How do you convert repeating decimals into fractions?

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How do you convert repeating decimals into fractions?

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Answer 1 · 2018-03-30T11:42:00

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Explanation:

For example,

Basic repeating decimals

$0 . ¯ ¯¯ ¯ 18$ $0.bar(18)$

$0 . ¯ ¯¯ ¯ 18 = \frac{18}{99}$ $0.bar(18)=18/99$
$0 . ¯ ¯¯ ¯ 18 = \frac{2}{11}$ $color(white)(0.bar(18))=2/11$

$0 . ¯ ¯¯¯¯ ¯ 123$ $0.bar(123)$

$0 . ¯ ¯¯¯¯ ¯ 123 = \frac{123}{999}$ $0.bar(123)=123/999$
$0 . ¯ ¯¯¯¯ ¯ 123 = \frac{41}{333}$ $color(white)(0.bar(123))=41/333$

$0 . ¯ ¯¯¯¯¯¯ ¯ 4680$ $0.bar(4680$

$0 . ¯ ¯¯¯¯¯¯ ¯ 4680 = \frac{4680}{9999}$ $0.bar(4680)=4680/9999$
$0 . ¯ ¯¯¯¯¯¯ ¯ 4680 = \frac{52}{1111}$ $color(white)(0.bar(4680))=52/1111$

Not-so-basic repeating decimals

$1.0 ¯ ¯¯ ¯ 42$ $1.0bar(42)$

Let $x = 1.0 ¯ ¯¯ ¯ 42$ $x=1.0bar(42)$

$1000 x - 10 x = 1042 . ¯ ¯¯ ¯ 42 - 10 . ¯ ¯¯ ¯ 42$ $1000x-10x=1042.bar(42)-10.bar(42)$
$\times \times / 990 x = 1032$ $color(white)(xxxx//)990x=1032$
$\times \times \times x . x = \frac{1032}{990}$ $color(white)(xxxxxxx.)x=1032/990$
$\times \times x 1.0 ¯ ¯¯ ¯ 42 = \frac{1032}{990}$ $color(white)(xxxxx)1.0bar(42)=1032/990$
$\times \times x 1.0 ¯ ¯¯ ¯ 42 = \frac{172}{165}$ $color(white)(xxxxx1.0bar(42))=172/165$

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How do you convert repeating decimals into fractions?

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