To convert the repeating decimal [tex]\( z = 0.142857142857142857 \ldots \)[/tex] into a fraction, let’s follow a step-by-step approach:
1. Let [tex]\( x = 0.142857142857 \ldots \)[/tex]
2. Multiply both sides of the equation by [tex]\( 10^6 \)[/tex] to shift the repeating block of the decimal to the left of the decimal point:
[tex]\[
10^6 x = 142857.142857 \ldots
\][/tex]
Let's denote this new equation as:
[tex]\[
y = 142857.142857 \ldots
\][/tex]
3. Subtract the original equation [tex]\( x \)[/tex] from this new equation [tex]\( y \)[/tex] to eliminate the repeating part:
[tex]\[
y - x = 142857.142857 \ldots - 0.142857 \ldots
\][/tex]
This simplifies to:
[tex]\[
10^6 x - x = 142857
\][/tex]
So,
[tex]\[
999999x = 142857
\][/tex]
4. Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[
x = \frac{142857}{999999}
\][/tex]
5. To simplify the fraction [tex]\( \frac{142857}{999999} \)[/tex], find the greatest common divisor (GCD) of the numerator and the denominator. For this fraction, the GCD is 1, so it can't be simplified further.
Thus, the fraction for the repeating decimal [tex]\( z = 0.142857142857 \ldots \)[/tex] is already in its simplest form:
[tex]\[
\frac{2573485501354569}{18014398509481984}
\][/tex]
Therefore, the repeating decimal [tex]\( z = 0.142857142857 \ldots \)[/tex] can be written as the simplified fraction:
[tex]\[
z = \frac{2573485501354569}{18014398509481984}
\][/tex]
