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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

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.

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.**

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.

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**Algebraic Application:**

**Numerical Application:**

**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:

\(6(14.5)+168.25\)

\(=87.0+168.25\)

\(=255.25\)

Frequently Asked Questions on Addition and Multiplication

There are four properties of addition for whole numbers:

- The Additive Identity Property
- The Associative Property
- The Distributive Property
- The Closure Property

Multiplication has the distributive, the commutative, the associative, the removing common factor and the neutral element properties.

The grouping property states that the order in which addends are grouped has no effect on the result of the addition.