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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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mathsDecimalsConversion of Decimals

### Conversion of Repeating Decimals to Fractions

This page gives a brief overview of conversion of numbers having repeating digits after decimal point into equivalent fractions.

click on the content to continue..

Convert 1/3 into a decimal. The long-division method to convert the fraction into a decimal is illustrated in the figure.

• 0.3
• 0.333cdots
• 0.333cdots

The answer is "0.333cdots". It is noted that the decimal number does not end and number 3 repeats.

Convert 23/9 into a decimal

• 2.555cdots
• 2.555cdots
• 0.2555cdots

The answer is "2.555cdots".

Convert 371/990 into a decimal.

• 0.37474 cdots
• 0.37474 cdots
• 3.7474cdots

The answer is "0.37474cdots". It is noted that the decimal number does not end and number 7 and 4 repeats.

371/990 = 0.37474cdots

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

•  is 4 repeated? like 0.37474444444cdots

•  is 74 repeated? like 0.37474747474cdots

•  is 37474 repeated? like 0.3747437474cdots

To avoid the confusion the following representation is adapted. The number is given as
0.3bar(74)

The line over 74 represents that 74 is repeated.

Convert 1/3 into a fraction.

• 0.bar(3)
• 0.333cdots
• both the above
• both the above

The answer is "both the above".

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

convert 0.bar(4) into a fraction

• 4/10 = 2/5
• 4/9
• 4/9

The answer is "4/9".

This is explained in the next page.

To convert 0.bar(4) into a fraction, the following steps are used

x=0.bar(4)
10x = 4.bar(4)

Subtracting the two equations
9x = 4
x=4/9

Convert 2.bar(4) into a fraction.

• 2 2/5
• 2 4/9
• 2 4/9

The answer is "2 4/9".

x=2.bar(4)
10x = 24.bar(4)

subtracting the two above
9x = 22
x=22/9
x=2 4/9

convert 2.23bar(43) into a fraction

• 2212/990
• 223/100 + 43/9900
• both the above
• both the above

The answer is "both the above". This is explained in the next page.

To convert 2.23bar(43) into a fraction:

x=2.23bar(43)
100x = 223.43bar(43)
Subtracting the above,
99x = 221.20
990x = 2212
x = 2212/990.

Another method is as follows.
100x = 223 + 0.bar(43)
10000x =223xx100 + 43.bar(43)
Subtracting the two equations
9900x = 223xx (100-1) + 43
x=223/100 + 43/9900

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.bar(3) is represented as 10x-x = 3.bar(3) - 0.bar(3). The value of x is derived as fraction 1/3.

slide-show version coming soon