1. **State the problem:** Add the fractions $\frac{1}{6}$ and $\frac{1}{10}$. 2. **Formula used:** To add fractions, use the formula $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$. 3. **Find the common denominator:** The denominators are 6 and 10. The least common denominator (LCD) is the least common multiple (LCM) of 6 and 10. - Prime factors: 6 = 2 \times 3, 10 = 2 \times 5 - LCM = 2 \times 3 \times 5 = 30 4. **Rewrite each fraction with denominator 30:** $$\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}$$ $$\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}$$ 5. **Add the fractions:** $$\frac{5}{30} + \frac{3}{30} = \frac{5 + 3}{30} = \frac{8}{30}$$ 6. **Simplify the fraction:** The numerator and denominator share a common factor of 2. $$\frac{\cancel{8}^{4}}{\cancel{30}^{15}} = \frac{4}{15}$$ 7. **Final answer:** $$\frac{1}{6} + \frac{1}{10} = \frac{4}{15}$$