Joan and Mike are measuring the volume of a box. They can’t remember the exact formula, but they do know that you have to multiply the dimensions of the box. Joan says, “We can multiply the length times the width and then, multiply by the height.” Mike says, “No, we have to multiply the width times the height and then, multiply by the length.” Who is correct?

In this concept, you will learn to apply the multiplication properties in decimal operations.

### Properties of Multiplying Decimals

A **property** is a rule that makes a statement about the way that numbers interact with each other during certain operations. The key thing to remember about a property is that the statement is true for all numbers.

The **commutative property of multiplication** states that the order in which you multiply numbers does not matter as the product will be the same.

When working with decimals and whole numbers, the product will be the same regardless of which number is multiplied first.

\begin{align*}4.5(7) = 7(4.5)\end{align*}

Check by multiplying.

The commutative property of multiplication also applies to multiplication problems with variables. Remember that a **variable** is a letter used to represent an unknown number. A variable next to a number tells you to multiply.

\begin{align*}5.6a = a5.6\end{align*}

The product of \begin{align*}5.5a\end{align*}

If

is equal to 3, find the product.

The product of \begin{align*}a = 3\end{align*}

The **associative property of multiplication** states that the order in which you group numbers in multiplication does not matter as the product will be the same. Remember that grouping refers to the use of parentheses or brackets.

The same applies when multiplying decimals.

You can change the grouping of the numbers and the product will remain the same. Multiply the numbers in parentheses first.

This is also true with variables.

Once again, you can change the grouping of the numbers and variables and the product will remain the same.

### Examples

#### Example 1

Earlier, you were given a problem about Joan and Mike trying to calculate volume.

Joan says, “We can multiply the length times the width and then, multiply by the height.” Mike says, “No, we have to multiply the width times the height and then, multiply by the length.” Who is correct? Find the volume of the box.

First, write the equation to find the volume using Joan’s method and Mike’s method.

Then, look at each of their equations. Remember that the commutative and associative properties of multiplication tell you that the order and grouping of numbers do not matter in multiplication. The products for both Joan and Mike will be equal.

\begin{align*}(12.5 \ in. \times 4 \ in.)(5 \ in.) = (4 \ in. \times 5 \ in.)(12.5 \ in.)\end{align*}

Joan and Mike are both correct.

Next, multiply the three numbers to find the volume of the box.

The volume of the box is 250 cubic inches.

#### Example 2

Identify the property.

The only thing that changed in this problem is the order of the values being multiplied.

This demonstrates the commutative property of multiplication.

#### Example 3

Identify the property.

The order of the numbers is the same, but the groupings are different.

This demonstrates the associative property of multiplication.

#### Example 4

Identify the property.

The order of the numbers being multiplied has changed.

This demonstrates the commutative property of multiplication.

#### Example 5

Identify the property.

The order of the numbers being multiplied has changed.

This demonstrates the commutative property of multiplication.

### Review

Identify the property illustrated in each problem.

- \begin{align*}7(8x) = (7 \times 8)x\end{align*}