In this chapter we will discuss important properties of multiplication of rational numbers with solved examples.

These properties are important as they will help you to solve different algebraic expression faster.

## Multiplication property of rational numbers

Here we have described following properties;

(a) Closure properties

(b) Commutative property

(c) Associative property

(d) Distributive property

(e) Multiplication with 0

(f) Multiplicative inverse

Closure property of Multiplication of Rational numbers

The property states that ” multiplication of two rational number always results in another rational number “

For example;

Let 1/5 and 2/7 are two rational numbers.

It’s multiplication is given as;

\mathtt{\Longrightarrow \ \frac{1}{5} \times \frac{2}{7}}\\\ \\ \mathtt{\Longrightarrow \frac{2}{35}}

Here the final result 2/35 is also a rational number.

### Commutative property of multiplication of rational number

The property states that “** in multiplication, changing the order of numbers will not affect the final result** “.

Let a/b and c/d be the given rational numbers.

Then according to commutative property.

\mathtt{\Longrightarrow \frac{a}{b} \times \frac{c}{d} \ =\ \frac{c}{d} \times \frac{a}{b}} **For example;**

Let 3/5 and 11/7 are the two rational numbers.

Multiplying the numbers we get;

\mathtt{\Longrightarrow \frac{3}{5} \times \frac{11}{7} \ =\ \frac{33}{35}}

Now **interchange the position of numbers and multiply**.

\mathtt{\Longrightarrow \frac{11}{7} \times \frac{3}{5} \ =\ \frac{33}{35}}

Note that even after interchanging the position of numbers we get the same result.

Hence, commutative property applies in multiplication of rational numbers.

### Associative Property of Multiplication of Rational number

If three of more rational numbers are present in multiplication then **changing the association (or group) will not change the end result**.

If a/b, c/d & e/f are the three rational numbers.

Then according to associative property.

\mathtt{\left(\frac{a}{b} \times \frac{c}{d}\right) \times \frac{e}{f} \ =\ \frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f}\right)} **For example;**

Let 1/2 , 4/5 and 13/9 are the rational numbers.

Multiplying the numbers we get;

\mathtt{\Longrightarrow \left(\frac{1}{2} \times \frac{4}{5}\right) \times \frac{13}{9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{10} \times \frac{13}{9}}\\\ \\ \mathtt{\Longrightarrow \frac{52}{90}}

Now multiplying the number by forming different group.

\mathtt{\Longrightarrow \frac{1}{2} \times \left(\frac{4}{5} \times \frac{13}{9}\right)}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2} \times \frac{52}{45}}\\\ \\ \mathtt{\Longrightarrow \frac{52}{90}}

Note that changing the group of multiplication results in same numbers.

Hence, associative property works in multiplication of rational numbers.

### Distributive property of Multiplication of Rational number

According to the property the **multiplication with sum of two rational number is equal to the sum of multiplication of rational numbers.**

The distributive property can be represented as;**a/b ( c/d + e/f ) = a/b . c/d + a/b .e/f**

Let us understand the property with yeh help of example.

Let 1/2 , 3/5 and 7/5 are the three rational numbers.

According to distributive property of multiplication.

\mathtt{\frac{1}{2} .\left(\frac{3}{5} +\frac{7}{5}\right) \ =\frac{1}{2} .\frac{3}{5} +\frac{1}{2} .\frac{7}{5}}\\\ \\ \mathtt{\frac{1}{2} .\frac{10}{5} =\frac{3}{10} +\frac{7}{10}}\\\ \\ \mathtt{\frac{10}{10} =\frac{10}{10}}\\\ \\ \mathtt{1=1}

Since, LHS = RHS.

The distributive property is verified.

### Multiplication property of rational number with 0

Any number multiplied with 0 results in 0.

If a/b be the given rational number. The multiplication with number 0 will result in 0.

\mathtt{\frac{a}{b} \times 0=0}

### Multiplicative inverse of rational number

Multiplicative inverse is a **number which when multiplied with original numbers gives result 1.**

The multiplicative inverse of rational number can be found by **taking its reciprocal.**

If a/b is the given rational number.

Then its multiplicative inverse will be b/a.

If we multiply the original number with its inverse, we will get 1 as a solution.

\mathtt{\Longrightarrow \ \frac{a}{b} \times \frac{b}{a}}\\\ \\ \mathtt{\Longrightarrow \ 1} **For example;**

Let the given rational number is -4/5

The multiplicative inverse is -5/4.

Multiplying the original number & its multiplicative inverse, we get;

\mathtt{\Longrightarrow \ \frac{-4}{5} \times \frac{-5}{4}}\\\ \\ \mathtt{\Longrightarrow \ 1}