Associative and Commutative Properties For Fraction Products

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Welcome to Buzzfeedng.com. In today’s post, we are going to be considering Associative and Commutative Properties For Fraction Products Using Commutative and Associative Properties of Multiplication with Fractions. The commutative property of multiplication states that when finding a product, changing the order of the factors will not change their product.

In symbols, the commutative property of multiplication says that for numbers a and b:

a⋅b=b⋅a



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The associative property of multiplication states that when finding a product, changing the way factors are grouped will not change their product. In symbols, the associative property of multiplication says that for numbers a,b and c:

(a⋅b)⋅c=a⋅(b⋅c)

The commutative property of multiplication can be useful when multiplying more than two fractions. It allows you to reorder the fractions in order to simplify before multiplying.

Here is an example.

Find the product of 68⋅12⋅1618.

First, notice that the first fraction and the third fraction have common factors. Use the commutative property to switch the second and third fractions.

68⋅12⋅1618 is equivalent to 68⋅1618⋅12.

Now, focus on the first two fractions. Look for common factors along the diagonals in the numerators and the denominators. Notice that 6 and 18 both have a factor of 6. You can divide both the 6 and the 18 by 6.

68⋅1618⋅12=18⋅163⋅12

Similarly, both the 16 and the 8 have a factor of 8. You can divide both the 16 and the 8 by 8.

18⋅163⋅12=11⋅23⋅12

Now you can multiply the fractions. Use what you have learned about fraction multiplication.

11⋅23⋅12=1⋅2⋅11⋅3⋅2=26

Finally, you can simplify your answer.

26=13

The answer is 68⋅12⋅1618=13.

Both the commutative property of multiplication and the associative property of multiplication can be useful in simplifying expressions involving fractions. The commutative property of multiplication allows you to reorder factors while the associative property of multiplication allows you to regroup factors.

Here is an example where the commutative property is useful.

Simplify 23⋅x⋅78.

First, use the commutative property of multiplication to reorder the factors.

23⋅x⋅78 is equivalent to 23⋅78⋅x.

Next, simplify 23⋅78⋅x. Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.

23⋅78=1424=712

23⋅78⋅x simplifies to 712x.

The answer is that 23⋅x⋅78 simplifies to 712x.

Here is an example where the associative property is useful.

Simplify (x⋅12)⋅35.

First, use the associative property of multiplication to regroup the factors.

(x⋅12)⋅35 is equivalent to x⋅(12⋅35).

Now, simplify x⋅(12⋅35). Multiply the fractions in the parentheses using what you have learned about fraction multiplication.

12⋅35=310

x⋅(12⋅35) simplifies to x⋅310.

The answer is that (x⋅12)⋅35 simplifies to x⋅310 or 310x.

Examples

Example 1

Earlier, you were given a problem about Elaine and Connie and their bracelets.

Connie is going to give Elaine 15 of the money she makes. Elaine is going to put 12 of that money into a savings account at the bank. Elaine wants to know how she could write an expression to represent the fraction of money that Connie earns that Elaine will be putting into the bank.

First, Elaine can write an expression and then she can simplify it. She doesn’t know how much money Connie will earn, so that unknown will be her variable. Let x be the amount of money Connie earns.

Elaine will be getting 15 of the money Connie earns, so Elaine will be getting 15⋅x. Elaine is going to put 12 of that money in the bank. So the money Elaine will be putting in the bank can be represented by the expression.

12⋅(15⋅x)

Now Elaine can simplify the expression. She can use the associative property of multiplication to regroup the factors.

12⋅(15⋅x) is equivalent to (12⋅15)⋅x.

Next, she can simplify (12⋅15)⋅x. She can multiply the fractions in the parentheses using what she has learned about fraction multiplication.

12⋅15=110

(12⋅15)⋅x simplifies to 110x.

The answer is that Elaine will be putting 110x dollars in the bank, where x is the amount of money Connie earns selling bracelets.

Example 2

Multiply 916⋅12⋅815.

First, notice that the first fraction and the third fraction have common factors. Use the commutative property to switch the second and third fractions.

916⋅12⋅815 is equivalent to 916⋅815⋅12.

Now, focus on the first two fractions. Look for common factors along the diagonals in the numerators and the denominators. Notice that 9 and 15 both have a factor of 3. You can divide both the 9 and the 15 by 3.

916⋅815⋅12=316⋅85⋅12

Similarly, both the 8 and the 16 have a factor of 8. You can divide both the 8 and the 16 by 8.

316⋅85⋅12=32⋅15⋅12

Now, you can multiply the fractions. Use what you have learned about fraction multiplication.

32⋅15⋅12=320

The answer is 916⋅12⋅815=320.

Example 3

Simplify (x⋅45)⋅12.

First, use the associative property of multiplication to regroup the factors.

(x⋅45)⋅12 is equivalent to x⋅(45⋅12).

Now, simplify x⋅(45⋅12). Multiply the fractions in the parentheses using what you have learned about fraction multiplication.

45⋅12=410=25

x⋅(45⋅12) simplifies to x⋅25.

The answer is that (x⋅45)⋅12 simplifies to x⋅255 or 25x.

Example 4

Simplify 67⋅x⋅13.

First, use the commutative property of multiplication to reorder the factors.

67⋅x⋅13 is equivalent to 67⋅13⋅x.

Next, simplify 67⋅13⋅x. Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.

67⋅13=621=27

67⋅13⋅x simplifies to 27x.

The answer is that 67⋅x⋅13 simplifies to 27x.

Example 5

Simplify 23⋅x⋅49.

First, use the commutative property of multiplication to reorder the factors.

23⋅x⋅49 is equivalent to 23⋅49⋅x.

Next, simplify 23⋅49⋅x. Multiply the fractions using what you have learned about fraction multiplication. Simplify your result.

23⋅49=827

23⋅49⋅x simplifies to 827x.

The answer is that 23⋅x⋅49 simplifies to 827x.

That is it on Associative and Commutative Properties For Fraction Products. I hope this article was helpful. Kindly share!!!

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