330}$$ 5. **Compare numerators:** $33 < 150 < 220$ 6. **Order from least to greatest:** $$\frac{1}{10} < \frac{5}{11} < \frac{2}{3}$$ **Final answer:** $\boxed{\frac{1}{10}, \frac{5}{11}, \frac{2}{3}}$