To rewrite the repeating decimal [tex]\(2.1\overline{6}\)[/tex] as a simplified fraction, we'll go through the following steps:
1. Express the repeating decimal as a variable.
Let's call the repeating decimal [tex]\( x \)[/tex].
[tex]\[
x = 2.1\overline{6}
\][/tex]
2. Eliminate the repeating part by shifting the decimal point.
Since the repeating part is a single digit "6", we multiply [tex]\( x \)[/tex] by 10 to move one decimal place to the right:
[tex]\[
10x = 21.6\overline{6}
\][/tex]
3. Subtract the original equation from the new equation.
Subtract [tex]\( x \)[/tex] from [tex]\( 10x \)[/tex] to eliminate the repeating part:
[tex]\[
10x - x = 21.6\overline{6} - 2.1\overline{6}
\][/tex]
Simplifying this:
[tex]\[
9x = 21.6 - 2.1
\][/tex]
[tex]\[
9x = 19.5
\][/tex]
4. Solve for [tex]\( x \)[/tex].
Divide both sides by 9:
[tex]\[
x = \frac{19.5}{9}
\][/tex]
5. Simplify the fraction.
To simplify [tex]\(\frac{19.5}{9}\)[/tex], first express the numerator as an integer:
[tex]\[
19.5 = \frac{195}{10}
\][/tex]
So,
[tex]\[
x = \frac{\frac{195}{10}}{9} = \frac{195}{90}
\][/tex]
6. Find the greatest common divisor (GCD) of 195 and 90.
To simplify [tex]\(\frac{195}{90}\)[/tex], we need to find the GCD. The prime factorizations are:
- 195: [tex]\(195 = 3 \times 5 \times 13\)[/tex]
- 90: [tex]\(90 = 2 \times 3 \times 3 \times 5\)[/tex]
The common factors are [tex]\(3\)[/tex] and [tex]\(5\)[/tex], so:
[tex]\[
\text{GCD}(195, 90) = 3 \times 5 = 15
\][/tex]
7. Divide both the numerator and denominator by their GCD:
[tex]\[
\frac{195 \div 15}{90 \div 15} = \frac{13}{6}
\][/tex]
So, the repeating decimal [tex]\( 2.1\overline{6} \)[/tex] can be expressed as the simplified fraction:
[tex]\[
2.1\overline{6} = \frac{13}{6}
\][/tex]
