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Answer :
To determine which numbers are equivalent to [tex]\(0.\overline{6}\)[/tex], we should recognize that this repeating decimal can be expressed in fraction form as [tex]\(\frac{2}{3}\)[/tex]. Let's evaluate each option:
A. [tex]\(\frac{6}{3}\)[/tex]
Simplifying [tex]\(\frac{6}{3}\)[/tex] gives [tex]\(2\)[/tex], which is not equivalent to [tex]\(\frac{2}{3}\)[/tex].
B. [tex]\(60\%\)[/tex]
Converting [tex]\(60\%\)[/tex] to a decimal gives [tex]\(0.60\)[/tex], which is not equivalent to [tex]\(0.\overline{6}\)[/tex].
C. [tex]\(0.666\ldots\)[/tex]
This is a shortened representation of the decimal that keeps repeating, just like [tex]\(0.\overline{6}\)[/tex], but without specifying it repeats. This is not precisely equal to [tex]\(0.\overline{6}\)[/tex].
D. [tex]\(\frac{6}{10}\)[/tex]
Simplifying [tex]\(\frac{6}{10}\)[/tex] gives [tex]\(0.6\)[/tex], which is not equivalent to [tex]\(0.\overline{6}\)[/tex].
E. [tex]\(\frac{2}{3}\)[/tex]
This is exactly equal to [tex]\(0.\overline{6}\)[/tex].
F. [tex]\(66.\overline{6}\%\)[/tex]
Converting [tex]\(66.\overline{6}\%\)[/tex] to a decimal, first recognize it as repeating 66.666...%, which as a decimal is [tex]\(0.6666\ldots\)[/tex], equivalently [tex]\(0.\overline{6}\)[/tex].
Therefore, the numbers equivalent to [tex]\(0.\overline{6}\)[/tex] are Option E ([tex]\(\frac{2}{3}\)[/tex]) and Option F ([tex]\(66.\overline{6}\%\)[/tex]).
A. [tex]\(\frac{6}{3}\)[/tex]
Simplifying [tex]\(\frac{6}{3}\)[/tex] gives [tex]\(2\)[/tex], which is not equivalent to [tex]\(\frac{2}{3}\)[/tex].
B. [tex]\(60\%\)[/tex]
Converting [tex]\(60\%\)[/tex] to a decimal gives [tex]\(0.60\)[/tex], which is not equivalent to [tex]\(0.\overline{6}\)[/tex].
C. [tex]\(0.666\ldots\)[/tex]
This is a shortened representation of the decimal that keeps repeating, just like [tex]\(0.\overline{6}\)[/tex], but without specifying it repeats. This is not precisely equal to [tex]\(0.\overline{6}\)[/tex].
D. [tex]\(\frac{6}{10}\)[/tex]
Simplifying [tex]\(\frac{6}{10}\)[/tex] gives [tex]\(0.6\)[/tex], which is not equivalent to [tex]\(0.\overline{6}\)[/tex].
E. [tex]\(\frac{2}{3}\)[/tex]
This is exactly equal to [tex]\(0.\overline{6}\)[/tex].
F. [tex]\(66.\overline{6}\%\)[/tex]
Converting [tex]\(66.\overline{6}\%\)[/tex] to a decimal, first recognize it as repeating 66.666...%, which as a decimal is [tex]\(0.6666\ldots\)[/tex], equivalently [tex]\(0.\overline{6}\)[/tex].
Therefore, the numbers equivalent to [tex]\(0.\overline{6}\)[/tex] are Option E ([tex]\(\frac{2}{3}\)[/tex]) and Option F ([tex]\(66.\overline{6}\%\)[/tex]).
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