1. **State the problem:** Add the fractions $\frac{1}{2}$, $\frac{2}{5}$, and $\frac{9}{10}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. Find the least common denominator (LCD) of 2, 5, and 10. 3. The denominators are 2, 5, and 10. The LCD is 10 because 10 is the smallest number divisible by 2, 5, and 10. 4. Convert each fraction to have denominator 10: $$\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}$$ $$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}$$ $$\frac{9}{10} = \frac{9}{10}$$ 5. Add the fractions: $$\frac{5}{10} + \frac{4}{10} + \frac{9}{10} = \frac{5+4+9}{10} = \frac{18}{10}$$ 6. Simplify the fraction $\frac{18}{10}$ by dividing numerator and denominator by their greatest common divisor (2): $$\frac{\cancel{18}^{9}}{\cancel{10}^{5}} = \frac{9}{5}$$ 7. The improper fraction $\frac{9}{5}$ can be written as the mixed number $1 \frac{4}{5}$. **Final answer:** $\frac{1}{2} + \frac{2}{5} + \frac{9}{10} = \frac{9}{5} = 1 \frac{4}{5}$