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Simplify the sums given below by following the examples.

Mathematics

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Here are the solutions for simplifying the sums and differences of fractions:

1. 23+23\frac{2}{3} + \frac{2}{3}

  1. The denominators are already the same.

  2. Add the numerators.

    23+23=2+23=43\frac{2}{3} + \frac{2}{3} = \frac{2+2}{3} = \frac{4}{3}

    The simplified sum is 43\boxed{\frac{4}{3}}.

    2. 45+715\frac{4}{5} + \frac{7}{15}

  3. Find the least common multiple (LCM) of the denominators 5 and 15, which is 15.

  4. Convert 45\frac{4}{5} to an equivalent fraction with a denominator of 15.

    45=4×35×3=1215\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}

  5. Add the fractions.

    1215+715=12+715=1915\frac{12}{15} + \frac{7}{15} = \frac{12+7}{15} = \frac{19}{15}

    The simplified sum is 1915\boxed{\frac{19}{15}}.

    3. 27+614\frac{2}{7} + \frac{6}{14}

  6. Find the LCM of the denominators 7 and 14, which is 14.

  7. Convert 27\frac{2}{7} to an equivalent fraction with a denominator of 14.

    27=2×27×2=414\frac{2}{7} = \frac{2 \times 2}{7 \times 2} = \frac{4}{14}

  8. Add the fractions.

    414+614=4+614=1014\frac{4}{14} + \frac{6}{14} = \frac{4+6}{14} = \frac{10}{14}

  9. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

    10÷214÷2=57\frac{10 \div 2}{14 \div 2} = \frac{5}{7}

    The simplified sum is 57\boxed{\frac{5}{7}}.

    4. 58−716\frac{5}{8} - \frac{7}{16}

  10. Find the LCM of the denominators 8 and 16, which is 16.

  11. Convert 58\frac{5}{8} to an equivalent fraction with a denominator of 16.

    58=5×28×2=1016\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16}

  12. Subtract the fractions.

    1016−716=10−716=316\frac{10}{16} - \frac{7}{16} = \frac{10-7}{16} = \frac{3}{16}

    The simplified difference is 316\boxed{\frac{3}{16}}.

    7. 35+715\frac{3}{5} + \frac{7}{15}

  13. Find the LCM of the denominators 5 and 15, which is 15.

  14. Convert 35\frac{3}{5} to an equivalent fraction with a denominator of 15.

    35=3×35×3=915\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}

  15. Add the fractions.

    915+715=9+715=1615\frac{9}{15} + \frac{7}{15} = \frac{9+7}{15} = \frac{16}{15}

    The simplified sum is 1615\boxed{\frac{16}{15}}.

    8. 12+210\frac{1}{2} + \frac{2}{10}

  16. Find the LCM of the denominators 2 and 10, which is 10.

  17. Convert 12\frac{1}{2} to an equivalent fraction with a denominator of 10.

    12=1×52×5=510\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}

  18. Add the fractions.

    510+210=5+210=710\frac{5}{10} + \frac{2}{10} = \frac{5+2}{10} = \frac{7}{10}

    The simplified sum is 710\boxed{\frac{7}{10}}.

    10. 57−914\frac{5}{7} - \frac{9}{14}

  19. Find the LCM of the denominators 7 and 14, which is 14.

  20. Convert 57\frac{5}{7} to an equivalent fraction with a denominator of 14.

    57=5×27×2=1014\frac{5}{7} = \frac{5 \times 2}{7 \times 2} = \frac{10}{14}

  21. Subtract the fractions.

    1014−914=10−914=114\frac{10}{14} - \frac{9}{14} = \frac{10-9}{14} = \frac{1}{14}

    The simplified difference is 114\boxed{\frac{1}{14}}.

    11. 45−815\frac{4}{5} - \frac{8}{15}

  22. Find the LCM of the denominators 5 and 15, which is 15.

  23. Convert 45\frac{4}{5} to an equivalent fraction with a denominator of 15.

    45=4×35×3=1215\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}

  24. Subtract the fractions.

    1215−815=12−815=415\frac{12}{15} - \frac{8}{15} = \frac{12-8}{15} = \frac{4}{15}

    The simplified difference is 415\boxed{\frac{4}{15}}.

    13. 23+29\frac{2}{3} + \frac{2}{9}

  25. Find the LCM of the denominators 3 and 9, which is 9.

  26. Convert 23\frac{2}{3} to an equivalent fraction with a denominator of 9.

    23=2×33×3=69\frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9}

  27. Add the fractions.

    69+29=6+29=89\frac{6}{9} + \frac{2}{9} = \frac{6+2}{9} = \frac{8}{9}

    The simplified sum is 89\boxed{\frac{8}{9}}.

    14. 35+710\frac{3}{5} + \frac{7}{10}

  28. Find the LCM of the denominators 5 and 10, which is 10.

  29. Convert 35\frac{3}{5} to an equivalent fraction with a denominator of 10.

    35=3×25×2=610\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}

  30. Add the fractions.

    610+710=6+710=1310\frac{6}{10} + \frac{7}{10} = \frac{6+7}{10} = \frac{13}{10}

    The simplified sum is 1310\boxed{\frac{13}{10}}.

    16. 37−421\frac{3}{7} - \frac{4}{21}

  31. Find the LCM of the denominators 7 and 21, which is 21.

  32. Convert 37\frac{3}{7} to an equivalent fraction with a denominator of 21.

    37=3×37×3=921\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}

  33. Subtract the fractions.

    921−421=9−421=521\frac{9}{21} - \frac{4}{21} = \frac{9-4}{21} = \frac{5}{21}

    The simplified difference is 521\boxed{\frac{5}{21}}.

    17. 23+112\frac{2}{3} + \frac{1}{12}

  34. Find the LCM of the denominators 3 and 12, which is 12.

  35. Convert 23\frac{2}{3} to an equivalent fraction with a denominator of 12.

    23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

  36. Add the fractions.

    812+112=8+112=912\frac{8}{12} + \frac{1}{12} = \frac{8+1}{12} = \frac{9}{12}

  37. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 3.

    9÷312÷3=34\frac{9 \div 3}{12 \div 3} = \frac{3}{4}

    The simplified sum is 34\boxed{\frac{3}{4}}.

    19. 58−316\frac{5}{8} - \frac{3}{16}

  38. Find the LCM of the denominators 8 and 16, which is 16.

  39. Convert 58\frac{5}{8} to an equivalent fraction with a denominator of 16.

    58=5×28×2=1016\frac{5}{8} = \frac{5 \times 2}{8 \times 2} = \frac{10}{16}

  40. Subtract the fractions.

    1016−316=10−316=716\frac{10}{16} - \frac{3}{16} = \frac{10-3}{16} = \frac{7}{16}

    The simplified difference is 716\boxed{\frac{7}{16}}.

    20. 45−610\frac{4}{5} - \frac{6}{10}

  41. Find the LCM of the denominators 5 and 10, which is 10.

  42. Convert 45\frac{4}{5} to an equivalent fraction with a denominator of 10.

    45=4×25×2=810\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}

  43. Subtract the fractions.

    810−610=8−610=210\frac{8}{10} - \frac{6}{10} = \frac{8-6}{10} = \frac{2}{10}

  44. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 2.

    2÷210÷2=15\frac{2 \div 2}{10 \div 2} = \frac{1}{5}

    The simplified difference is 15\boxed{\frac{1}{5}}.

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