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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.



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Convert `1/3` into a decimal.repeating 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`.

                            
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