Write a repeating decimal that is between 9/7 and 10/7. Justify your answer.

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Write a repeating decimal that is between 9/7
and 10/7.
Justify your answer.

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  1. Given:

    Two numbers are [tex]\dfrac{9}{7}[/tex]  and [tex]\dfrac{10}{7}[/tex].

    To find:

    A repeating decimal that is between [tex]\dfrac{9}{7}[/tex]  and [tex]\dfrac{10}{7}[/tex].

    Solution:

    Using calculator, we get

    [tex]\dfrac{9}{7}=1.285714285714…[/tex]

    [tex]\dfrac{9}{7}=1.\overline{285714}[/tex]

    and,

    [tex]\dfrac{10}{7}=1.42857142857
    1…[/tex]

    [tex]\dfrac{10}{7}=1.\overline{428571}[/tex]

    Now, the repeating decimal that is between [tex]\dfrac{9}{7}[/tex]  and [tex]\dfrac{10}{7}[/tex] be x. So,

    [tex]1.\overline{285714}<x<1.\overline{428571}[/tex]

    On analyzing the numbers to hundredth places, we get 1.28 < 1.33 < 1.42, therefore

    [tex]1.\overline{285714}<1.3333…<1.\overline{428571}[/tex]

    [tex]1.\overline{285714}<1.\overline{3}<1.\overline{428571}[/tex]

    And we know that [tex]1.\overline{3}[/tex] is the decimal form of [tex]\dfrac{4}{3}[/tex].

    Therefore, the required number is [tex]1.\overline{3}[/tex] .

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