Fraction Operations E73Af8

1. We are asked to perform addition and subtraction of fractions for four separate problems. 2. The general formula for adding or subtracting fractions is $$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$ where $a,b,c,d$ are integers and $b,d \neq 0$. 3. Important rules: - Find a common denominator (usually the least common denominator). - Convert each fraction to an equivalent fraction with the common denominator. - Add or subtract the numerators. - Simplify the resulting fraction if possible. **Problem 8:** $$\frac{5}{3} + \frac{4}{9} + \frac{6}{18}$$ 4. Find the least common denominator (LCD) of 3, 9, and 18. - $\mathrm{LCD}(3,9,18) = 18$ 5. Convert each fraction: - $$\frac{5}{3} = \frac{5 \times 6}{3 \times 6} = \frac{30}{18}$$ - $$\frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18}$$ - $$\frac{6}{18}$$ is already with denominator 18. 6. Add the numerators: - $$\frac{30}{18} + \frac{8}{18} + \frac{6}{18} = \frac{30 + 8 + 6}{18} = \frac{44}{18}$$ 7. Simplify $$\frac{44}{18}$$: - Both numerator and denominator divisible by 2. - $$\frac{44}{18} = \frac{\cancel{44}^{22}}{\cancel{18}^{9}} = \frac{22}{9}$$ **Problem 9:** $$\frac{5}{4} + \frac{7}{8} + \frac{1}{16}$$ 8. Find LCD of 4, 8, and 16. - $\mathrm{LCD}(4,8,16) = 16$ 9. Convert each fraction: - $$\frac{5}{4} = \frac{5 \times 4}{4 \times 4} = \frac{20}{16}$$ - $$\frac{7}{8} = \frac{7 \times 2}{8 \times 2} = \frac{14}{16}$$ - $$\frac{1}{16}$$ already has denominator 16. 10. Add numerators: - $$\frac{20}{16} + \frac{14}{16} + \frac{1}{16} = \frac{20 + 14 + 1}{16} = \frac{35}{16}$$ 11. $$\frac{35}{16}$$ is an improper fraction and cannot be simplified further. **Problem 10:** $$\frac{5}{8} - \frac{1}{4}$$ 12. Find LCD of 8 and 4. - $\mathrm{LCD}(8,4) = 8$ 13. Convert $$\frac{1}{4}$$ to denominator 8: - $$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$$ 14. Subtract numerators: - $$\frac{5}{8} - \frac{2}{8} = \frac{5 - 2}{8} = \frac{3}{8}$$ **Problem 11:** $$\frac{5}{12} - \frac{7}{24}$$ 15. Find LCD of 12 and 24. - $\mathrm{LCD}(12,24) = 24$ 16. Convert $$\frac{5}{12}$$ to denominator 24: - $$\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}$$ 17. Subtract numerators: - $$\frac{10}{24} - \frac{7}{24} = \frac{10 - 7}{24} = \frac{3}{24}$$ 18. Simplify $$\frac{3}{24}$$: - Both divisible by 3. - $$\frac{3}{24} = \frac{\cancel{3}^{1}}{\cancel{24}^{8}} = \frac{1}{8}$$ **Final answers:** - Problem 8: $$\frac{22}{9}$$ - Problem 9: $$\frac{35}{16}$$ - Problem 10: $$\frac{3}{8}$$ - Problem 11: $$\frac{1}{8}$$