To express the repeating decimal [tex]\( 3.\overline{1} \)[/tex] as a simplified fraction, follow these steps:
1. Assign a Variable: Let [tex]\( x = 3.\overline{1} \)[/tex].
2. Express the Repeating Decimal: Recognize that [tex]\( 3.\overline{1} = 3.1111...\)[/tex].
3. Move the Decimal Point: Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 31.1111...
\][/tex]
4. Subtract Equations: Subtract the original [tex]\( x = 3.1111... \)[/tex] from the new equation [tex]\( 10x = 31.1111... \)[/tex] to eliminate the repeating decimal:
[tex]\[
10x - x = 31.1111... - 3.1111...
\][/tex]
This simplifies to:
[tex]\[
9x = 28
\][/tex]
5. Solve for [tex]\( x \)[/tex]: Divide both sides of the equation by 9 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{28}{9}
\][/tex]
Therefore, the repeating decimal [tex]\( 3.\overline{1} \)[/tex] can be written as the simplified fraction [tex]\(\frac{28}{9}\)[/tex].
So, [tex]\( 3.\overline{1} = \frac{28}{9} \)[/tex].
