# 2.12: Associative and Commutative Properties of Multiplication with Decimals

**At Grade**Created by: CK-12

**Practice**Commutative Property of Multiplication with Decimals

Marigold has a lot of tomato plants in her vegetable garden. Marigold is planning to pick the ripe tomatoes and make salsa with them. She looks up a basic recipe. The recipe says that for every cup of tomatoes, she will need 0.5 onion. For every onion, she needs 4 cloves of garlic. How can Marigold determine how many cloves of garlic she needs in terms of the number of cups of tomatoes she picks?

In this concept, you will learn to identify and use the commutative and associative properties of multiplication with decimals.

### Commutative and Associative Properties of Multiplication with Decimals

The **Commutative Property of Multiplication** states that when finding a product, changing the order of the factors will not change their product. In symbols, the Commutative Property of Multiplication says that for numbers

Here is an example using simple whole numbers.

Show that

First, find

Next, find

Notice that both products are 8.

The answer is that because both

The **Associative Property of Multiplication** states that when finding a product, changing the way factors are grouped will not change their product. In symbols, the Associative Property of Multiplication says that for numbers

Here is an example using simple whole numbers.

Show that

First find

Next, find

Notice that both products are 60.

The answer is that because both

Both the Commutative Property of Multiplication and the Associative Property of Multiplication can be useful in simplifying expressions. The Commutative Property of Multiplication allows you to reorder factors while the Associative Property of Multiplication allows you to regroup factors.

Here is an example.

Simplify

First, use the Associative Property of Multiplication to regroup the factors.

Now, simplify

The answer is that

Here is another example.

Simplify

First, use the Commutative Property of Multiplication to reorder the factors.

Next, use the Associative Property of Multiplication to regroup the factors.

Now, simplify

The answer is that

### Examples

#### Example 1

Earlier, you were given a problem about Marigold, who is planning to make salsa.

Her recipe says that for every cup of tomatoes she will need 0.5 onion, and for every onion she will need 4 cloves of garlic. Marigold wants to figure out how many cloves of garlic she will need in terms of the number of cups of tomatoes she picks.

First, Marigold should write an expression for this situation. She should start by defining her variable. She doesn’t know how many cups of tomatoes she will have, so that unknown quantity will be her variable.

Let

Now, the problem says that she will need 0.5 onion for every tomato. So the number of onions she needs is

Next, the problem says that she will need 4 cloves of garlic for every onion. Since she will have

Now, Marigold can simplify the expression.

First, she can use the Commutative Property of Multiplication to reorder the factors.

Next, she can use the Associative Property of Multiplication to regroup the factors.

Now, she can simplify \begin{align*}(4 \cdot 0.5)x\end{align*}. She can use what she learned about decimal number multiplication to multiply the numbers in parentheses.

\begin{align*}& \quad 0.5\\ & \underline{\times \quad 4}\\ & \quad 2.0\end{align*}

\begin{align*}(4 \cdot 0.5)x\end{align*} simplifies to \begin{align*}2x\end{align*}.

The answer is that Marigold will need 2 cloves of garlic for every cup of tomatoes she picks.

#### Example 2

Simplify the following expression.

\begin{align*}4.5(9.2y)\end{align*}

First, use the Associative Property of Multiplication to regroup the factors.

\begin{align*}4.5(9.2y)\end{align*} is equivalent to \begin{align*}(4.5 \cdot 9.2)y\end{align*}.

Now, simplify \begin{align*}(4.5 \cdot 9.2)y\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \quad \ \ 4.5 \\ & \underline{\times \ \ \ 9.2 \;\;}\\ & \qquad 90\\ & \underline{+ \ 4050 \;\;}\\ & \ \ \ 41.50\end{align*}

\begin{align*}(4.5 \cdot 9.2)y\end{align*} simplifies to \begin{align*}41.4y\end{align*}.

The answer is that \begin{align*}4.5(9.2y)\end{align*} simplifies to \begin{align*}41.4y\end{align*}.

#### Example 3

Simplify \begin{align*}4.8(3.1k)\end{align*}.

First, use the Associative Property of Multiplication to regroup the factors.

\begin{align*}4.8(3.1k)\end{align*} is equivalent to \begin{align*}(4.8 \cdot 3.1)k\end{align*}.

