Associative Property Formulas | List of Associative Property Formulas You Should Know - BYJUS

# Associative Property Formulas

Associative Property is one of the basic properties of Mathematics. Here we will learn the Associative Property Formula and its application in problems based on multiplication and addition operations....Read MoreRead Less

### What is Associative Property?

The associative property states that changing the grouping in which 3 or more numbers are added or multiplied does not alter the result. Here grouping refers to the way the numbers in the equation are placed within the parentheses.

This property can be applied to complex equations based on addition or multiplication operations, which makes the calculation easier.

### Formula for the Associative Property of Addition

The formula for the associative property of addition is:

$$(a + b) + c = a + (b + c)$$

Here, a, b and c are the 3 numbers being added. ### Formula for the Associative Property of Multiplication

The formula for the associative property of multiplication is:

$$(a \times b) \times c = a \times (b \times c)$$ ### Solved Examples

Example 1: To find the sum of $$78 + (45 +23)$$, Mark added $$78$$ and $$45$$ first and then added $$23$$. Which property did Mark use and why?

Solution:

The property used in this scenario is the associative property of addition. Usually, in such cases the expressions within the parenthesis are simplified first. However, in the question there is only addition as an operation.

Hence, the order in which the numbers are added does not change the value of the final result. This means that Mark will obtain the right result regardless of the order used in this addition operation.

Example 2: If $$8 \times 30 = 240$$, then find the value of $$16 \times 15$$.

Solution:

$$8 \times 30 = 240$$                       [Given]

$$8 \times (15 \times 2) = 240$$             [Write $$30$$ as $$15 \times 2$$]

$$(8 \times 2) \times 15 = 240$$             [Associative Property of Multiplication]

$$16\times 15 = 240$$                     [Multiply $$8$$ by $$2$$]

Hence, $$16\times 15 = 240$$.

Example 3: Find $$12 + 104$$.

Solution:

Write $$104$$ as a sum of $$100$$ and $$4$$.

$$12 + 104$$

$$= 12 + (100 + 4)$$

$$= (12 + 4) + 100$$             [Apply associative property of addition]

$$= 16 + 100$$                       [Add]

$$= 116$$                               [Add]

Hence, $$12 + 4$$ is $$116$$.

Example 4: Annie bought 10 packets of glossy ribbons. Each packet had $$8$$ ribbons and the cost of each ribbon was $$\ 2$$. How much did Annie spend in total?

Solution:

Amount of money spent by Annie $$=$$ Total number of ribbons $$\times$$ Cost of each ribbon

Total number of ribbons $$=$$ Number of packets $$\times$$ Number of ribbons in each packet

So,

Amount of money spent by Annie $$=$$ (Number of packets $$\times$$ Number of ribbons in each packet) $$\times$$ Cost of each ribbon

Substituting the values,

Amount of money spent by Annie $$= (10 \times 8) \times 2$$

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 10 \times (8 \times 2)$$              [Apply Associative Property of Multiplication]

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 10 \times 16$$                      [Multiply]

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 160$$                             [Multiply]

Hence, Annie spent a total of $$\ 160$$ on the ribbons.