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Properties of Multiplication in Decimal Operations

Associative and Commutative rules true for decimals

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Properties of Multiplication in Decimal Operations
License: CC BY-NC 3.0

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.

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

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.

\begin{align*}\begin{array}{rcl} 4.5\\ \underline{\times \;\; 7}\\ 31.5 \end{array}\end{align*} 

\begin{align*}\begin{array}{rcl} & \quad \ \ 7\\ & \underline{\times 4.5}\\ & \quad 35\\ & \underline{+ \;28\;\;\;}\\ & \ 31.5 \end{array}\end{align*} 

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*} is the same as the product of \begin{align*}a5.6\end{align*}. Note that when using variables to express multiplication, the number is written before the variable.

\begin{align*}5a\end{align*} 

If \begin{align*}a\end{align*} is equal to 3, find the product.

\begin{align*}\begin{array}{rcl} 5.6\\ \underline{\times \;\; 3}\\ 16.8 \end{array}\end{align*}

The product of \begin{align*}5a\end{align*}, if \begin{align*}a = 3\end{align*}, is 16.8.

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.

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

The same applies when multiplying decimals.

\begin{align*}6(3.4 \times 2) = (6 \times 3.4)2\end{align*}

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

\begin{align*}\begin{array}{rcl} 6(6.8) & = & 20.4(2)\\ 40.8 & = & 40.8 \end{array}\end{align*}

This is also true with variables.

\begin{align*}\begin{array}{rcl} 5(6a) & = & (5 \times 6)a\\ 30a & = & 30a \end{array}\end{align*}

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.

\begin{align*}\begin{array}{rcl} & \text{Joan}\qquad \qquad \qquad \qquad \qquad \qquad\qquad\qquad\qquad \text{Mike}\\ & \text{Volume} = (\text{length} \times \text{width})(\text{height}) \qquad \qquad \text{Volume} = (\text{width} \times \text{height})(\text{length})\\ & \text{Volume} = (12.5 \ in. \times 4 \ in.)(5 \ in.) \qquad \qquad \ \ \text{Volume} = (4 \ in. \times 5 \ in.)(12.5 \ in.) \end{array}\end{align*}

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.

\begin{align*}\begin{array}{rcl} \text{Volume} & = & 12.5 \ in. \times 4 \ in. \times 5 \ in.\\ \text{Volume} & = & 250 \ in.^3 \end{array}\end{align*}

The volume of the box is 250 cubic inches.

Example 2

Identify the property.

\begin{align*}4(3.67) = 3.67(4)\end{align*}

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.

\begin{align*}4.5(5a) = (4.5 \times 5)a\end{align*}

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.

\begin{align*}6.7(4) = 4(6.7)\end{align*}

The order of the numbers being multiplied has changed.

This demonstrates the commutative property of multiplication.

Example 5

Identify the property.

\begin{align*}5.4a = a5.4\end{align*}

The order of the numbers being multiplied has changed.

This demonstrates the commutative property of multiplication.

Review

Identify the property illustrated in each problem.

  1. \begin{align*}4.6a = a4.6\end{align*}
  2. \begin{align*}(4a)(b) = 4(ab)\end{align*}
  3. \begin{align*}(5.5a)(c) = 5.5(ac)\end{align*}
  4. \begin{align*}ab = ba\end{align*}
  5. \begin{align*}6ab = ab(6)\end{align*}
  6. \begin{align*}6 \times 4 = 4 \times 6\end{align*}
  7. \begin{align*}5(ab) = (5a) \times b\end{align*}
  8. \begin{align*}7(8x) = (7 \times 8)x\end{align*}
  9. \begin{align*}2xy = 2yx\end{align*}
  10. \begin{align*}3(4a) = (3 \times 4)a\end{align*}
  11. \begin{align*}6 \times 7 \times 4 = 4 \times 7 \times 6\end{align*}
  12. \begin{align*}abc = cab\end{align*}
  13. \begin{align*}xy(az) = x(yaz)\end{align*}
  14. \begin{align*}abcd = dcab\end{align*}
  15. \begin{align*}2a(bc) = (2a)bc\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.3. 

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Vocabulary

Estimate

To estimate is to find an approximate answer that is reasonable or makes sense given the problem.

multiplication

Multiplication is a simplified form of repeated addition. Multiplication is used to determine the result of adding a term to itself a specified number of times.

Product

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

Properties

Properties are rules that work for given sets of numbers.

The Associative Property of Multiplication

The associative property of multiplication states that regardless of how you group numbers in a multiplication problem, the product will be the same.

The Commutative Property of Multiplication

The commutative property of multiplication states that regardless of the order in which you multiply numbers in a multiplication problem, the product will be the same.

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  1. [1]^ License: CC BY-NC 3.0

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