Now, simplify \begin{align*}(4.8 \cdot 3.1)k\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \quad \ \ \ 4.8 \\ & \underline{\times \quad 3.1 \;\;}\\ & \qquad 48\\ & \underline{+ \ 1440 \;\;}\\ & \ \ \ 14.88\end{align*}

\begin{align*}(4.8 \cdot 3.1)k\end{align*} simplifies to \begin{align*}14.88k\end{align*}.

The answer is that \begin{align*}4.8(3.1k)\end{align*} simplifies to \begin{align*}14.88k\end{align*}.

#### Example 4

Simplify \begin{align*}(3.45p) \cdot 2.3\end{align*}.

First, use the Commutative Property of Multiplication to reorder the factors.

\begin{align*}(3.45p) \cdot 2.3\end{align*} is equivalent to \begin{align*}2.3 \cdot (3.45p)\end{align*}.

Next, use the Associative Property of Multiplication to regroup the factors.

\begin{align*}2.3 \cdot (3.45p)\end{align*} is equivalent to \begin{align*}(2.3 \cdot 3.45)p\end{align*}.

Now, simplify \begin{align*}(2.3 \cdot 3.45)p\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \quad \ \ \ 3.45 \\ & \underline{\times \quad 2.3 \;\;\;}\\ & \quad \ \ 1035 \\ & \underline{+ \ \ \ 6900 \;\;\;}\\ & \quad \ 7.935\end{align*}

\begin{align*}(2.3 \cdot 3.45)p\end{align*} simplifies to \begin{align*}7.935p\end{align*}.

The answer is that \begin{align*}(3.45p) \cdot 2.3\end{align*} simplifies to \begin{align*}7.935p\end{align*}.

#### Example 5

Simplify \begin{align*}1.98 \cdot (a \cdot 6.4)\end{align*}.

First, use the Commutative Property of Multiplication to reorder the factors within the parentheses.

\begin{align*}1.98 \cdot (a \cdot 6.4)\end{align*} is equivalent to \begin{align*}1.98 \cdot (6.4 \cdot a)\end{align*}.

Next, use the Associative Property of Multiplication to regroup the factors.

\begin{align*}1.98 \cdot (6.4 \cdot a)\end{align*} is equivalent to \begin{align*}(1.98 \cdot 6.4) \cdot a\end{align*}

Now, simplify \begin{align*}(1.98 \cdot 6.4) \cdot a\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \qquad 1.98 \\ & \underline{\times \quad \ 6.4 \;\;\;}\\ & \quad \quad\ 792 \\ & \underline{+ \ \ 11880 \;\;\;}\\ & \quad 12.672\end{align*}

\begin{align*}(1.98 \cdot 6.4) \cdot a\end{align*} simplifies to \begin{align*}12.672a\end{align*}.

The answer is that \begin{align*}1.98 \cdot (a \cdot 6.4)\end{align*} simplifies to \begin{align*}12.672a\end{align*}.

### Review

Simplify the following expressions.

- \begin{align*}(4.21 \times 8.8) \times p\end{align*}
- \begin{align*}16.14 \times q \times 6.2\end{align*}
- \begin{align*}3.6(91.7x)\end{align*}
- \begin{align*}5.3r(2.8)\end{align*}
- \begin{align*}5.6x(3.8)\end{align*}
- \begin{align*}2.4y(2.8)\end{align*}
- \begin{align*}6.7x(3.1)\end{align*}
- \begin{align*}8.91r(2.3)\end{align*}
- \begin{align*}5.67y(2.8)\end{align*}
- \begin{align*}4.53x(2.2)\end{align*}
- \begin{align*}5.6(2.8x)\end{align*}
- \begin{align*}9.2y(3.2)\end{align*}
- \begin{align*}4.5x(2.3)\end{align*}
- \begin{align*}15.4x(12.8)\end{align*}
- \begin{align*}18.3y(14.2)\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.12.

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Associative Property

The associative property states that you can change the groupings of numbers being added or multiplied without changing the sum. For example: (2+3) + 4 = 2 + (3+4), and (2 X 3) X 4 = 2 X (3 X 4).Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not affect the sum or product. For example .Estimation

Estimation is the process of finding an approximate answer to a problem.Product

The product is the result after two amounts have been multiplied.### Image Attributions

In this concept, you will learn to identify and use the commutative and associative properties of multiplication with decimals.

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