To write the repeating decimal [tex]\(0 . \overline{7}\)[/tex] as a fraction, follow these steps:
1. Set up an equation for the repeating decimal:
Let [tex]\( x = 0.\overline{7} \)[/tex]
2. Multiply both sides of the equation by 10:
Multiplying by 10 shifts the decimal point one place to the right.
[tex]\[
10x = 7.7777\ldots
\][/tex]
3. Subtract the original equation from the multiplied equation:
By performing this subtraction, the repeating parts (the [tex]\(0.7777\ldots\)[/tex]) will cancel out.
[tex]\[
10x - x = 7.7777\ldots - 0.7777\ldots
\][/tex]
Simplifying this, we get:
[tex]\[
9x = 7
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 9:
[tex]\[
x = \frac{7}{9}
\][/tex]
Thus, the repeating decimal [tex]\(0 . \overline{7}\)[/tex] can be written as the fraction [tex]\(\frac{7}{9}\)[/tex].
