Show that [tex]$0.\overline{3} = 0.333\ldots$[/tex] can be expressed in the form [tex]$\frac{p}{q}$[/tex], where [tex][tex]$p$[/tex][/tex] and [tex]$q$[/tex] are integers and [tex]$q \neq 0$[/tex].



Answer :

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].