Certainly! Let's express the repeating decimal [tex]\( 3.\overline{5} \)[/tex] as a fraction in a simplified form step-by-step.
1. Define the repeating decimal:
Let [tex]\( x = 3.\overline{5} \)[/tex], which means [tex]\( x = 3.5555\ldots \)[/tex].
2. Create an equation by multiplying by 10:
To eliminate the repeating part, we'll multiply [tex]\( x \)[/tex] by 10:
[tex]\[
10x = 35.5555\ldots
\][/tex]
3. Set up the subtraction:
Now, subtract the original [tex]\( x \)[/tex] from this equation to get rid of the repeating part:
[tex]\[
10x - x = 35.5555\ldots - 3.5555\ldots
\][/tex]
Simplifying the left side:
[tex]\[
9x
\][/tex]
Simplifying the right side:
[tex]\[
35.5555\ldots - 3.5555\ldots = 32
\][/tex]
So we have:
[tex]\[
9x = 32
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{32}{9}
\][/tex]
5. Simplify the fraction:
The fraction [tex]\( \frac{32}{9} \)[/tex] is already in its simplest form since the greatest common divisor (GCD) of 32 and 9 is 1.
Therefore, the repeating decimal [tex]\( 3.\overline{5} \)[/tex] as a simplified fraction is [tex]\( \frac{32}{9} \)[/tex].
