To convert the recurring decimal [tex]\(0.1\dot{5}\)[/tex] into a fraction, we can follow a systematic algebraic approach. Here are the step-by-step details:
1. Define the recurring decimal as a variable:
Let [tex]\(x\)[/tex] be the recurring decimal.
[tex]\[
x = 0.155555 \ldots
\][/tex]
2. Multiply the variable by a power of 10 to shift the decimal point to the right:
Since the decimal repeats every digit after the initial '1,' we multiply by 10 (one place to the right).
[tex]\[
10x = 1.55555\ldots
\][/tex]
3. Set up an equation by subtracting the original decimal from the shifted decimal:
Subtract the original [tex]\(x\)[/tex] from this new equation to eliminate the repeating part.
[tex]\[
10x - x = 1.55555\ldots - 0.155555\ldots
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
9x = 1.4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1.4}{9}
\][/tex]
To simplify [tex]\(\frac{1.4}{9}\)[/tex], we first convert the decimal to a fraction:
[tex]\[
1.4 = \frac{14}{10}
\][/tex]
Now substitute [tex]\(\frac{14}{10}\)[/tex] into the fraction:
[tex]\[
x = \frac{14/10}{9} = \frac{14}{10} \cdot \frac{1}{9} = \frac{14}{90}
\][/tex]
5. Simplify the fraction [tex]\(\frac{14}{90}\)[/tex]:
Determine the greatest common divisor (GCD) of 14 and 90, which is 2.
Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{14 \div 2}{90 \div 2} = \frac{7}{45}
\][/tex]
Therefore, the recurring decimal [tex]\(0.1\dot{5}\)[/tex] as a fraction in its simplest form is:
[tex]\[
\boxed{\frac{7}{45}}
\][/tex]
