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Commutative Property of Addition with Fractions

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Commutative Property of Addition with Fractions

Have you ever combined ingredients to make a cake? Sometimes it matters which order you put them in, but often if you keep the dry ingredients separate from the wet ingredients, then it doesn't make a difference to the final product.

Sammi is baking a cake. She begins by combining the dry ingredients. She adds $\frac{1}{2}$ cup of white flour and then rereads the recipe to see that she need another $2 \frac{1}{4}$ cups of white flour. The dough looks sticky, so she puts a bit of whole wheat flour in too. It is just a small bit to even out the recipe.

Can you write an expression to show how Sammi added the flour? Can you simplify it?

This Concept will show you how to use the Commutative Property of Addition in fraction operations. You will be able to write an expressions and simplify it by the end of the Concept.

Guidance

Now that we know the basics of adding fractions, we can use two mathematical properties of addition to help us solve more complicated problems.

The Commutative Property of Addition states that the order of the addends does not change the sum. Let’s test the property using simple whole numbers.

$& 4 + 5 + 9 = 18 && 5 + 4 + 9 = 18 && 9 + 5 + 4 = 18\\& 4 + 9 + 5 = 18 && 5 + 9 + 4 = 18 && 9 + 4 + 5 = 18$

As you can see, we can add the three addends (4, 5, and 9) in many different orders. The Commutative Property of Addition works also works for four, five, six addends.

It works for fraction addends, too. This means that the order that you add fractions in does not change the sum of the fractions.

Parentheses are grouping symbols used in math to let us know which operations to complete first. The order of operations tells us that operations in parentheses must be completed before any other operation. The Associative Property of Addition states that the way in which addends are grouped does not change the sum. Once again, let’s test the property using simple whole numbers.

$& (4 + 5) + 9 = 18 && (5 + 4) + 9 = 18 && (9 + 5) + 4 = 18$

Clearly, the different way the addends are grouped has no effect on the sum. The Associative Property of Addition works for multiple addends as well as fraction addends.

How do we use these properties when adding fractions?

These two properties are extremely useful when adding fractions. If you are adding three fractions and two of the fractions have like denominators, you can add those two fractions together and then find a common denominator with the third. This can be a big time saver.

When you are working with variable expressions or with expressions which contain an algebraic unknown (like $x$ ) you can use the commutative and associative properties of addition to simplify the expression.

Simplify the following variable expressions using the associative and commutative properties of addition.

$3 \frac{2}{3} + x + \frac{1}{3}$

To simplify means to make smaller. We are going to simplify this expression. We use the Commutative Property of Addition to do this.

That’s a great question. If you think about it, because it doesn’t matter which order you add fractions in, you can add the mixed number and the fraction and ignore the $x$ . This will help you to simplify the expression. Let’s add the two numbers together.

$3 \frac{2}{3}+\frac{1}{3}=4$

One-third plus two-thirds is three-thirds which is the same as one. We add one to the whole number three and get four.

Our simplified expression is $4 + x$ .

$\frac{3}{10} + \left(\frac{1}{4} + x\right)$

To simplify this expression, we are going to use the Associative Property of Addition. The hint is that there are parentheses in this expression.

That’s a good question. We use that property because we can change the grouping. You know from the order of operations that operations inside the parentheses are done first. Well, you can’t complete this operation in the grouping symbol because you don’t know what $x$ is. But if you change the grouping, you can simplify the expression.

$\left(\frac{3}{10} + \frac{1}{4}\right) + x$

Now we add three-tenths and one-fourth.

$\frac{3}{10} &= \frac{6}{20}\\ \frac{1}{4} &= \frac{5}{20}\\ \frac{6}{20}+\frac{5}{20} &= \frac{11}{20}$

Our answer is $\frac{11}{20}+x$ .

Simplify each expression using the Commutative and Associative Properties of Addition. Be sure your answers are in simplest form.

Example A

$\frac{2}{3}+ y+ \frac{1}{5}$

Solution: $\frac{13}{15}+y$

Example B

$\frac{1}{2}+ \left(\frac{1}{2}+ x\right)$

Solution: $1 + x$

Example C

$x+ \frac{4}{9}+\frac{2}{9}$

Solution: $x + \frac{2}{3}$

Here is the original problem once again.

Sammi is baking a cake. She begins by combining the dry ingredients. She adds $\frac{1}{2}$ cup of white flour and then rereads the recipe to see that she need another $2 \frac{1}{4}$ cups of white flour. The dough looks sticky, so she puts a bit of whole wheat flour in too. It is just a small bit to even out the recipe.

Can you write an expression to show how Sammi added the flour? Can you simplify it?

Now let's write an expression including the two fractions and the little bit of whole wheat flour is x because the measurement is not clear.

$\frac{1}{2}+2 \frac{1}{4} + x$

Next, we can simplify it by adding the fractions.

$2 \frac{3}{4} + x$

This is our answer.

Vocabulary

Fraction
a part of a whole
Commutative Property of Addition
states that the order in which you add values does not change the sum of the values.
Associative Property of Addition
the way that you group numbers does not change the sum of the numbers being added.

Guided Practice

Here is one for you to try on your own.

Simplify. Be sure your answer is in simplest form.

$\frac{3}{7} + y + \frac{2}{7}$

To simplify this, using the commutative property, we add the fractions.

$\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$

Now we put the variable into the final expression.

$y + \frac{5}{7}$

This is our answer.

Practice

1. $\frac{1}{6}+\frac{2}{6}+\frac{3}{7}$

2. $\frac{1}{4}+ 3 \frac{5}{8} + 4 \frac{3}{4}$

3. $\frac{2}{9} + \left(\frac{1}{3} + \frac{5}{9}\right)$

4. $\left(2 \frac{7}{8} + \frac{2}{3}\right) + 1 \frac{1}{8}$

Directions: Simplify the expressions using the associative and commutative properties of addition.

5. $x + 3 \frac{2}{3} + 5 \frac{1}{6}$

6. $\frac{1}{4} + x + \frac{5}{8}$

7. $\left(\frac{1}{9} + x\right) + \frac{2}{9}$

8. $2 \frac{1}{14} + \left(x + 3 \frac{5}{7}\right)$

9. $3 \frac{1}{4} + \left(x + 1 \frac{2}{3}\right)$

10. $2 \frac{1}{10} + \left(x + 3 \frac{1}{3}\right)$

11. $4 \frac{1}{2} + \left(x + 2 \frac{1}{6}\right)$

12. $3 \frac{1}{9} + \left(x + 2 \frac{2}{18}\right)$

Directions : Write an expression or an equation for each situation and then solve. There are two parts to each problem.

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 $3 \frac{1}{5}$ cups of chopped pinecones with $1 \frac{1}{2}$ cups of mud. For the snail sauce on top, she uses another $1 \frac{3}{8}$ 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 $9 \frac{1}{6}$ feet of gold ribbon. Estelle contributes $\frac{3}{4}$ foot of red ribbon. Aaron brings in $5 \frac{2}{7}$ feet of gold ribbon at the last minute. How much ribbon does the cheerleading squad have?

16. Marcus and Anita are trying to complete the project. They finished $\frac{1}{4}$ of the project on Saturday. They got another four-sevenths of the project done on Tuesday. If the project needs to be finished by Thursday, write an algebraic expression that shows how much of the project needs to be completed, how much they have done on Saturday, how much they have done on Tuesday and how much they need to do on Wednesday.