
Fractions 



Proper and improper fractions 
Mixed numbers 
Converting mixed numbers to
improper fractions 
Converting
improper fractions to mixed numbers 
Equivalent fractions 
Converting and reducing fractions 
Lowest terms 
Comparing fractions 
Decimal fractions 
Adding and subtracting like
fractions 
Adding and subtracting fractions 
Multiplying fractions 
Reciprocal fractions 
Dividing
fractions 
Simplifying complex fractions 





Proper and improper fractions 
A
fraction is an expression representing a part, or several equal parts,
of a unit. 

Thus,
for example in the fraction 3/4 the denominator 4 tells us how
many parts the whole is divided into, and the numerator 3 tells us
how many of those parts we're dealing with. 


When the numerator is
less than the denominator, the fraction is proper, i.e. less than unity. 

When the numerator is greater than or equal to denominator, the fraction is
improper, i.e. greater than unity: 




A fraction greater than 1 can also be written as a
mixed number,
i.e. as the sum of a positive integer and a proper fraction: 


Converting mixed numbers to
improper fractions 
To
convert a mixed number to an improper fraction, multiply the
whole number by the denominator of the fraction, add the
numerator of the fraction to the product and write the sum over
the given denominator. 

Converting
improper fractions to mixed numbers 
To
convert an improper fraction to a mixed number divide the
fraction's numerator by its denominator. The integer part of the
division is the integer part of the mixed number. The remainder
of the division is the numerator of the resulting fraction, as
is shown in the above example. 

Converting and reducing fractions 
Fractions that represent the same number are called
equivalent fractions. 
For any fraction, multiplying the numerator and
denominator by the same nonzero number gives an equivalent fraction. 

A fraction is in
lowest terms when the greatest common factor of its numerator and denominator is 1. 
To
reduce a fraction to lowest terms, divide the numerator and denominator by their
greatest common factor. 

equivalent fractions: 

and 




Comparing fractions 
To compare fractions with the
same denominator, look at their numerators. The larger fraction is the one with the larger numerator. 
If the
numerators of two fractions are the same, the fraction with the
smaller denominator is the larger fraction. 

Example:
To
compare fractions with different denominators, e.g. 4/5 and
7/9, multiply the numerator 
and the denominator of both fractions, the first fraction by the
denominator of the second fraction and the second fraction by the
denominator of the first fraction. We get equivalent fractions with the same
denominators, thus 


Decimal fractions 
Decimal
fractions are vulgar fractions whose denominator is a power of
ten. Or, it is a fraction written in the placevalue notation with base 10
preceded by dot, after which
each successive digit indicates a multiple of the successive
negative powers of 10. 

Example: 



Adding and subtracting like
fractions 
To add or subtract two fractions with
the same denominator, add or subtract the numerators and write the
sum over the common denominator. 

Examples: 



Adding and subtracting fractions 
To add or subtract fractions with different denominators: First find
the least common denominator (the smallest number that can be divided by each denominator). Write equivalent fractions using this
denominator. 
Then add or subtract the fractions. 

Examples: 




Multiplying fractions 
To multiply a whole number by a fraction means to
multiply it by a numerator and to divide a product by a denominator. 


When two fractions are
multiplied, the result is a fraction with a numerator that is the product of the fractions' numerators and
a denominator that is the product of the fractions' denominators. 



Examples: 



Reciprocal fractions 
The fractions a/b
and b/a
are reciprocals. Reciprocal fractions are two fractions
whose product is 1. 




Dividing
fractions 
To divide a number by a fraction, multiply the number by the
reciprocal of the fraction. 
To divide mixed numbers, you should
always convert them to improper fractions, then multiply the first
number by the reciprocal of the second. 
To divide a fraction by a
whole number, write the whole number as an improper fraction with a denominator of
1, then multiply as fractions. 



Simplifying complex fractions 
Complex
fractions or compound fractions are fractions in which the
numerator and/or denominator contain fractions. 
To simplify complex fractions, change the complex fraction into a division problem, i.e., divide the numerator by
the denominator. 

Examples: 















Beginning
Algebra Contents A 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 