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What is the fractional form of [tex]$0.\overline{8}$[/tex]?

A. [tex]$\frac{8}{10}$[/tex]
B. [tex][tex]$\frac{1}{8}$[/tex][/tex]
C. [tex]$\frac{8}{9}$[/tex]
D. [tex]$\frac{8}{11}$[/tex]


Sagot :

To determine the fractional form of the repeating decimal [tex]\( 0.\overline{8} \)[/tex], follow these steps:

1. Let the repeating decimal be represented by [tex]\( x \)[/tex]:
[tex]\[ x = 0.\overline{8} \][/tex]

2. Express the repeating decimal by multiplying [tex]\( x \)[/tex] by 10:
[tex]\[ 10x = 8.8888\ldots \][/tex]
Here, the decimal part is again [tex]\( 0.\overline{8} \)[/tex].

3. Subtract the original equation from this new equation:
[tex]\[ 10x = 8.8888\ldots \][/tex]
[tex]\[ - (x = 0.8888\ldots) \][/tex]
[tex]\[ 9x = 8 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{8}{9} \][/tex]

Thus, the fractional form of [tex]\( 0.\overline{8} \)[/tex] is:
[tex]\[ \frac{8}{9} \][/tex]

The correct answer is:
C. [tex]\(\frac{8}{9}\)[/tex]