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Conversion of Repeating Decimals to Fractions


    what you'll learn...

overview

This page covers

 •  Some fractions result in decimals with repeating digits.

 •  Representation of Repeating Decimals.

 •  Conversion of Repeating Decimals to equivalent Fractions : A decimal has two parts, a non-repeating part at the beginning and repetitive part. With some simple arithmetics, equivalent fraction is derived.

eg: 0.3¯ is represented as 10x-x=3.3¯-0.3¯. The value of x is derived as fraction 13.

never ending...

repeating

Convert 13 into a decimal. The answer is "0.333". It is noted that the decimal number does not end and number 3 repeats. The long-division method to convert the fraction into a decimal is illustrated in the figure.


Convert 239 into a decimal.
The answer is "2.555".
Note: is used to represent that the digits are repeating.


Convert 371990 into a decimal.
The answer is "0.37474". It is noted that the decimal number does not end and number 7 and 4 repeats.

representing the repetition

371990=0.37474

In the above representation with , it is not clear which part of the digits are repeating

 •  is 4 repeated? like 0.37474444444

 •  is 74 repeated? like 0.37474747474

 •  is 37474 repeated? like 0.3747437474

To avoid the confusion the following representation is adapted. The number is given as
0.374¯

The line over 74 represents that 74 is repeated.


Convert 13 into a fraction.
The answer is 0.3¯

Representation of Repeating Decimals : The repetitive pattern in decimal digits is represented with an over-line.

convert back to fraction

How do we convert 0.4¯ into a fraction?

To convert 0.4¯ into a fraction, the following steps are used

x=0.4¯
10x=4.4¯

Subtracting the two equations
9x=4
x=49

examples

Converting 2.4¯ into a fraction.
x=2.4¯
10x=24.4¯

subtracting the two above
9x=22
x=229
x=249


To convert 2.2343¯ into a fraction:

x=2.2343¯
100x=223.4343¯
Subtracting the above,
99x=221.20
990x=2212
x=2212990.

Another method is as follows.
100x=223+0.43¯
10000x=223×100+43.43¯
Subtracting the two equations
9900x=223×(100-1)+43
x=223100+439900

summary

Representation of Repeating Decimals : The repetitive pattern in decimal digits is represented with an over-line.

Conversion of Repeating Decimals to equivalent Fractions : A decimal has two parts, a non-repeating part at the beginning and repetitive part. With some simple arithmetics, equivalent fraction is derived.

eg: 0.3¯ is represented as 10x-x=3.3¯-0.3¯. The value of x is derived as fraction 13.

Outline

The outline of material to learn "decimals" is as follows.

Note: goto detailed outline of Decimals

•   Decimals - Introduction

    →   Decimals as Standard form of Fractions

    →   Expanded form of Decimals

•   Decimals - Conversion

    →   Conversion between decimals and fractions

    →   Repeating decimals

    →   Irrational Numbers

•   Decimals - Arithmetics

    →   Comparing decimals

    →   Addition & Subtraction

    →   Multiplication

    →   Division

•   Decimals - Expressions

    →   Expression Simplification

    →   PEMA / BOMA