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# 2.6: Associative and Commutative Properties of Addition with Decimals

Difficulty Level: At Grade Created by: CK-12
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Practice Commutative Property of Addition with Decimals

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Jeff’s dad works at an architectural firm in Boston. He has the same commute to work every day. First he drives 3.4 miles to the train station. Then he takes the train for 15.6 miles into the city. He’s lucky because the train station is right next to his office. To get home at the end of the day, Jeff’s dad does the reverse. How many miles does Jeff’s dad travel on his way home from work?

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

### Using Commutative and Associative Properties of Addition with Decimals

The Commutative Property of Addition states that when finding a sum, changing the order of the addends will not change their sum. In symbols, the commutative property of addition says that for numbers a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}:

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

Here is an example using simple whole numbers.

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

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

2+4=6\begin{align*}2 + 4 = 6\end{align*}

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

4+2=6\begin{align*}4 + 2 = 6\end{align*}

Notice that both sums are 6.

The answer is that because both 2+4\begin{align*}2 + 4\end{align*} and 4+2\begin{align*}4 + 2\end{align*} are equal to 6, they are equal to each other. 2+4=4+2\begin{align*}2 + 4 = 4 + 2\end{align*}.

The Associative Property of Addition states that when finding a sum, changing the way addends are grouped will not change their sum. In symbols, the associative property of addition says that for numbers a,b\begin{align*}a, b\end{align*} and c\begin{align*}c\end{align*}:

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

Here is an example using simple whole numbers.

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

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

\begin{align*}\begin{array}{rcl} (2+5) + 6 &=& 7+6 \\ &=& 13 \end{array}\end{align*}

Next, find \begin{align*}2+(5+6)\end{align*}. Again, start by adding the numbers in parentheses. Then add 2 to the result.

\begin{align*}\begin{array}{rcl} 2+(5+6) &=& 2+11 \\ &=& 13 \end{array}\end{align*}

Notice that both sums are 13.

The answer is that because both \begin{align*}(2+5)+6\end{align*} and \begin{align*}2+(5+6)\end{align*} are equal to 13, they are equal to each other. \begin{align*}(2+5)+6=2+(5+6)\end{align*}.

Both the commutative property of addition and the associative property of addition can be useful in solving equations.

Here is an example.

Find the value of \begin{align*}x\end{align*} in the following equation.

\begin{align*}71.321+42.29=x+71.321\end{align*}

First, notice that both sides of the equation are the sum of two values. Also, both sides of the equation include the number 71.321 as one of the terms.

By the commutative property of addition, the second term must also be the same on both sides of the equation. \begin{align*}x\end{align*} must be equal to 42.29.

\begin{align*}71.321+42.29=42.29+71.321\end{align*}

The answer is \begin{align*}x=42.29\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Jeff and his dad’s commute home from work.

To get to work, Jeff’s dad drives for 3.4 miles and then takes the train for 15.6 miles. To get home, he does the reverse. Jeff wants to know how many miles his dad travels to get home.

First, figure out how many miles Jeff’s dad travels to get to work. You know that to get to work Jeff’s dad travels 3.4 miles plus 15.6 more miles.

\begin{align*}3.4+15.6=19\end{align*}

This means Jeff’s dad travels 19 miles to get to work.

Now, to get home Jeff’s dad travels in reverse. By the commutative property of addition, you know that changing the order that you add two numbers won’t change the answer.

\begin{align*}3.4+15.6=15.6+3.4\end{align*}

Therefore, Jeff’s dad must travel 19 miles to get home from work just like he travels 19 miles to get to work.

The answer is Jeff’s dad travels 19 miles to get home from work.

#### Example 2

Find the value of \begin{align*}x\end{align*} in the following equation by using properties of addition.

\begin{align*}0.14 + 2.58 = (x + 1.45) + 1.13\end{align*}

First, notice that the right side of the equation has two addends grouped together. Use the associative property of addition to rewrite the right side of the equation by changing the way the addends are grouped.

\begin{align*}0.14+2.58=x+(1.45+1.13)\end{align*}

Now, combine the terms in parentheses on the right side of the equation.

\begin{align*}0.14+2.58=x+2.58\end{align*}

Next, notice that both sides of the equation now have two terms that are added together. Also, both sides of the equation have a 2.58 as one of the terms. This means the second terms on each side of the equation must be equal. \begin{align*}x\end{align*} must be equal to 0.14.

The answer is \begin{align*}x=0.14\end{align*}.

#### Example 3

Identify which property is shown below.

\begin{align*}3.4 + 5.6 = 5.6 + 3.4\end{align*}

First, notice that both sides of the equation have the same two addends, but the order has been changed. This is an example of the commutative property of addition.

#### Example 4

Identify which property is shown below.

\begin{align*}(3.4 + 3.2) + 5.6 = 3.4 + (3.2 + 5.6)\end{align*}

First, notice that both sides of the equation have the same three addends in the same order, but the grouping has been changed. This is an example of the associative property of addition.

#### Example 5

Find the value of \begin{align*}x\end{align*} in the following equation by using properties of addition.

\begin{align*}1.94 + 0.38 = x + 1.94\end{align*}

First, notice that both sides of the equation are the sum of two values. Also, both sides of the equation include the number 1.94 as one of the terms.

By the commutative property of addition, the second term must also be the same on both sides of the equation. \begin{align*}x\end{align*} must be equal to 0.38.

\begin{align*}1.94+0.38=0.38+1.94\end{align*}

The answer is \begin{align*}x=0.38\end{align*}.

### Review

Identify the property illustrated in each equation.

1. \begin{align*}4.5 + (x + y) + 2.6 = (4.5 + x) + y + 2.6\end{align*}
2. \begin{align*}3.2 + x + y + 5.6 = x + 3.2 + y + 5.6\end{align*}
3. \begin{align*}1.5 + (2.3 + y) + 5.6 = (1.5 + 2.3) + y + 5.6\end{align*}
4. \begin{align*}3.2 + 5.6 + 1.3 + 2.6 = 3.2 + 2.6 + 5.6 + 1.3\end{align*}
5. \begin{align*}4.5 + 15.6 = 15.6 + 4.5\end{align*}
6. \begin{align*}(x + y) + 5.6 = x + (y + 5.6)\end{align*}
7. \begin{align*}17.5 + 18.9 + 2 = 2 + 17.5 + 18.9\end{align*}
8. \begin{align*}(x + y) + z = x + (y + z)\end{align*}

Find the value of \begin{align*}x\end{align*} in the following equations by using properties of addition.

1. \begin{align*}.8603 + .292 = x + .8603\end{align*}
2. \begin{align*}(2.65 + x) + 19.35 = 22 + 2.115\end{align*}
3. \begin{align*}.306 + 1.076 = (.782 + x) + .294\end{align*}
4. \begin{align*}6.174 + 76.41 = 76.41 + x\end{align*}

Solve each problem.

1. Lamont measures the amount of water he drinks. His results for four consecutive days were as follows: 0.6 liters, 0.72 liters, 0.84 liters, 0.96 liters. If the pattern continues, how much water will Lamont drink on the fifth day?
2. Barbara is starting a jewelry making enterprise. At the supply store, she spent $19.19 on beads and$6.81 on wire and clasps. If she left the store with \$24.50, how much did she start with?
3. A rectangular playground has a length of 18.36 yards and a width of 12.24 yards. What is the perimeter of the playground?

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### Vocabulary Language: English

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

Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).

Estimate

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

Rounding

Rounding is reducing the number of non-zero digits in a number while keeping the overall value of the number similar.

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