Let's find the fraction form of the repeating decimal [tex]\(0.\overline{9}\)[/tex].
1. Express the repeating decimal:
Let [tex]\( x = 0.\overline{9} \)[/tex], where the bar indicates that 9 is repeating indefinitely.
2. Set up an equation to eliminate the repeating part:
Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 9.999999\ldots
\][/tex]
3. Subtract the original equation from this new equation:
[tex]\[
10x - x = 9.999999\ldots - 0.999999\ldots
\][/tex]
Simplify the equation:
[tex]\[
9x = 9
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 9:
[tex]\[
x = 1
\][/tex]
5. Convert to fraction form:
The number [tex]\( x = 1 \)[/tex] can be expressed as a fraction:
[tex]\[
x = \frac{1}{1}
\][/tex]
Therefore, the fraction form [tex]\(\frac{p}{q}\)[/tex] of the number [tex]\( 0.\overline{9} \)[/tex] is:
[tex]\[
\frac{1}{1}
\][/tex]
So, [tex]\( p = 1 \)[/tex] and [tex]\( q = 1 \)[/tex].
Final result: The [tex]\(\frac{p}{q}\)[/tex] form of the number [tex]\( 0 . \overline{9} \)[/tex] is [tex]\( \frac{1}{1} \)[/tex].
