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Mathematics

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Step-by-step solution

5 steps

Here are the comparisons for the given fractions:

  1. Compare 34\frac{3}{4} and 45\frac{4}{5}.

    To compare these fractions, find a common denominator, which is 20.

    34=3×54×5=1520\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} 45=4×45×4=1620\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}

    Since 15<1615 < 16, we have 1520<1620\frac{15}{20} < \frac{16}{20}.
    Therefore, 34<45\frac{3}{4} \boxed{<} \frac{4}{5}.

  2. Compare 25\frac{2}{5} and 14\frac{1}{4}.

    To compare these fractions, find a common denominator, which is 20.

    25=2×45×4=820\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} 14=1×54×5=520\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}

    Since 8>58 > 5, we have 820>520\frac{8}{20} > \frac{5}{20}.
    Therefore, 25>14\frac{2}{5} \boxed{>} \frac{1}{4}.

  3. Compare 12\frac{1}{2} and 48\frac{4}{8}.

    Simplify the fraction 48\frac{4}{8}.

    48=4÷48÷4=12\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}

    Since both fractions are equal to 12\frac{1}{2}.
    Therefore, 12=48\frac{1}{2} \boxed{=} \frac{4}{8}.

  4. Compare 23\frac{2}{3} and 46\frac{4}{6}.

    Simplify the fraction 46\frac{4}{6}.

    46=4÷26÷2=23\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}

    Since both fractions are equal to 23\frac{2}{3}.
    Therefore, 23=46\frac{2}{3} \boxed{=} \frac{4}{6}.

  5. Compare 34\frac{3}{4} and 23\frac{2}{3}.

    To compare these fractions, find a common denominator, which is 12.

    34=3×34×3=912\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} 23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

    Since 9>89 > 8, we have 912>812\frac{9}{12} > \frac{8}{12}.
    Therefore, 34>23\frac{3}{4} \boxed{>} \frac{2}{3}.

    Here are the final answers:

    • 34<45\frac{3}{4} \boxed{<} \frac{4}{5}
    • 25>14\frac{2}{5} \boxed{>} \frac{1}{4}
    • 12=48\frac{1}{2} \boxed{=} \frac{4}{8}
    • 23=46\frac{2}{3} \boxed{=} \frac{4}{6}
    • 34>23\frac{3}{4} \boxed{>} \frac{2}{3}
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