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# Fractions

## Identity, inverse, and zero product properties

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Identify and Apply Number Properties in Fraction Operations

### [Figure1] License: CC BY-NC 3.0

The United States flag is typically 1910\begin{align*}1\frac{9}{10}\end{align*} as long as it is wide. The length of a particular United States flag is 5 feet. How wide is it?

In this concept, you will learn to identify and apply number properties in fraction operations.

### Number Properties for Fractions

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

Here are a few properties you need to keep in mind when applying number properties to fraction operations:

The Additive Identity Property states that the sum of any number and zero is that number. For example:

311+0=311\begin{align*}\frac{3}{11} + 0 = \frac{3}{11}\end{align*}

The Additive Inverse Property states that the sum of any number and its inverse is zero. For example:

34+(34)=0\begin{align*}\frac{3}{4} + \left ( -\frac{3}{4} \right ) = 0\end{align*}

Let's consider which of the following shows the Additive Inverse Property.

a. xy+0=0\begin{align*}\frac{x}{y} + 0= 0\end{align*}

b. xy+0=xy\begin{align*}\frac{x}{y} + 0 = \frac{x}{y}\end{align*}

c. xy+(xy)=0\begin{align*}\frac{x}{y} + \left ( -\frac{x}{y} \right ) = 0\end{align*}

Consider choice a. This equation states that a number added to zero is equal to zero. This is not necessarily correct, unless xy\begin{align*}\frac{x}{y}\end{align*} 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.

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

The Multiplicative Identity Property states that the product of any number and one is that number. For example:311×1=311\begin{align*}\frac{3}{11} \times 1 = \frac{3}{11}\end{align*}The Zero Property states that the product of any number and zero is zero. For example:47×0=0\begin{align*}\frac{4}{7} \times 0 = 0\end{align*}The Multiplicative Inverse Property states that the product of any number and its reciprocal is one. For example:34×43=1\begin{align*}\frac{3}{4} \times \frac{4}{3} = 1\end{align*}Which of the following shows the Multiplicative Inverse Property?

a. xy×0=xy\begin{align*}\frac{x}{y} \times 0 = \frac{x}{y}\end{align*}

b. xy×yx=0\begin{align*}\frac{x}{y} \times \frac{y}{x} = 0\end{align*}

c. xy×yx=1\begin{align*}\frac{x}{y} \times \frac{y}{x} = 1\end{align*}

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.

Let's look at an example.

Simplify.

18+(14+38)\begin{align*}\frac{1}{8} + \left ( \frac{1}{4} + \frac{3}{8} \right )\end{align*}

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

First, apply the commutative property.

18+(38+14)\begin{align*}\frac{1}{8} + \left ( \frac{3}{8} + \frac{1}{4} \right )\end{align*}

Next, apply the associate property.

(18+38)+14\begin{align*}\left ( \frac{1}{8} + \frac{3}{8} \right ) + \frac{1}{4}\end{align*}

Then, simplify to find the sum.

(18+38)+14====48+1448+286834\begin{align*}\begin{array}{rcl} \left ( \frac{1}{8} + \frac{3}{8} \right ) + \frac{1}{4} & = & \frac{4}{8} + \frac{1}{4}\\ \\ & = & \frac{4}{8} + \frac{2}{8}\\ \\ & = & \frac{6}{8}\\ \\ & = & \frac{3}{4} \end{array}\end{align*}

You could also use properties when working with a variable. Let's look at an example.

23×(a×32)\begin{align*}\frac{2}{3} \times \left ( a \times\frac{3}{2} \right )\end{align*}

First, apply the commutative property.

23×(32×a)\begin{align*}\frac{2}{3} \times \left ( \frac{3}{2} \times a \right )\end{align*}

Next, apply the multiplication inverse property.

23×32=1\begin{align*}\frac{2}{3} \times \frac{3}{2} = 1\end{align*}

23×(32×a)==1×aa\begin{align*}\begin{array}{rcl} \frac{2}{3} \times \left ( \frac{3}{2} \times a \right ) & = & 1 \times a\\ & = & a \end{array}\end{align*}

The answer is a\begin{align*}a\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about the United States flag and its standard ratio.

Remember, it is 5 feet long and the length is 1910\begin{align*}1 \frac{9}{10}\end{align*} times the width.

First, turn the mixed number into an improper fraction.

