Mathematical equations have their own manipulative principles. These principles or properties help us to solve such equations. Basically, there are three properties which outline the backbone of mathematics. They are:

• Associative property

• Commutative property

• Distributive property

### ASSOCIATIVE PROPERTY

Associative as the name implies, means grouping. Origin of the term associative is from the word “associate”. Basic mathematical operations which can be performed using associate property are addition and multiplication. This is normally applicable to more than 2 numbers.

As in case of Commutative property, the order of grouping does not matter in Associative property. It will not alter the result. The grouping of number can be done in parenthesis irrespective of the order of terms. Thus, the associative law expresses that it doesn’t make a difference which part of the operation is carried out first; the answer will be the same.

**Associative property for addition:** Addition follows associative property i.e. regardless of how numbers are parenthesized the final sum of the numbers will be same. In associative property of addition states that

\((x~+~y)~+~z\)

Let us say, we want to add \(5~+~10~+~4\)

\(⇒~~~~~~~~~(5~+~10)~+~4\)

Now, let’s regroup the terms like \(10\)

\(⇒~~~~~5~+~(10~+~4)\)

Yes, it can be seen that the sum in both case are same. This is associative property of addition.

**Associative property for Multiplication:** Rule for associative property of multiplication is

\((xy)~ z\)

On solving \(5 \times 3 \times 2\)

\(⇒ ~~~~~~~~(5~\times~3)~\times~2\)

After regrouping,

\(⇒ ~~~~~~~~5~\times~(3~\times~2)\)

Products will be the same.

Thus, addition and multiplication are associative in nature but subtraction and division are not associative.

For example, divide \(100~ ÷ ~10~ ÷~ 5\)

\(⇒~~~~~~~~(100~ ÷ ~10)~ ÷ ~5 ~≠~100~÷~(10 ~÷ ~5)\)

\(⇒~~~~(10)~ ÷~ 5~ ≠ ~100~ ÷ ~(2)\)

\(⇒~~~ 2~≠~50\)

Subtract, \(3-2-1\)

\(⇒ ~~~~~(3~-~2)~-~1~≠~3~ -~(2~-~1)\)

\(⇒~~~~~~~(1)~ – ~1~ ≠ ~3~-~ (1)\)

\(⇒~~~~0~≠~2\)

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