Decimal Representation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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Convert

0.\overline{142857}

to a fraction in simplest form.

1
Let $x = 0.142857142857\ldots$ The repeating block has $6$ digits, so multiply by $10^6 = 1{,}000{,}000$ : $1{,}000{,}000x = 142857.\overline{142857}$ .
2
Subtract: $1{,}000{,}000x - x = 142857$ , so $999{,}999x = 142857$ .
3
Solve: $x = \dfrac{142857}{999999}$ . Simplify by dividing both by $142857$ : $x = \dfrac{1}{7}$ .

Answer

0.\overline{142857} = \dfrac{1}{7}

Every repeating decimal is rational. To convert, multiply by

10^n

(where

n

is the length of the repeating block), subtract the original equation to eliminate the repeating part, then solve and simplify.

About Decimal Representation

Writing fractions as digits to the right of a decimal point, using place values of tenths, hundredths, thousandths, etc.

Convert [formula] to a decimal by long division, and determine whether it terminates or repeats.

Write each fraction as a decimal and classify as terminating or repeating: (a) [formula], (b) [formu

Convert [formula] to a fraction in simplest form. (Note: only the [formula] repeats.)