What are the Properties of Addition and Multiplication? (Types & Examples) - BYJUS

# Properties of Addition and Multiplication

Addition and multiplication are two of the four basic operations in math. Addition and multiplication are related to each other; operations that involve repeated addition can be simplified using multiplication. Learn some interesting properties of addition and multiplication that will help you solve problems easily....Read MoreRead Less

## Properties of Addition and Multiplication

In simple terms, addition is a concept that involves combining or adding things together. It is a mathematical operation that involves action with numbers in which we add numbers together.

## What is Multiplication?

When you combine equal groups of objects, multiplication gives you the total number of objects. Multiplication of whole numbers is nothing but the repetitive addition of that whole number to itself. For example, $$2+2+2+2+2=2\times~5$$ or 10

Let us have a look at some of the properties related to addition and multiplication and let us explore if they can be applied to algebraic expressions.

## What are Properties of Addition and Multiplication?

Equivalent expressions are those that produce the same number for any value of each variable. A student can use the commutative and the associative properties to write equivalent expressions.

## Here are the Properties for Addition and Multiplication

1. The commutative property According to the commutative property of multiplication, the order in which numbers or terms of an algebraic expression are multiplied or added, does not affect the final product or sum. For Multiplication:
• Algebraic Application:
The commutative property for multiplication is $$a\times b=b\times a$$
• Numerical Application:
Let us consider two numbers, 7 and 9, to verify the above property.       So, $$7\times 9=9\times 7$$          or, 63 = 63 For Addition:
• Algebraic Application:
The commutative property for addition is $$a+b=b+a$$
• Numerical Application:
Let us consider two numbers, 7 and 9, to verify the above property.      So, 7 + 9 = 9 + 7         or, 16 = 16 2. The associative property According to the associative property of multiplication, while performing addition or multiplication, regrouping numbers or terms of an algebraic expression, would not affect the final sum or product.
• Algebraic Application:
The associative property for multiplication is $$(a\times b)\times c=a\times(b\times c)$$
• Numerical Application:
Let us consider three numbers, 1, 9 and 2, to verify the above property. So, $$(1\times 9)\times 2=1\times(9\times 2)$$ or,  $$9\times 2=1\times 18$$ or, 18 = 18 For Addition:
• Algebraic Application:
$$(a+b)+c=a+(b+c)$$
• Numerical Application:
Let us consider three numbers, 7, 9 and 1, to verify the above property. So, ( 7 + 9 ) + 1 = 7 + ( 9 + 1 ) or, 16 + 1 = 7 + 10 or, 17 = 17 3. The distributive property According to the distributive property of multiplication, while multiplying a number or a term of an algebraic expression to the sum or difference of two numbers or terms, we distribute the multiplicand to the addends or subtrahend and minuend. It can be seen as follows:
• Algebraic Application:
$$(a+b)\times c=a\times c+a\times b$$
• Numerical Application:
Let us consider three numbers, 1, 9 and 2, to verify the above property.      So, $$(1+9)\times 2=2\times 1+2\times 9$$ $$=2+18$$ $=20$ 4. The addition property of zero       According to this property, the addition of any number with zero will always be equal to that number.
• Algebraic Application:
$$0+(x) =x$$
• Numerical Application:
Let us consider a number 4.       So, 0 + ( 4 ) = 4 5. The multiplication properties of zero and one       According to this property, the multiplication of any number by zero will always be equal to zero and multiplication of any number by one will always be equal to that number.
• Algebraic Application:
The multiplication property of zero and one is $$x\times 0=0$$ and $$x\times 1=x$$
• Numerical Application:
Now, assume a number 5.      So, $$5\times 0=0$$ and $$5\times 1=5$$

## Properties of Addition and Multiplication Examples

Example 1: Solve the given algebraic expression using the properties of addition and multiplication.

(a)$$(2\times 3+x)+4\times 2$$

(b) $$3\times (8a)$$

(c) $$3\times a\times 1$$

Solution:

Part (a)

we have $$(2\times 3+x)+4\times 2$$

$$=(6+x)+8$$          As,$$2\times 3=6$$ and $$4\times 2=8$$

$$=(x+6)+8$$          Using the commutative property of addition

$$=x+(6+8)$$          Using the associative property of addition

$$=x+14$$

Part (b)

We have: $$3\times (8a)$$

$$=(3\times 8)\times a$$         Using the associative property of multiplication

$$=24a$$

Part (c)

We have: $$3\times a~\times 1$$

$$=3\times (a\times 1)$$       Using the associative property of multiplication

$$=3\times a$$                 Using the multiplication property of 1

$$=3a$$

Example 2: You are a member of the volleyball team, which consists of six players. The summer league has a registration fee of $100. You also need $$x$$ dollars for buying each player a new shirt that has the school logo. The team also needs an additional$68.25 for buying new volleyballs. Write an expression to represent the total amount that the principal needs to allot to the school volleyball team. Also, if each T-shirt costs $$(x)$$ \$14.50, calculate the total amount needed.

Solution:

Write an expression that represents the sum of the league fee, the cost of the T-shirts, and the cost of the basketballs using a verbal model.

League registration fee + (Number of T-shirts $$\times$$  cost per T-shirt) + Cost of buying volleyballs.

We can form an expression using the above explanation:

$$=(100+6\times x)+68.25$$

$$=(100+6x)+68.25$$

$$=(6x+100)+68.25$$    Using the commutative property of addition

$$=6x+(100+68.25)$$    Using the associative property of addition

$$=6x+168.25$$

Now, if the value of $$x$$ is 14.5, the total cost will be:

$$6(14.5)+168.25$$

$$=87.0+168.25$$

$$=255.25$$