Table of Contents

If you ever need assistance solving the Rational Expressions Involving Division you can have a look at the Properties of Division of Rational Numbers prevailing. Make use of the Rational Numbers Division Properties to simplify the expressions during your calculations. Each of the Properties is explained in detail taking enough examples. To gain adequate knowledge and solve related problems on your own.

### Closure Property of Division of Rational Numbers

Rational Numbers are closed under Division Except for Zero. Let us assume two rational numbers a/b, c/d where c/d ≠0 then (a/b ÷ c/d) is also a Rational Number.

**Example**

(i) 2/3÷ 4/5 = 2/3*5/4

= (2*5)/3*4

= 10/12

Therefore 2/3÷ 4/5 i.e. 10/12 is also a Rational Number.

(ii) 3/7 ÷ -5/4

Clearly -5/4 ≠0

= 3/7*-4/5

= 3*-4/7*5

= -12/35

Therefore, 3/7 ÷ -5/4 i.e. -12/35 is also a Rational Number.

Closure Property is true for division except for zero.

### Commutative Property of Division of Rational Numbers

Division of Rational Numbers isn’t commutative. Consider two rational number a/b, c/d then a/b÷c/d ≠ c/d÷a/b

**Example**

1/4÷3/2 = 1/4*2/3 = 1*2/4*3 = 2/12 = 1/6

3/2÷1/4 = 3/2*4/1 = 3*4/2*1 = 12/2 = 6

Therefore, 1/4÷3/2 ≠3/2÷1/4

Thus, Commutative Property is not true for Division.

### Associative Property of Division of Rational Numbers

Usually, the Division of Rational Numbers doesn’t obey the Associative Property. Let us consider a/b, c/d, e/f be three Rational Numbers then a/b ÷ (c/d ÷ e/f) ≠ (a/b ÷ c/d) ÷ e/f

2/5÷(4/5 ÷ 1/9) = 2/5÷(4/5*9/1) = 2/5÷(36/5) = 2/5*5/36 = 2*5/5*36 = 10/180 = 1/18

(2/5÷4/5) ÷ 1/9 = (2/5*5/4)÷ 1/9 = (2*5/5*4) ÷ 1/9 = 10/20÷ 1/9 = 10/20*9/1 = 10*9/20*1 = 90/20 = 9/2

2/5÷(4/5 ÷ 1/9) ≠ (2/5÷4/5) ÷ 1/9

### Property of 1 of Division of Rational Numbers

For every rational number a/b we have (a/b÷1) = a/b

**Example**

(i) 12/5÷1 = 12/5

(ii) 4/-7÷1= 4/-7

(iii) 3/9÷1= 3/9

### Property 1

For any non zero rational number a/b we have (a/b÷a/b) =1

**Example**

(i) 3/4÷3/4 = 3/4*4/3 = 3*4/4*3 = 12/12 =1

(ii) 4/7÷4/7 = 4/7*7/4 = 4*7/7*4 = 28/28 = 1