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Mathematics

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Step-by-step solution

32 steps

Here are the comparisons for each pair of fractions:

  • a) 12\frac{1}{2} >\boxed{>} 14\frac{1}{4}

  1. To compare, convert 12\frac{1}{2} to an equivalent fraction with a denominator of 4.

    $$
\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}
$$

  2. Compare 24\frac{2}{4} and 14\frac{1}{4}.

    Since 2>12 > 1, 24>14\frac{2}{4} > \frac{1}{4}.

    • b) 12\frac{1}{2} =\boxed{=} 28\frac{2}{8}

  3. To compare, convert 12\frac{1}{2} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}
$$

  4. Compare 48\frac{4}{8} and 28\frac{2}{8}.

    Since 4>24 > 2, 48>28\frac{4}{8} > \frac{2}{8}.
    Correction: The student's answer in the image is =\boxed{=}. Let's re-evaluate.

  5. Simplify 28\frac{2}{8}.

    $$
\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}
$$

  6. Compare 12\frac{1}{2} and 14\frac{1}{4}.

    Since 12\frac{1}{2} is larger than 14\frac{1}{4}, the correct symbol is >>. 12>28 \frac{1}{2} \boxed{>} \frac{2}{8}

    • c) 14\frac{1}{4} <\boxed{<} 38\frac{3}{8}

  7. To compare, convert 14\frac{1}{4} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}
$$

  8. Compare 28\frac{2}{8} and 38\frac{3}{8}.

    Since 2<32 < 3, 28<38\frac{2}{8} < \frac{3}{8}.

    • d) 12\frac{1}{2} =\boxed{=} 24\frac{2}{4}

  9. Simplify 24\frac{2}{4}.

    $$
\frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2}
$$

  10. Compare 12\frac{1}{2} and 12\frac{1}{2}.

    They are equal.

    • e) 22\frac{2}{2} >\boxed{>} 18\frac{1}{8}

  11. Simplify 22\frac{2}{2}.

    $$
\frac{2}{2} = 1
$$

  12. Compare 1 and 18\frac{1}{8}.

    Since 1 is greater than any proper fraction, 1>181 > \frac{1}{8}.

    • f) 58\frac{5}{8} >\boxed{>} 24\frac{2}{4}

  13. To compare, convert 24\frac{2}{4} to an equivalent fraction with a denominator of 8.

    $$
\frac{2}{4} = \frac{2 \times 2}{4 \times 2} = \frac{4}{8}
$$

  14. Compare 58\frac{5}{8} and 48\frac{4}{8}.

    Since 5>45 > 4, 58>48\frac{5}{8} > \frac{4}{8}.

    • g) 24\frac{2}{4} <\boxed{<} 88\frac{8}{8}

  15. Simplify both fractions.

    $$
\frac{2}{4} = \frac{1}{2}
$$
$$
\frac{8}{8} = 1
$$

  16. Compare 12\frac{1}{2} and 1.

    Since 12<1\frac{1}{2} < 1.

    • h) 14\frac{1}{4} <\boxed{<} 48\frac{4}{8}

  17. Simplify 48\frac{4}{8}.

    $$
\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}
$$

  18. Compare 14\frac{1}{4} and 12\frac{1}{2}.

    Since 14\frac{1}{4} is smaller than 12\frac{1}{2}, 14<12\frac{1}{4} < \frac{1}{2}.

    • i) 48\frac{4}{8} =\boxed{=} 12\frac{1}{2}

  19. Simplify 48\frac{4}{8}.

    $$
\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}
$$

  20. Compare 12\frac{1}{2} and 12\frac{1}{2}.

    They are equal.

    • j) 12\frac{1}{2} <\boxed{<} 44\frac{4}{4}

  21. Simplify 44\frac{4}{4}.

    $$
\frac{4}{4} = 1
$$

  22. Compare 12\frac{1}{2} and 1.

    Since 12<1\frac{1}{2} < 1.

    • k) 12\frac{1}{2} >\boxed{>} 18\frac{1}{8}

  23. To compare, convert 12\frac{1}{2} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}
$$

  24. Compare 48\frac{4}{8} and 18\frac{1}{8}.

    Since 4>14 > 1, 48>18\frac{4}{8} > \frac{1}{8}.

    • l) 58\frac{5}{8} >\boxed{>} 12\frac{1}{2}

  25. To compare, convert 12\frac{1}{2} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}
$$

  26. Compare 58\frac{5}{8} and 48\frac{4}{8}.

    Since 5>45 > 4, 58>48\frac{5}{8} > \frac{4}{8}.

    • m) 12\frac{1}{2} <\boxed{<} 78\frac{7}{8}

  27. To compare, convert 12\frac{1}{2} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}
$$

  28. Compare 48\frac{4}{8} and 78\frac{7}{8}.

    Since 4<74 < 7, 48<78\frac{4}{8} < \frac{7}{8}.

    • n) 34\frac{3}{4} >\boxed{>} 18\frac{1}{8}

  29. To compare, convert 34\frac{3}{4} to an equivalent fraction with a denominator of 8.

    $$
\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
$$

  30. Compare 68\frac{6}{8} and 18\frac{1}{8}.

    Since 6>16 > 1, 68>18\frac{6}{8} > \frac{1}{8}.

    • o) 14\frac{1}{4} <\boxed{<} 68\frac{6}{8}

  31. To compare, convert 14\frac{1}{4} to an equivalent fraction with a denominator of 8.

    $$
\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}
$$

  32. Compare 28\frac{2}{8} and 68\frac{6}{8}.

    Since 2<62 < 6, 28<68\frac{2}{8} < \frac{6}{8}.

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