To convert the repeating decimal [tex]\( 1.\overline{6} \)[/tex] into a simplified fraction, follow these steps:
1. Let [tex]\( x = 1.\overline{6} \)[/tex]. This means [tex]\( x \)[/tex] is equal to 1.666666..., where the digit 6 repeats indefinitely.
2. Multiply both sides of this equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 16.\overline{6}
\][/tex]
This means [tex]\( 10x \)[/tex] is equal to 16.666666... with the digit 6 repeating indefinitely.
3. Subtract the original equation [tex]\( x = 1.\overline{6} \)[/tex] from this new equation:
[tex]\[
10x - x = 16.\overline{6} - 1.\overline{6}
\][/tex]
On the left side, this simplifies to:
[tex]\[
9x
\][/tex]
On the right side, the repeating decimals cancel out, leaving:
[tex]\[
16 - 1 = 15
\][/tex]
So, we have:
[tex]\[
9x = 15
\][/tex]
4. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{15}{9}
\][/tex]
5. Simplify the fraction [tex]\(\frac{15}{9}\)[/tex] by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 15 and 9 is 3.
Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{15 \div 3}{9 \div 3} = \frac{5}{3}
\][/tex]
Thus, the simplified fraction of the repeating decimal [tex]\( 1.\overline{6} \)[/tex] is:
[tex]\[
\boxed{\frac{5}{3}}
\][/tex]
