Commutative Property with
Products of Fractions
Jen Kershaw
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Printed: November 20, 2014
AUTHOR
Jen Kershaw
www.ck12.org
Chapter 1. Commutative Property with Products of Fractions
C HAPTER
1
Commutative Property with
Products of Fractions
Here you’ll learn to identify and apply the commutative and associative properties of multiplication in fraction
operations with variable and numerical expressions.
Now back to the bake sale. Have you ever made a casserole?
Richard is baking 2 21 casseroles for the bake sale. Victoria does not think this will be enough food. She thinks he
should bake at least 6 87 times this amount.
How many casseroles does Victoria think Richard needs to bake?
To figure this out, you will need to write an equation and solve it. Does the order of the fractions matter in this
equation?
Why or why not?
This Concept will teach you about the commutative and associative properties of multiplication. You will be
able to complete this dilemma by the end of the Concept.
Guidance
Previously we worked with commutative and associative properties of addition. Knowing how the mechanism
of addition works helped us solve more complicated addition problems involving fractions. The properties of
multiplication are a lot like the properties of addition. In this lesson, we are going to discover how to use the
commutative property of multiplication and the associative property of multiplication.
The Commutative Property of Multiplication states that the order of the factors does not change the product.
Let’s test the property using simple whole numbers.
1·2·3 = 6
2·1·3 = 6
2·3·1 = 6
3·2·1 = 6
3·1·2 = 6
1·3·2 = 6
As you can see, we can multiply the three factors (1, 2, and 3) in many different orders. The Commutative Property
of multiplication works also works for four, five, six factors. It works for fraction addends, too.
The Associative Property of Multiplication states that the way in which factors are grouped does not change the
product. Notice that we use parentheses as the grouping symbol just as we did with addition. Once again, let’s test
the property using simple whole numbers.
(1 · 3) · 2 = 6
(2 · 3) · 1 = 6
(2 · 1) · 3 = 6
Clearly, the different way the factors are grouped has no effect on the final product. The associative property of
multiplication works for multiple factors as well as fraction factors.
These two properties are extremely useful when multiplying fractions. If you are multiplying three fractions
and two of the fractions contain factors that you can cancel out, you can multiply those two fractions together
and have a new fraction in simplest terms, then simply multiply your new simpler fraction with the third
fraction.
1
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6
8
· 21 · 16
18
Here the easiest thing to do is to simplify the first and the third fraction. We can rearrange the fractions thanks to the
Commutative Property to make our work simpler.
6
8
16 1
· 18
·2
Now we can simplify on the diagonals of the first two fractions.
6 and 18 have the greatest common factor of 6.
8 and 16 have the greatest common factor of 8.
1
1
· 32 · 12
Now we multiply across.
2
6
Finally we simplify.
Our final answer is 13 .
Did you notice that we could have simplified the 2’s on the diagonal?
How do we apply this to variable expressions?
When you are working with variable expressions or expressions which contain an algebraic unknown (like x)
you can use the commutative and associative properties of multiplication to simplify the expression. Let’s see
how it works.
2
3
· x · 87
We can use the Commutative Property of Multiplication to move the fractions together. Then we can find the product
of the two fractions and then we will have simplified the expression. Notice that we can’t solve the expression
because we don’t know the value of x.
2
3
· 87 · x
We can simplify the two and the eight on the diagonals before we multiply.
1
3
· 47 · x
Our simplified expressions is
7
12
· x.
Here is one where the associative property will be very useful.
x · 21 · 35
Here we can move the grouping symbol or the parentheses to include the two fractions. Then we can multiply the
two fractions and that will give us our simplified expression. Notice that we can’t solve this because we don’t know
the value of the variable.
x · 21 · 35
3
Our simplified expression is x · 10
.
Use the Commutative Property and the Associative Property to simplify each expression.
Example A
x · 54 · 12
Solution: 25 x
2
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Chapter 1. Commutative Property with Products of Fractions
Example B
6
7
· x · 31
Solution: 27 x
Example C
Solve this problem. Be sure your answer is in simplest form.
2
3
× 94
Solution:
8
27
Here is the original problem once again.
Richard is baking 2 12 casseroles for the bake sale. Victoria does not think this will be enough food. She thinks he
should bake at least 6 87 times this amount.
How many casseroles does Victoria think Richard needs to bake?
To figure this out, you will need to write an equation and solve it. Does the order of the fractions matter in this
equation?
Why or why not?
First, let’s write an equation to solve the problem. We want to know the product of the two values, so we are going
to multiply.
2 12 × 6 78 = x
Does it matter the order of the values?
Because of the commutative property of multiplication, the order that we write the values does not matter.
Now we can solve it.
First, change each mixed number to an improper fraction and rewrite the problem.
5
2
× 55
8
Now we multiply.
275
16
Next, we divide to figure out the mixed number.
3
17 16
casseroles
This is our final answer.
Vocabulary
Fraction
a part of a whole.
Mixed Number
a whole number and a fraction
Variable Expression
an expression that uses numbers, operations and variables.
3
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Commutative Property of Multiplication
the order that you multiply numbers does not affect the product.
Associative Property of Multiplication
the grouping of the numbers does not affect the product of those numbers.
Guided Practice
Here is one for you to try on your own.
Solve and simplify.
2
3
9 6
· 12
·7
Answer
To solve this problem, we are going to multiply these three fractions together. If we cross simplify first, we end up
with the following problem.
1
1
· 13 · 17
Now we multiply across.
3
7
This is our answer.
Video Review
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/54833
This is a video on the commutative law of multiplication.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/57619
This is a video on the associative law of multiplication
Explore More
Directions: Multiply.
1.
4
2
3
9 6
· 12
·7
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2.
1
3
4.
10
12
5.
1
3
4
5
1
3
Chapter 1. Commutative Property with Products of Fractions
· 1 54 · 34
3. 49 · 58 · 37
6.
7.
7
· 3 51 · 10
9 6
· 18
·7
9 1
· 20
·2
4 2
· 12
·9
Directions: Simplify the following expressions using the commutative and associative properties of multiplication.
8.
7
8
· x · 54
9. x · 2 32 · 56
10.
5
8
· 1 32 · x
11.
6
8
· x · 32
12.
5
4
8 ·x· 5
4
3
10 · x · 5
13.
Directions: Solve each problem.
14. Crazy Sal’s is having a Delirious Discount Sale. He is selling everything in his store for 38 of the marked price.
Rowena finds a t-shirt that is marked at $36. How much will she pay for the shirt at the discounted price?
15. Dan is cutting plywood for his science fair project. He cuts a board that is 3 14 feet long. After he cuts it, he
realizes that he really needs a piece about 32 of this length. How long will the new piece of wood that Dan cuts be?
5