# 1.6.8: Associative and Commutative Property with Fractions

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## Associative and Commutative Properties of Addition with Fractions

Lisa is making bread and weighing the ingredients as she goes. She starts with 16(3/4) ounces of flour. Then, she adds water. Finally, she adds 1/4 ounce of yeast. All together her dough weighs 27 ounces. She forgot to look at the weight of the water after she added it. How can Lisa figure out how much water she added to her dough?

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

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

Recall that 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 and b:

a+b=b+a

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 and c:

(a+b)+c=a+(b+c)

Both the commutative property of addition and the associative property of addition can be useful in simplifying expressions involving fractions and mixed numbers. The commutative property of addition allows you to reorder terms while the associative property of addition allows you to regroup terms.

Here is an example.

Simplify the following expression.

3(2/3)+x+1/3

First, use the commutative property of addition to reorder the terms.

3(2/3)+x+1/3 is equivalent to 3(2/3)+1/3+x.

Next, simplify by combining 3(2/3) and 1/3. Use what you have learned about adding fractions and mixed numbers.

3(2/3)+1/3=4

3(2/3)+1/3+x simplifies to 4+x.

The answer is 3(2/3)+x+1/3 simplifies to 4+x.

Here is another example.

Simplify the following expression.

3/10+(1/4+x)

First, notice that this expression has parentheses. That's a clue that you can use the associative property to rewrite it. Use the associative property to regroup the terms.

3/10+(1/4+x) is equivalent to (3/10+1/4)+x.

Next, simplify by combining 3/10 and 1/4. Use what you have learned about adding fractions. You will need to find a common denominator and rewrite each fraction. Then, add the equivalent fractions.

3/10=(3×2)/(10×2)=6/20

1/4=(1×5)/(4×5)=5/20

6/20+5/20=11/20

(3/10+1/4)+x simplifies to 11/20+x.

The answer is 3/10+(1/4+x) simplifies to 11/20+x.

### Examples

Example $$\PageIndex{1}$$

She started with 16(3/4) ounces of flour, then she added water, then she added 1/4 ounce of yeast. All together her dough weighed 27 ounces. Lisa forgot to weigh the water she added individually. She wants to figure out how much water she added to her dough.

Solution

First, Lisa should write an equation to represent this situation. She doesn't know the weight of the water, so let the variable x be equal to the weight of the water. She knows the sum of the flour, water, and yeast is equal to 27.

16(3/4)+x+1/4=27

Next, Lisa can simplify the left side of her equation. She can use the commutative property to reorder the terms.

16(3/4)+x+1/4 is equivalent to 16(3/4)+1/4+x.

Now, Lisa can simplify by adding the mixed number and fraction.

16(3/4)+1/4+x simplifies to 17+x.

Next, Lisa can rewrite her equation.

16(3/4)+x+1/4=27 is equivalent to 17+x=27.

Finally, Lisa can solve her equation by subtracting 17 from both sides.

17+x−17=27−17

x=10

Example $$\PageIndex{1}$$

Simplify the following expression.

3/7+y+2/7

Solution

First, use the commutative property of addition to reorder the terms.

3/7+y+2/7 is equivalent to 3/7+2/7+y.

Next, simplify by combining 37 and 27. Use what you have learned about adding fractions.

3/7+2/7=5/7

3/7+2/7+y simplifies to 5/7+y.

The answer is 3/7+y+2/7 simplifies to 5/7+y.

Example $$\PageIndex{1}$$

Simplify the following expression.

2/3+y+1/5

Solution

First, use the commutative property of addition to reorder the terms.

23+y+15 is equivalent to 23+15+y.

Next, simplify by combining 23 and 15. Use what you have learned about adding fractions. You will need to find a common denominator and rewrite each fraction. Then, add the equivalent fractions.

2/3=(2×5)/(3×5)=10/15

1/5=(1×3)/(5×3)=3/15

(10/15)+(3/15)=13/15

2/3+1/5+y simplifies to 13/15+y.

The answer is 2/3+y+1/5 simplifies to 13/15+y.

Example $$\PageIndex{1}$$

Simplify the following expression.

1/2+(1/2+x)

Solution

First, use the associative property to regroup the terms.

1/2+(1/2+x) is equivalent to (1/2+1/2)+x.

Next, simplify by combining 1/2 and 1/2. Use what you have learned about adding fractions.

1/2+1/2=1

(1/2+1/2)+x simplifies to 1+x.

The answer is 1/2+(1/2+x)simplifies to 1+x.

Example $$\PageIndex{1}$$

Simplify the following expression.

(x+4/9)+2/9

Solution

First, use the associative property to regroup the terms.

(x+4/9)+2/9 is equivalent to x+(4/9+2/9).

Next, simplify by combining 4/9 and 2/9. Use what you have learned about adding fractions. Rewrite your result in simplest form.

4/9+2/9=6/9=2/3

x+(4/9+2/9) simplifies to x+2/3.

The answer is (x+4/9)+2/9 simplifies to x+2/3.

### Review

Simplify each expression using the commutative and associative properties of addition.

1. 1/6+y+2/6
2. 1/4+x+4(3/4)
3. 2/9+y+5/9
4. 2(7/8)+x+1(1/8)
5. (x+3(2/3))+5(1/6)
6. 1/4+x+5/8
7. (1/9+x)+2/9
8. 2(1/14)+(x+3(5/7))
9. 3(1/4)+(x+1(2/3))
10. 2(1/10)+(x+3(1/3))
11. 4(1/2)+(x+2(1/6))
12. 3(1/9)+(x+2(2/18))
13. One-third of the CDs in Joseph’s CD collection are classical music CDs. Two-sevenths of the CDs are hip-hop CDs. What fraction of Joseph’s collection is classical and hip-hop?
14. Naira is making pinecone stew. First, she mixes 3(1/5) cups of chopped pinecones with 1(1/2) cups of mud. For the snail sauce on top, she uses another 1(3/8) cups of chopped pinecones. How much chopped pinecone does Naira use for her recipe?
15. Jennifer is trying to determine if the cheerleading squad has enough ribbon for the pep rally on Friday. Kurt contributes 9(1/6) feet of gold ribbon. Estelle contributes 3/4 foot of red ribbon. Aaron brings in 5(2/7) feet of gold ribbon at the last minute. How much ribbon does the cheerleading squad have?

### Vocabulary

Term Definition
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 and\,(a)(b)=(b)(a).
fraction A fraction is a part of a whole. A fraction is written mathematically as one value on top of another, separated by a fraction bar. It is also called a rational number.