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2.12: Associative and Commutative Properties of Multiplication with Decimals

Difficulty Level: At Grade Created by: CK-12
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Let’s Think About It

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.

Guidance

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 \begin{align*}a\end{align*} and \begin{align*}b\end{align*}:

\begin{align*}a \cdot b=b \cdot a\end{align*}

Here is an example using simple whole numbers.

Show that \begin{align*}2 \cdot 4 = 4 \cdot 2\end{align*}.

First, find \begin{align*}2 \cdot 4\end{align*}.

\begin{align*}2 \cdot 4 = 8\end{align*}

Next, find \begin{align*}4 \cdot 2\end{align*}.

\begin{align*}4 \cdot 2 = 8\end{align*}

Notice that both products are 8.

The answer is that because both \begin{align*}2 \cdot 4\end{align*} and \begin{align*}4 \cdot 2\end{align*} are equal to 8, they are equal to each other.

\begin{align*}2 \cdot 4 = 4 \cdot 2\end{align*}

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 \begin{align*}a\end{align*},\begin{align*}b\end{align*} and \begin{align*}c\end{align*}:

\begin{align*}(a \cdot b) \cdot c=a \cdot (b \cdot c)\end{align*}

Here is an example using simple whole numbers.

Show that \begin{align*}(2 \cdot 5) \cdot 6=2 \cdot (5 \cdot 6)\end{align*}.

First find \begin{align*}(2 \cdot 5) \cdot 6\end{align*}. Start by multiplying the numbers in parentheses. Then multiply the result with 6.

\begin{align*}\begin{array}{rcl} (2 \cdot 5) \cdot 6 &=& 10 \cdot 6\\ &=& 60 \end{array}\end{align*}

Next, find \begin{align*}2 \cdot (5 \cdot 6)\end{align*}. Again, start by multiplying the numbers in parentheses. Then multiply 2 by the result.

\begin{align*}\begin{array}{rcl} 2 \cdot (5 \cdot 6)&=&2 \cdot 30\\ &=&60 \end{array}\end{align*}

Notice that both products are 60.

The answer is that because both \begin{align*}(2 \cdot 5) \cdot 6\end{align*} and \begin{align*}2 \cdot (5 \cdot 6)\end{align*} are equal to 60, they are equal to each other.

\begin{align*}(2 \cdot 5) \cdot 6=2 \cdot (5 \cdot 6)\end{align*}

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 \begin{align*}29.3(12.4x)\end{align*}.

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

\begin{align*}29.3(12.4x)\end{align*} is equivalent to \begin{align*}(29.3 \cdot 12.4)x\end{align*} .

Now, simplify \begin{align*}(29.3 \cdot 12.4)x\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \quad \ \ 29.3\\ & \underline{\times \ \ \ 12.4 \;\;}\\ & \quad \ \ 1172\\ & \quad \ \ 5860\\ & \underline{+ \ 29300}\\ & \ \ \ 363.32\end{align*}

\begin{align*}(29.3 \cdot 12.4)x\end{align*} simplifies to \begin{align*}363.32x\end{align*}.

The answer is that \begin{align*}29.3(12.4x)\end{align*} simplifies to \begin{align*}363.32x\end{align*}.

Here is another example.

Simplify \begin{align*}(0.3x) \cdot 0.4\end{align*}.

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

\begin{align*}(0.3x) \cdot 0.4\end{align*} is equivalent to \begin{align*}0.4 \cdot (0.3x)\end{align*}.

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

\begin{align*}0.4 \cdot (0.3x)\end{align*} is equivalent to \begin{align*}(0.4 \cdot 0.3)x\end{align*}.

Now, simplify \begin{align*}(0.4 \cdot 0.3)x\end{align*}. Multiply the numbers in parentheses. Use what you have learned about decimal number multiplication.

\begin{align*}& \quad 0.4\\ & \underline{\times \ 0.3 \;\;}\\ & \quad 0.12\end{align*}

\begin{align*}(0.4 \cdot 0.3)x\end{align*} simplifies to \begin{align*}0.12x\end{align*}.

The answer is that \begin{align*}(0.3x) \cdot 0.4\end{align*} simplifies to \begin{align*}0.12x\end{align*}.

Guided Practice

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*}.

Examples

Example 1

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 2

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 3

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*}.

Follow Up

Remember 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 \begin{align*}x\end{align*} equal the number of cups of tomatoes Marigold picks.

Now, the problem says that she will need 0.5 onion for every tomato. So the number of onions she needs is \begin{align*}0.5x\end{align*}.

Next, the problem says that she will need 4 cloves of garlic for every onion. Since she will have \begin{align*}0.5x\end{align*} onions, she will need \begin{align*}(0.5x) \cdot 4\end{align*} cloves of garlic.

Now, Marigold can simplify the expression.

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

\begin{align*}(0.5x) \cdot 4\end{align*} is equivalent to \begin{align*}4 \cdot (0.5x)\end{align*}.

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

\begin{align*}4 \cdot (0.5x)\end{align*} is equivalent to \begin{align*}(4 \cdot 0.5)x\end{align*}.

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.

Video Review

https://www.youtube.com/watch?v=ENKH97PYssg

https://www.youtube.com/watch?v=5RzDVNob0-0

Explore More

Simplify the following expressions.

  1. \begin{align*}(4.21 \times 8.8) \times p\end{align*}
  2. \begin{align*}16.14 \times q \times 6.2\end{align*}
  3. \begin{align*}3.6(91.7x)\end{align*}
  4. \begin{align*}5.3r(2.8)\end{align*}
  5. \begin{align*}5.6x(3.8)\end{align*}
  6. \begin{align*}2.4y(2.8)\end{align*}
  7. \begin{align*}6.7x(3.1)\end{align*}
  8. \begin{align*}8.91r(2.3)\end{align*}
  9. \begin{align*}5.67y(2.8)\end{align*}
  10. \begin{align*}4.53x(2.2)\end{align*}
  11. \begin{align*}5.6(2.8x)\end{align*}
  12. \begin{align*}9.2y(3.2)\end{align*}
  13. \begin{align*}4.5x(2.3)\end{align*}
  14. \begin{align*}15.4x(12.8)\end{align*}
  15. \begin{align*}18.3y(14.2)\end{align*}

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Vocabulary

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 a+b=b+a \text{ and\,} (a)(b)=(b)(a).

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.

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