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Please study

Fractions to Decimals before repeating decimal

if you have not already done so.
That knowledge is a prerequisite here.

Terminating Decimals

While expressing a fraction into a decimal by
the division method, if the division comes
to an end after a finite number of steps,
then such a decimal is a Terminating Decimal.

Examples : 1⁄4 = 0.25, 3⁄4 = 0.75, 1⁄16 = 0.0625

Non-Terminating Decimals

While expressing a fraction into a decimal by
the division method, if the division process
continues indefinitely, and zero remainder is
never obtained then such a decimal
is called Non-Terminating Decimal.

Examples :
1⁄13 = 0.076923....,
1⁄17 = 0.05882352.....

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Recurring or Repeating Decimals

If in a decimal, a digit or a set of digits
in the decimal part is repeated continuously,
then such a number is called a Recurring or Repeating Decimal.

The repeating digit(s) are expressed by putting a bar on the set.

Examples :
(i) 2⁄3 = 0.666.... = 0.6
(ii) 5⁄6 = 0.8333.... = 0.83
(iii) 3⁄11 = 0.272727.... = 0.27
(iv) 9⁄37 = 0.243243243.... = 0.243

Pure recurring decimals

A decimal in which all the digits in the decimal part
are repeated, is called a Pure recurring decimal.

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Mixed recurring decimals :

A decimal in which some of the digits in the decimal
part are repeated and the rest are not repeated,
is called a Mixed recurring decimal.

In the Examples given above for repeating decimal,
(i), (iii), (iv) are pure recurring decimals
and (ii) is Mixed recurring decimal.

Pure recurring decimal to fraction

Write the repeated digits only once in the numerator
and take as many nines in the denominator
as is the number of repeating digits.

Examples :
0.6 = 6⁄9 = 2⁄3
0.27 = 27⁄99 = 3⁄11
0.243 = 243⁄999 = 27⁄111 = 9⁄37

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Mixed recurring decimal to fraction : Repeating Decimal

In the numerator, take the difference between the number
formed by all the digits in the decimal part ( taking
repeated digits only once) and the number formed by the
digits which are not repeated.

In the denominator, take the number formed by as many nines
as there are repeating digits followed by as many zeros
as is the number of non-repeating digits.

Examples :
0.83 = (83 - 8)⁄90 = 75⁄90 = 5⁄6
0.307 = (307 - 3)⁄990 = 304⁄990 = 152⁄495
7.536 = 7 + 0.536 = 7 + (536 - 53)⁄900 = 7 + 483⁄900 = 7 + 161⁄300 = 7  161300

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