# Properties of Division

In this post we will understand some important properties of division.

These properties will help you solve problems of advanced algebra, calculus and other Math topics in less time.

Make sure to remember the properties as the topic is directly asked in grade 6 Math exams.

## List of Important Division Properties

### Identity Property of Division

The property says that if you divide any number by 1 you will get the same number.

Where A can be any possible real number.

But why this property is called “Identity” Property?

Because after division, the identity of the number remain unchanged.

Identity of number is preserved as we have the same number before and after the division operation.

Examples of Identity Property of Division

Example 01
Divide 42 by 1

Hence, on dividing 42 with 1 you will get 42 as solution

Example 02
Divide 983 with 1

Hence, division of 983 with 1 results in 983 as solution

### Zero Property of Division

There are two concepts in the property:
(a) Dividing zero by any number
(b) Dividing any number by zero

(a) Dividing zero by any number
Division of zero with any number results in zero.

This concept is expressed as:

Where A can be any real number.

Given below are some examples:
\mathtt{( a) \ 0\ \div \ 42\ =\ 0}\\\ \\ \mathtt{( b) \ 0\ \div \ 1256\ =\ 0}

(b) Dividing any number by zero
You cannot divide any number with zero.

In Math, Zero means nothing.
Dividing number with nothing will not give you any result.

Technically you can say that dividing any number with zero results in infinity.

### Division by Itself

Division of a number with same number results in number 1.

The property is expressed as:

Where A can be any possible real number except 0.

Examples
(a) 65 ÷ 65

(b) 954 ÷ 954

Conclusion
Division of number with itself results in number 1

### Division property of Equality

According to the property, in a given balanced equation if you divide the number on both sides, the equation will still remain balanced and valid.

Let the given equation is:
6 = 6

Divide both sides by 3, we get:
\mathtt{\frac{6}{3} \ =\ \frac{6}{3}}\\\ \\ \mathtt{2\ =\ 2}\\\ \\

Generally the property can be expressed as:

Note:
Remember to divide the number on both sides otherwise the equation will not remain valid.

Division property of Equality examples

Example 01
Find the value of x in given equation
3x = 60

Divide both the sides by 3
\mathtt{\frac{3x}{3} \ =\ \frac{60}{3}}\\\ \\ \mathtt{x\ =\ 20}

Hence, the value of x is 20

Example 02
Fine the value of x in given equation
4x – 4 = 36

Add 4 on both sides of equation
4x – 4 + 4 = 36 + 4
4x = 40

Divide both sides by 40
\mathtt{\frac{4x}{4} \ =\ \frac{40}{4}}\\\ \\ \mathtt{x\ =\ 10}

Hence, value of x is 10

### Distributive Property of Division

Distributive property helps to reduce complex equation into simple one.

The expression for the property is given as:

Property Verification
Let A = 2, B = 4 and C = 6

Calculating ( B + C ) / A
⟹ ( 4 + 6 ) / A
⟹ 10 / 2
⟹ 5

Now Calculate B/A + C/A
⟹ 4/2 + 6/2
⟹2 + 3
⟹ 5

Hence, ( B + C ) / A = B/A + C/A

### Division Algorithm

When you divide whole number a with whole number b, the following relation occurs:

Dividend = (Divisor x Quotient) + Remainder

Where:
Dividend: The number to be divided is called divided. Here a is dividend.

Divisor: The number by which dividend is divided is called divisor. Here b is divisor.

Quotient: The solution of the division is called quotient. Here q is the quotient.

Remainder: The number leftover after the division is called remainder. Here r is remainder.

Verification of Division Algorithm
Divide number 60 by 2

Observe the below division

Here:
Dividend = 60
Divisor = 2
Quotient (q) = 30
Remainder (r) = 0

Applying the division algorithm

Dividend = (Divisor x Quotient) + Remainder

60 = (2 x 30) + 0

60 = 60

L.H.S = R.H.S
Hence the division algorithm is verified.

### Division and Multiplication in algebraic equation

If in algebraic equation a ÷ b = c
then, a = b x c is also true.

This property is very helpful to solve algebraic equation fast.
When you are giving competition exams like American Regions Mathematics League (ARML), SAT etc. solving questions using conventional will not let you go far. Hence, you the shortcuts to have a good score.

Verification of the property

\mathtt{Its\ given\ that:}\\\ \\ \mathtt{\frac{a}{b} \ =\ c}\\\ \\ \mathtt{Multiply\ b\ on\ both\ sides}\\\ \\ \mathtt{\frac{a\times b}{b} \ =\ c\ \times \ b}\\\ \\ \mathtt{a\ =\ c\ \times \ b}\\\ \\

How to remember the property?

Just remember that division becomes multiplication while moving to other side of the algebraic equation.

Property Examples

\mathtt{( a) \ \frac{5}{3} \ y\ =\ 2}\\\ \\ \mathtt{5y\ =\ 2\ \times \ 3}\\\ \\ \mathtt{5y\ =\ 6}\

\mathtt{( b) \ \frac{12}{11} \ y\ =\ 4}\\\ \\ \mathtt{12y\ =\ 4\ \times \ 11}\\\ \\ \mathtt{12y\ =\ 44}

## Frequently asked questions- Property of Division

Will Commutative property works in division?

Commutative property says that changing the order of number in math operation will not affect the end result.

This is not the case for division.

Consider the below case:
a ÷ b ÷ c

If you change the order of numbers like b ÷ c ÷ a, the end result would be completely different.

Conclusion: Commutative property does not work in division.

Does Associative property works in Division?

No!!

Associative property says that forming different group will not affect the end result.

But such is not the case with division.

For example:
The result of (A ÷ B) ÷ C is not same as A ÷ (B ÷ C)

Hence in division, associative property does not work.

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