Proper, Improper, Mixed, Complex, Continued, Decimal Fraction: Formula and Tricks with Examples

A fraction is known as a rational number and written in the form of p/q where p and q are integers and q ≠ 0.

The lower number ‘q’ is known as denominator and the upper number ‘p’ is known as numerator.

Type of Fractions

Proper Fraction:

The fraction in which numerator is less the denominator is e.g. 3/11,15/17,21/43 etc. are called a proper fraction.

Improper fraction:

The fraction in which numerator is greater than the denominator e.g. 11/7,17/13,18/14 are is called improper fraction.

Mixed fraction:

Mixed fraction is a composition of fraction and whole number. 

Complex fraction:

A complex fraction is that fraction in which numerator or denominator or both are fractions.

Like Fractions

those fractions, whose denominators are same, are known as like fractions. e.g.7/19,1/19 are like fractions.

Unlike Fractions

whose fractions, whose denominators are different ,are known as unlike fractions. e.g. 11/13 ,4/17 etc. are unlike fractions.

Inverse (Reciprocal) Fraction

It we inverse the numerator and the denominator of a fraction, then the resultant fraction will be the inverse (reciprocal)fraction of the original fraction. eg. If fraction =3/8,
then its inverse (reciprocal) fraction =8/3

Continued fraction:

 Fractions which contain addition or subtraction of fractions or a series of fractions generally in denominator (sometimes in numerator also) are called continued fractions.

It is also defined as a fraction whose numerator is an integer and whose denominator is an integer plus a fraction.

Decimal fraction:

The fraction whose denominator is 10 or its higher power, is called a decimal fraction.

e.g. 3/10,7/1000 etc.

Type of  Decimal Fractions

(i) Recurring Decimal Fraction

A fraction in which digits after decimal are repeated in a manner is called a recurring decimal fraction. To represent it, draw a line on recurring digits.
e.g. 1/3=0.333 …= 0.3

(ii) Pure Recurring Decimal Fraction

A fraction in which all digits after decimal are repeated in a manner, is called a pure recurring decimal fraction.
e.g. 0.476 etc.
To convert these fractions into simple fractions, write all repeated digits once in numerator and place as many nines in the denominator as the number of digits repeating.
e.g. 0.57=57/99

(iii) Mixed Recurring Decimal Fraction


A fraction in which some digits are repeated and some are not repeated after decimal, is called as mixed recurring decimal fraction.
e-g 1.369898.. = 1.3698

Comparison of Fractions

If the denominators of all the given fractions are equal then the fraction of greater numerator will be the greater fraction.

If the numerators of all the given fractions are equal then the fraction of smaller denominator will be greater fraction.

When numerator is greater than denominator and the differences of numerator and denominator are equal, then the fraction of smaller numerator will be the greater faction.

Quicker Method (Cross Multiplication)

This is a short-cut method to compare fractions. Using this method we can compare all types of fractions.

The fraction whose numerator is in the greater product is greater.

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