Show that $0.142857142857 \ldots = \frac{1}{7}$.
(N/A) Let $x = 0.142857142857 \ldots$ $(1)$
Since the repeating block has $6$ digits,multiply both sides by $1,000,000$:
$1,000,000 x = 142857.142857142857 \ldots$ $(2)$
Subtracting equation $(1)$ from equation $(2)$:
$1,000,000 x - x = 142857.142857 \ldots - 0.142857 \ldots$
$999,999 x = 142857$
Solving for $x$:
$x = \frac{142857}{999999}$
Dividing both numerator and denominator by $142857$:
$x = \frac{1}{7}$
Thus,$0.142857142857 \ldots = \frac{1}{7}$.
Since the repeating block has $6$ digits,multiply both sides by $1,000,000$:
$1,000,000 x = 142857.142857142857 \ldots$ $(2)$
Subtracting equation $(1)$ from equation $(2)$:
$1,000,000 x - x = 142857.142857 \ldots - 0.142857 \ldots$
$999,999 x = 142857$
Solving for $x$:
$x = \frac{142857}{999999}$
Dividing both numerator and denominator by $142857$:
$x = \frac{1}{7}$
Thus,$0.142857142857 \ldots = \frac{1}{7}$.
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