The properties of fractions include the same properties as the set of real numbers, which are Closure, Associative, Commutative, Identity and Inverse for the operations of addition and multiplication as well as the Distributive property.  Note: Subtraction is addition with negative values and division is multiplication by taking one over the value.  

The Closure property states that the result of the addition or multiplication of any two numbers in a set will produce another number in the same set. Since the sum of any two fractions results in a fraction, the property of Closure exists for the set of fractions for the operation of addition and multiplication. Note: 3 can be expressed as a fraction written as . Symbolically, for any a, b, c and d such that b or d 0;

The Associative property for the operation of addition states that the order in which we add three whole numbers does not matter since the result will always be the same sum. This property also works for the operation of multiplication. Therefore, for any values a, b, c, d, e and f such that d, e or f 0;

The Associative property is useful for computations. For example, the sum of three fractions is calculated by adding left to right following the order of operations: . However, it can also be found using the Associative property of addition: where regrouping not only gives the same result but is slightly easier to compute. The results are the same because the sum is equivalent no matter how the numbers are grouped.

The Commutative property provides the opportunity to change the order of terms under the operations of addition and multiplication. The order in which we add two fractions does not change the result. For example,

since

.

Likewise,

           

since ,


This seems trivial but it is a powerful property for algebra that allows one to simplify expressions.
The expression,
 can be written as
since,

by the Commutative property. Using the order of operations and combining like terms yields, 
2x+3

Symbolically, for any values a, b, c and d such that c or d 0;

The Identity property of addition identifies the numbers that do not change the value of a fraction under the operations of addition and multiplication. The additive identity is zero. For example, when you add zero to any fraction, the result is the original fraction,

The additive identity is used later in Algebra 1 when completing the square. 

The multiplicative identity is one. Any fraction multiplied by one will result in the original fraction with the exception of zero,
                         
The multiplicative identity is used when finding a common denominator. For example,
                     .
The use of the multiplicative identity becomes apparent in Algebra 1 when working with rational expressions.

Symbolically, for any values a and b;

.
 

The Inverse property of addition states that the sum of the original fraction and its inverse is zero. Finding an additive inverse is important when solving an equation. For example,

In this case the second line is applying the additive inverse property to isolate the variable x. The additive inverse of 2 is –2.

The Inverse property of multiplication states that the product of the original fraction and its inverse is one. Finding a multiplicative inverse is important when solving an equation. For example,

In this case the second line is applying the multiplicative inverse property to isolate the variable x. The multiplicative inverse of 2 is .

Symbolically, for any value a and b;

The Distributive property is a useful tool for the multiplication of rational numbers.  In the real world we frequently use this property for mental multiplication. For example, when finding a discount at a store such as 30% off the original price of $40 one might take,

Symbolically, for any values a, b, c, d, e and f  such that d, e or f 0;