1. **Problem Statement:** We are asked to find the lowest common denominator (LCD) for pairs of denominators and then compare fractions by converting them to equivalent fractions with common denominators. 2. **Finding the Lowest Common Denominator (LCD):** The LCD of two denominators is the smallest number that both denominators divide into evenly. It is the least common multiple (LCM) of the denominators. 3. **Example: Find the LCD for each pair:** - For 2 and 3: The multiples of 2 are 2, 4, 6, 8... and of 3 are 3, 6, 9... The smallest common multiple is 6. - For 4 and 10: Multiples of 4 are 4, 8, 12, 16, 20... and of 10 are 10, 20, 30... The LCD is 20. - For 25 and 10: Multiples of 25 are 25, 50, 75... and of 10 are 10, 20, 30, 40, 50... The LCD is 50. - For 7 and 14: Multiples of 7 are 7, 14, 21... and of 14 are 14, 28... The LCD is 14. - For 12 and 18: Multiples of 12 are 12, 24, 36... and of 18 are 18, 36, 54... The LCD is 36. 4. **Comparing Fractions by Common Denominators:** - (a) Compare $\frac{5}{8}$ and $\frac{7}{10}$. - LCD of 8 and 10 is 40. - Convert: $\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40}$ and $\frac{7}{10} = \frac{7 \times 4}{10 \times 4} = \frac{28}{40}$. - Since $25 < 28$, $\frac{5}{8} < \frac{7}{10}$. - (b) Compare $\frac{5}{6}$ and $\frac{7}{9}$. - LCD of 6 and 9 is 18. - Convert: $\frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18}$ and $\frac{7}{9} = \frac{7 \times 2}{9 \times 2} = \frac{14}{18}$. - Since $15 > 14$, $\frac{5}{6} > \frac{7}{9}$. - (c) Compare $-\frac{8}{14}$ and $-\frac{16}{35}$. - LCD of 14 and 35 is 70. - Convert: $-\frac{8}{14} = -\frac{8 \times 5}{14 \times 5} = -\frac{40}{70}$ and $-\frac{16}{35} = -\frac{16 \times 2}{35 \times 2} = -\frac{32}{70}$. - Since $-40 < -32$, $-\frac{8}{14} < -\frac{16}{35}$. 5. **Summary:** - LCD helps us compare fractions by converting them to equivalent fractions with the same denominator. - After conversion, compare numerators directly. Final answers: - LCDs: 6, 20, 50, 14, 36 - Comparisons: (a) $\frac{5}{8} < \frac{7}{10}$, (b) $\frac{5}{6} > \frac{7}{9}$, (c) $-\frac{8}{14} < -\frac{16}{35}$