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Decimals

Use associative, commutative, and distributive properties with decimals

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Identify and Apply Number Properties in Decimal Operations

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Three friends, Sandra, Janet, and Karen decide to go out for submarine sandwiches. They have $27.00 between the three of them. They each want to get a submarine sandwich and a drink for their lunch. Subs cost $5.89 plus 14% tax and the drinks are $2.00 with no tax. Do they have enough money?

In this concept, you will learn to identify and apply number properties in the addition and multiplication of decimals.

Number Property

A property is a rule that applies to mathematical statements.

Properties help us to understand certain ways of doing things in mathematics. These rules can always be used in solving math problems. The first two properties you will work with are the Associative Property of Addition and the Commutative Property of Addition.

The Associative Property of Addition states that when three or more numbers are added, the sum is the same no matter what order we add them. For example, \begin{align*}4+(2+8)=(4+2)+8\end{align*}.

The Commutative Property of Addition states that when two numbers are added, the order of the addition does not matter. For example, \begin{align*}4+8=8+4\end{align*}.

Let’s look at an example.

Which of the following shows the Commutative Property?

(a) \begin{align*}x+9.5=9.5x\end{align*}

(b) \begin{align*}x-9.5=9.5-x\end{align*}

(c) \begin{align*}x+9.5=9.5+x\end{align*}

You need to think about the definition of the commutative property. Remember with this property, the two numbers need to be added and the order they are added is not important.

Choice (a) has two numbers added on the left but they are multiplied on the right. This is not true.

Choice (b) is subtraction not addition. This is not the commutative property.

Choice (c) does show the commutative property as both sides show addition in different orders. This is the correct answer.

You can use properties to help you simplify numerical expressions. Using addition properties to reorganize expressions often makes it easier to simplify.

Let’s look at an example.

Simplify: \begin{align*}10.5+(3.2+4.5)\end{align*} 

First, apply the commutative property. You notice that there is 10.5 and 4.5 in the problem.

\begin{align*}10.5+(3.2+4.5)=10.5+(4.5+3.2)\end{align*}

Next, apply the associate property.

\begin{align*}10.5+(3.2+4.5)=(10.5+4.5)+3.2\end{align*}

Then, use mental math to find the sum.

\begin{align*}(10.5+4.5)+3.2=18.2\end{align*}

The answer is 18.2

Now, let’s look at multiplication. To work with multiplication of decimals, you are going to use a few other properties. These properties help in simplifying expressions.

The Associative Property of Multiplication states that when three or more numbers are multiplied, the product is the same no matter what order we multiply them. For example, \begin{align*}4 \times (2 \times 8)=(4 \times 2) \times 8\end{align*}.

The Commutative Property of Multiplication states that when two numbers are multiplied, the order of the multiplication does not matter. For example, \begin{align*}4 \times 8=8 \times 4\end{align*}.

The Distributive Property states that the product of a number and a sum is equal to the sum of the individual product of the addends and the number. For example, \begin{align*}3 \times (5+8)=3 \times 5+3 \times 8\end{align*}.

Let’s look at an example of the distributive property that includes variables.

Simplify: \begin{align*}2.5(2.1x+4.3y)\end{align*}

The addends inside the parentheses cannot be combined because two different variables (\begin{align*}x\end{align*} and \begin{align*}y\end{align*}) are being used, so you can use the distributive property to help you simplify the expression.

First, multiply the term outside the parentheses with both of the terms inside the parentheses.

\begin{align*}2.5(2.1x+4.3y)=(2.5 \times 2.1x)+(2.5 \times 4.3y)\end{align*} 

Next, simplify each set of brackets:

\begin{align*}2.5 \times 2.1x=5.25x \end{align*}

\begin{align*}2.5 \times 4.3y=10.75y\end{align*}

Then, combine:

\begin{align*}5.25x+10.75y\end{align*}

The answer is \begin{align*}5.25x+10.75y\end{align*}.

Examples

Example 1

Earlier, you were given a problem about three girls going out for lunch.

The girls have $27.00 between the three of them. The want to each buy a sub and a drink. Submarine sandwiches are $5.89 plus tax and the drinks are $2.00 with no tax. Let’s help them figure out their total bill and let them know if they have enough money.

