To express the repeating decimal [tex]$1 . \overline{5}$[/tex] as a fraction, let's follow these steps:
1. Let [tex]\( x \)[/tex] represent the repeating decimal:
[tex]\[
x = 1.5555\ldots
\][/tex]
2. Multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\[
10x = 15.5555\ldots
\][/tex]
3. Subtract the original equation (step 1) from this new equation (step 2):
[tex]\[
10x - x = 15.5555\ldots - 1.5555\ldots
\][/tex]
[tex]\[
9x = 14
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{14}{9}
\][/tex]
5. To express the fraction in its lowest terms, we need to simplify it:
- Find the greatest common divisor (GCD) of the numerator and the denominator. For 14 and 9, the GCD is 1.
- Divide both the numerator and the denominator by the GCD:
[tex]\[
\frac{14 \div 1}{9 \div 1} = \frac{14}{9}
\][/tex]
Therefore, the repeating decimal [tex]\(1 . \overline{5}\)[/tex] is equal to the fraction [tex]\(\frac{14}{9}\)[/tex] when expressed in its simplest form.
