### Decimal Fraction

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Definitions:

1) Fractions: fraction is an expression that indicates the quotient of two quantities.

Examples of fractions:

etc.

A fraction has two parts, Numerator and Denominator. Numerator is the number at the top of the fraction and denominator is the number at the bottom of the fraction.(denominator can not be equal to zero).

2) Decimal Fraction: A decimal fraction is a fraction in which denominator is an integer power of ten.

Example:  =  0.17

3) Generally, a zero is added to the left of the decimal point if a whole number is absent to draw the attention to the fact that the number is a decimal fraction.

Example:  .01 should be written as  0.01

Conversions:

1) From decimal to fraction : Put 1 in the denominator under the decimal point and add as many zeros as is the number of digits after the decimal point. Now, remove the decimal point and reduce the fraction to its lowest terms.

Example 0.5 =

Recurring decimal: If a number or a set of numbers is repeated continuously in a decimal fraction, it is called a recurring decimal.

A repeating decimal is denoted by placing a horizontal line above the repeated numerals.

Example:  = 0.333333….. here 3 is repeating continuously.

Representation =

Types of recurring decimals:

1) Pure Recurring Decimals

Pure Recurring Decimal is a decimal fraction in which all the figures after the decimal point are repeated.

= 0.333333…..  =

2) Mixed recurring decimals:

Mixed Recurring Decimal is a decimal fraction in which some figures are not repeated and some figures are repeated.

Example:  = 0.58333...  =

Convert a Pure Recurring Decimal into Vulgar Fraction :To convert a pure recurring decimal into a vulgar fraction

(1) Remove the number left to the decimal point, if any.

(2) Write the repeated figures only once in the numerator without the decimal point.

(3) Write as many nines in the denominator as the number of repeating figures.

(4) Add the number removed in step 1(if any) with the fraction obtained in the above

Example :

Convert a Mixed Recurring Decimal into Vulgar Fraction:  To convert a mixed recurring decimal into a vulgar fraction

(1) Remove the number left to the decimal point, if any.

(2) Numerator is the difference between the number formed by all the digits (taking repeated digits only once) and that formed by the digits which are not repeated.

(3) Denominator is the number formed by taking as many nines as the number of repeating figures followed by as many zeros as the number of non-repeating digits.

(4) Add the number removed in step 1(if any) with the fraction obtained in the above steps.

Example :

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