To convert the repeating decimal [tex]\( z = 0.142857142857\ldots \)[/tex] to a fraction, follow these steps:
1. Identify the repeating part: Note that the repeating sequence is [tex]\( 142857 \)[/tex], which has 6 digits.
2. Set up an equation: Let [tex]\( z = 0.142857142857\ldots \)[/tex].
3. Eliminate the repeating part: Multiply both sides of the equation by [tex]\( 10^6 \)[/tex] (because the repeating part has 6 digits):
[tex]\[
10^6z = 142857.142857142857\ldots
\][/tex]
4. Subtract the original equation from this new equation to eliminate the repeating decimals:
[tex]\[
10^6z - z = 142857.142857142857\ldots - 0.142857142857\ldots
\][/tex]
Simplifying this gives:
[tex]\[
999999z = 142857
\][/tex]
5. Solve for [tex]\( z \)[/tex] by dividing both sides of the equation by [tex]\( 999999 \)[/tex]:
[tex]\[
z = \frac{142857}{999999}
\][/tex]
6. Simplify the fraction: To simplify [tex]\( \frac{142857}{999999} \)[/tex], we find the greatest common divisor (GCD) of 142857 and 999999. It turns out that the GCD is 142857.
Dividing both the numerator and the denominator by 142857, we get:
[tex]\[
\frac{142857 \div 142857}{999999 \div 142857} = \frac{1}{7}
\][/tex]
Therefore, the repeating decimal [tex]\( z = 0.142857142857\ldots \)[/tex] can be written as the simplified fraction:
[tex]\[
z = \frac{1}{7}
\][/tex]
