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# Fractions

## Identity, inverse, and zero product properties

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Practice Fractions
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Identify and Apply Number Properties in Fraction Operations

Do you know how to simplify an expression with fractions by using properties? Take a look at this dilemma.

Simplify: $\frac{2}{3}\times\left(\frac{2}{7}\times\frac{3}{2}\right)$

To simplify this expression, you will need to know how to work with number properties and fractions. This Concept will show you how to accomplish this task successfully.

### Guidance

Now that we are working with fractions, you will have a chance to investigate how the different properties of addition and subtraction can help us when we work with fractions in expressions.

Here are a few properties.

The sum of any number and zero is that number: $\frac{3}{11}+0=\frac{3}{11}$

The sum of any number and its inverse is zero: $\frac{3}{4}+ \left( - \frac{3}{4}\right)=0$

Which of the following shows the Additive Inverse Property?

a. $\frac{x}{y}+0=0$

b. $\frac{x}{y}+0=\frac{x}{y}$

c. $\frac{x}{y}+\left(-\frac{x}{y}\right)=0$

Consider choice a.

This equation states that a number added to zero is equal to zero. This is not necessarily correct, unless $\frac{x}{y}$ is also equal to zero.

Consider choice b.

This equation states that the sum of a number and zero is equal to that number. This is correct, but illustrates the additive identity, not the additive inverse property.

Consider choice c.

This equation states that the sum of a number and its inverse is equal to zero. This illustrates the additive inverse property, so this is the correct equation.

We can also use these properties to help us when we simplify a numerical expression. Remember that a numerical expression is a group of numbers and operations. Because we are working with fractions, the numerical expressions in this section will be made up of fractions.

Simplify: $\frac{1}{8}+\left(\frac{1}{4}+\frac{3}{8}\right)$

You can use addition properties to reorganize this expression to make it easier to simplify.

First apply the commutative property: $\frac{1}{8}+\left(\frac{1}{4} + \frac{3}{8}\right)=\frac{1}{8}+\left(\frac{3}{8}+\frac{1}{4}\right)$

Then apply the associate property: $\frac{1}{8}+\left(\frac{3}{8} + \frac{1}{4}\right)=\left ( \frac{1}{8}+\frac{3}{8}\right)+\frac{1}{4}$

Now you can easily simplify to find the sum.

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

The answer is $\frac{3}{4}$ .

Using properties to reorganize fractions can help us to work with these fractions. Notice that we reorganized the common denominators together and this simplified our work.

Now let’s look at how the properties of multiplication and division can help us when working with fractions. You have already learned the Commutative Property, the Associative Property, and the Distributive Property.

Multiplicative Identity

The product of any number and one is that number: $\frac{3}{11}\times 1=\frac{3}{11}$

Zero Property

The product of any number and zero is zero: $\frac{4}{7}\times 0=0$

Multiplicative Inverse

The product of any number and its reciprocal is one: $\frac{3}{4}\times\frac{4}{3}=1$

Which of the following shows the Multiplicative Inverse Property?

a. $\frac{x}{y}\times 0=\frac{x}{y}$

b. $\frac{x}{y}\times \frac{y}{x}=0$

c. $\frac{x}{y}\times \frac{y}{x}=1$

Consider choice a.

This equation states that the product of a number and zero is equal to that number. This is not correct.

Consider choice b.

This equation states that the product of a number and its reciprocal is equal to zero. This is not correct.

Consider choice c.

This equation states that the product of a number and its reciprocal is equal to one. This illustrates the multiplicative inverse property, so this is the correct equation.

Take a few minutes to write these properties down in your notebooks. Be sure to include an example of each.

We could also use properties when working with a variable. Take a look at this one.

$\frac{2}{3}\times\left(a \times \frac{3}{2}\right)$

First, we can apply the commutative property: $\frac{2}{3} \times \frac{3}{2} \times a$

Now we apply the multiplication inverse property: $\frac{2}{3} \times \frac{3}{2} =1$

Our simplified expression is $a$ .

If we had a value substituted in for $a$ , then that would be our answer.

#### Example A

Name the property: $\frac{3}{8} \times 0$

Solution: Zero Property

#### Example B

Name the property: $\frac{5}{6} \times \frac{6}{5}$

Solution: Multiplicative Inverse

#### Example C

Simplify: $\frac{3}{4} (b \times \frac{4}{3})$

Solution: $a$

Now let's go back to the dilemma from the beginning of the Concept.

Simplify: $\frac{2}{3}\times\left(\frac{2}{7}\times\frac{3}{2}\right)$

You can use multiplication properties to reorganize this expression to make it easier to simplify.

First apply the commutative property: $\frac{2}{3}\times\left(\frac{2}{7}\times\frac{3}{2}\right)=\frac{2}{3} \times \left(\frac{3}{2} \times \frac{2}{7}\right)$

Then apply the associate property: $\frac{2}{3}\times\left(\frac{3}{2}\times\frac{2}{7}\right)=\left(\frac{2}{3} \times \frac{3}{2}\right) \times \frac{2}{7}$

Then apply the multiplicative inverse property: $\left(\frac{2}{3}\times\frac{3}{2}\right)\times\frac{2}{7}=1\times\frac{2}{7}$

Finally, apply the multiplicative identity property: $1 \times \frac{2}{7}=\frac{2}{7}$

### Vocabulary

any number plus zero is still that number.
any number plus it’s opposite or inverse is equal to 0.
Multiplicative Identity
any number times 1 is the same number.
Zero Property
any number times 0 is zero.
Multiplicative Inverse
any number times it’s reciprocal is 1.

### Guided Practice

Here is one for you to try on your own.

Simplify: $\frac{4}{5} + \frac{1}{2} + x$

Solution

First, we find a common denominator so that we can add the fractions. The lowest common denominator for 5 and 2 is 10. Let's rename both fractions in terms of tenths.

$\frac{8}{10} + \frac{5}{10} + x$

Now we can add the fractions.

$\frac{13}{10} + x$

Change the improper fraction to a mixed number.

$1 \frac{3}{10} + x$

This is our simplified expression.

### Practice

Directions : Identify each property shown below.

1. $\frac{3}{4} + 0 = \frac{3}{4}$

2. $\frac{3}{4} + -\frac{3}{4} = 0$

3. $\frac{6}{7} \times 0 = 0$

4. $\frac{5}{8} \times 0 = 0$

5. $\frac{6}{7} \times \frac{7}{6} = 1$

6. $\frac{3}{4} + x = x + \frac{3}{4}$

7. $\frac{1}{4} + y = y + \frac{1}{4}$

8. $\frac{1}{2} \times (x + 3)= \frac{1}{2}x + \frac{1}{2}(3)$

Directions : Simplify each expression.

9. $\frac{3}{4} + \frac{1}{4} + x$

10. $\frac{6}{7} \times \frac{1}{3} \times x$

11. $2 \times \frac{4}{8}x$

12. $3x \times \frac{6}{8}$

13. $\frac{4}{5} + \frac{1}{2} + \frac{6}{10}$

14. $\frac{6}{10} - \frac{1}{3}$

15. $\frac{1}{2} \times 3x$

### Vocabulary Language: English

The sum of any number and zero is the number itself.

The additive inverse or opposite of a number x is -1(x). A number and its additive inverse always sum to zero.
Multiplicative Identity

Multiplicative Identity

The multiplicative identity for multiplication of real numbers is one.
Multiplicative Inverse

Multiplicative Inverse

The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).
Zero Property

Zero Property

The zero property of multiplication says that the product of any number and zero is zero. The zero property of addition states that the sum of any number and zero is the number.