Sure, let's express the repeating decimal [tex]\( 0.\overline{3} \)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers, and [tex]\( q \neq 0 \)[/tex].
1. Represent the repeating decimal as a variable:
Let's denote the repeating decimal by [tex]\( x \)[/tex].
So, [tex]\( x = 0.\overline{3} \)[/tex], which means [tex]\( x = 0.3333... \)[/tex]
2. Multiply both sides by 10 to shift the decimal point one place:
This will help us align the repeating parts.
[tex]\[
10x = 3.3333...
\][/tex]
3. Set up the equation to eliminate the repeating part:
Now, subtract the original [tex]\( x = 0.3333... \)[/tex] from this new equation.
[tex]\[
10x = 3.3333...
\][/tex]
[tex]\[
x = 0.3333...
\][/tex]
[tex]\[
10x - x = 3.3333... - 0.3333...
\][/tex]
4. Simplify the equation:
[tex]\[
9x = 3
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3}{9}
\][/tex]
6. Simplify the fraction:
[tex]\[
x = \frac{1}{3}
\][/tex]
Therefore, the repeating decimal [tex]\( 0.\overline{3} \)[/tex] can be expressed in the form [tex]\(\frac{p}{q}\)[/tex], where [tex]\( p = 1 \)[/tex] and [tex]\( q = 3 \)[/tex]. So, [tex]\( 0.\overline{3} = \frac{1}{3} \)[/tex].