First, let’s set up the equation to find the total cost.

\begin{align*}&\text{Cost} = \text{subs} + \text{taxes} + \text{drinks} \\ &\text{Cost} = (3 \times 5.98) + (3 \times 5.89) \times 0.14 + (3 \times 2)\end{align*}

Next, you can use the associative property of addition to reorder these parentheses.

\begin{align*}\text{Cost}=(3 \times 5.89)+(3 \times 2)+(3 \times 5.89) \times 0.14\end{align*}

Then, multiply the numbers inside the parentheses.

\begin{align*}\text{Cost}=17.67+6+17.67 \times 0.14\end{align*}

Finally complete the addition and multiplications using the rules of PEDMAS.

 \begin{align*}\text{Cost} & =17.67+6+17.67 \times 0.14\\ & =23.67+2.47\\ & =26.14\end{align*}

The answer is $26.14.

The girls have $27.00 so they have enough for lunch.

Example 2

Simplify using the distributive property.

\begin{align*}4.5(2x+3)\end{align*}

First, multiply the term outside the parentheses with both of the terms inside the parentheses.

\begin{align*}4.5(2x+3)=(4.5 \times 2x)+(4.5 \times 3)\end{align*}

Next, simplify each set of brackets:

\begin{align*}(4.5 \times 2x)=9x\end{align*}

\begin{align*}(4.5 \times 3)=13.5\end{align*}

Then, combine: \begin{align*}9x+13.5\end{align*}

The answer is \begin{align*}9x+13.5\end{align*}.

Example 3

Simplify the expression: \begin{align*}3.1+2.7+4.3\end{align*}

First, apply the associate property. Notice that (0.7 from 2.7) and 0.3 (from 4.3) will add to 1.

\begin{align*}3.1+2.7+4.3=3.1+(2.7+4.3)\end{align*}

Next, add inside the parentheses.

\begin{align*}3.1+(2.7+4.3)=3.1+7\end{align*}

Then, finish the addition problem.

\begin{align*}3.1+7=10.1\end{align*}

The answer is 10.1.

Example 4

Simplify the expression: \begin{align*}6.2(4x-3)\end{align*}

First, apply the distributive property.

\begin{align*}6.2(4x-3)=6.2 \times 4x-6.2 \times 3\end{align*}

Next, multiply.

\begin{align*}6.2 \times 4x -6.2 \times 3=24.8x-18.6\end{align*}

The answer is \begin{align*}24.8x-18.6\end{align*}.

Example 5

Simplify the expression: \begin{align*}6(3 \times 4) \times 7\end{align*}

First, apply the associative property.

\begin{align*}6(3 \times 4) \times 7=(6 \times 3) \times (4 \times 7)\end{align*}

Next, multiply inside the parentheses.

\begin{align*}(6 \times 3) \times (4 \times 7) = 18 \times 28\end{align*}

Then, multiply to find the answer.

\begin{align*}18 \times 28=504\end{align*}

The answer is 504.

Review

Use the associative and commutative properties of addition to solve each problem.

1. \begin{align*}(7.2 + 9.1) + 3.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

2. \begin{align*}5.4 + 2.1 + 5.4 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

3. \begin{align*}(1.2 + 6.7) + 1.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

4. \begin{align*}(4.1 + 9.2) + 9.0 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

5. \begin{align*}(14.11 + 9.2) + 8.0 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Use what you have learned to solve each problem.

6. \begin{align*}(7 \times 9) + 3.2 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

7. \begin{align*}15.4 + 2.1 - 5.4 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

8. \begin{align*}(1.2 \times 6) + 1.3 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

9. \begin{align*}(14.7 \div 2) + 9.0 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

10. \begin{align*}(11.1 + 2) + 18.0 = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*} 

Use the distributive property to simplify each expression.

11. \begin{align*}3.2(2x+4) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

12. \begin{align*}5.2(3x - 2)= \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

13. \begin{align*}6.3(4y + 4)= \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

14. \begin{align*}2.2(9a - 1)= \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

15. \begin{align*}6.7(8x + 9) = \underline{\;\;\;\;\;\;\;\;\;\;}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.5.  

 

Vocabulary

Addend

An addend is the number being added in an addition problem.

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).

Property

A property is a rule that works for a given set of numbers.

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