Recurring Decimal-Converting a Mixed Recurring Decimal Into Vulgar Fraction

First of all, a mixed decimal number is said to those numbers which have fractional parts in them. The recurring word is derived from a Latin word that suggests a sense of ‘return to’. In mathematical consideration, recurring refers to the regular or periodic repetition of some digit. In this regard, a recurring decimal is a decimal fraction in which a digit or group of digits is repeated indefinitely. This article is based on the “conversion of particular mixed recurring decimals to vulgar fractions”.

Concept of decimal

A decimal can be defined as the number in which a numeral part and its decimal part can be easily separated or identified by the decimal point. In general, a decimal number can be expressed in decimal form. A dot “.” is usually used to indicate a decimal point. The digits before the decimal point have higher numeral values as compared to the digits after decimal that have relatively smaller values. It can be expressed with an example as 15.65, where 15 is a whole number having its significant value and digits after the decimal point, 48 is considered as a decimal part that has smaller values. 

Concept of fraction

A fraction can be represented as a part of a whole or any number of equal parts. It is justified by examples; ⅕, 80/100, where the first fraction denotes 1 part of whole 5 and secondly, 80 parts of whole 100 equal parts. Conversion of mixed recurring decimals to different forms of fractions requires concepts of a vulgar fraction.

Concept of mixed recurring decimal

A recurring decimal can usually be defined as the decimal number in which a digit or group of digits repeats forever. It is a rational number that generally contains a pattern of digits that repeats indefinitely after the decimal point. Generally, if a digit or group of digits repeats forever after decimal point then repeated or recurring parts can be separated from the whole number by using a decimal point. It is also found, in general, that in recurring decimals, digits are repeated at equal intervals after decimal points. It can be justified by providing some suitable examples as:

• ⅓ = 0.3333….. (Where a single digit “3” repeats forever)
• 1/7 = 0.142857142857…….. (Where a group of digits, “142857” repeats infinitely)
• 77/600 = 0.1283333333……. (Where 3 repeats forever)

On another hand, a digit or group of digits found to be repeating is represented by placing dots over the first and last digits of recurring patterns. In some cases, a line is also given over a digit or group of digits to represent its repeating nature. 

Converting a mixed recurring decimal into a fraction

A mixed recurring decimal is a decimal number in which some digit or digits after the decimal is fixed and after that digit, some digits are repeated forever. Conversion of different mixed recurring decimals to different forms of vulgar fractions is required following some specific steps.

• First of all, write a mixed recurring decimal number by removing bars from the top and then equate it with any variable x.
• Figure out several digits after decimal that does not contain a bar or are repeated.
• Multiply both sides by 10b, provided b is assumed as a non-recurring decimal digit, marked as equation (i). 
• Write repeating decimal at least two times
• Again multiply both sides by 10n if n-digits have a bar, marked as equation (ii). 
• Subtract equation (i) from (ii)
• Then, at last, divide both sides of equation by the coefficient of x.

The above steps can be justified by an example of conversion of  particular mixed recurring decimals to fractions: 0.17̅

:- x = 0.1777

10x = 1.77777 ……………………….. (i)

10 x 10x = 1.777777…….. x 10

100x = 17.77777………………………(ii)

Subtracting (i) from (ii)

100x – 10 x = 17.77… – 1.777….

90x = 16

x = 16/90

x = 8/42 

Conclusion

It is to be concluded that a decimal number that has the repetition of some digits after a decimal point is considered a mixed recurring decimal. Conversion of mixed recurring decimals to different types of fractions requires straight processes that help in better-solving problems related to mix recurring decimals. It is observed that mixed recurring decimals are a pure periodic form of decimal numbers. It refers to the repetition of digits or digits after a decimal point. It is also found that repeated digits of a recurring decimal number can be represented using a bar sign ( ̅ ) on the top of repeating digits or digits. Recurring decimal can also be applied to calculating weight and height and dealing with money apart from solving mathematical problems.