10×1+91910==191910\begin{align*}\begin{array}{rcl} 10 \times 1 + 9 & = & 19\\ 1 \frac{9}{10} & = & \frac{19}{10} \end{array}\end{align*}

Next, set up the equation to solve for the width.

1910×w1910×w==l5\begin{align*}\begin{array}{rcl} \frac{19}{10} \times w & = & l\\ \\ \frac{19}{10} \times w & = & 5 \end{array}\end{align*}

Then, use the multiplicative inverse to isolate w\begin{align*}w\end{align*} (get it on the left side of the equal sign alone).

(1910)×1910×w1×ww===5(1019)5(1019)5019\begin{align*}\begin{array}{rcl} \left ( \frac{19}{10} \right ) \times \frac{19}{10} \times w & = & 5 \left ( \frac{10}{19} \right )\\ \\ 1 \times w & = & 5 \left ( \frac{10}{19} \right )\\ \\ w & = & \frac{50}{19} \end{array}\end{align*}

Then put the improper fraction back into a mixed number.

\begin{align*}\begin{array}{rcl} w & = & \frac{50}{19}\\ \\ w & = & 2\frac{12}{19} \end{array}\end{align*}

The answer is \begin{align*}2\frac{12}{19}\end{align*}.

Therefore, the width of this particular flag is \begin{align*}2\frac{12}{19}\end{align*} feet.

#### Example 2

Simplify:

\begin{align*}\frac{4}{5} + \frac{1}{2} +x\end{align*}

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

\begin{align*}\frac{8}{10} + \frac{5}{10} + x\end{align*}

\begin{align*}\frac{8}{10} + \frac{5}{10} + x = \frac{13}{10} + x\end{align*}

Then, change the improper fraction to a mixed number.

\begin{align*}\frac{13}{10} + x = 1 \frac{3}{10} x\end{align*}

The answer is \begin{align*}1 \frac{3}{10} x\end{align*}.

#### Example 3

Name the property: \begin{align*}\frac{3}{8} \times 0\end{align*}

This is a product of a number and zero. It is the zero property.

#### Example 4

Name the property: \begin{align*}\frac{5}{6} \times \frac{6}{5}\end{align*}

This is a product of a number and its reciprocal. It is the multiplicative inverse property.

#### Example 5

Simplify: \begin{align*}\frac{3}{4} \left ( b \times \frac{4}{3} \right )\end{align*}

First, apply the commutative property.

\begin{align*}\frac{3}{4} \times \left ( \frac{4}{3} \times b \right )\end{align*}

Next, apply the multiplication inverse property.

\begin{align*}\frac{3}{4} \times \frac{4}{3} = 1\end{align*}

\begin{align*}\begin{array}{rcl} \frac{3}{4} \times \left ( \frac{4}{3} \times b \right ) & = & 1 \times b\\ & = & b \end{array}\end{align*}

The answer is \begin{align*}b\end{align*}.

### Review

Identify each of the properties below.

1. \begin{align*}\frac{3}{4} + 0 = \frac{3}{4}\end{align*}

2. \begin{align*}\frac{3}{4} + -\frac{3}{4}=0\end{align*}

3. \begin{align*}\frac{6}{7} \times 0 = 0\end{align*}

4. \begin{align*}\frac{5}{8} \times 0 = 0\end{align*}

5. \begin{align*}\frac{6}{7} \times \frac{7}{6}=1\end{align*}

6. \begin{align*}\frac{3}{4} +x = x+\frac{3}{4}\end{align*}

7. \begin{align*}\frac{1}{4} + y = y+\frac{1}{4}\end{align*}

8. \begin{align*}\frac{1}{2} \times (x+3) = \frac{1}{2} x + \frac{1}{2} (3)\end{align*}

Simplify each expression.

9. \begin{align*}\frac{3}{4} + \frac{1}{4} +x\end{align*}

10. \begin{align*}\frac{6}{7} \times \frac{1}{3} \times x\end{align*}

11. \begin{align*}2 \times \frac{4}{8}x\end{align*}

12. \begin{align*}3x \times \frac{6}{8}\end{align*}

13. \begin{align*}\frac{4}{5} + \frac{1}{2} +\frac{6}{10}\end{align*}

14. \begin{align*}\frac{6}{10} - \frac{1}{3}\end{align*}

15. \begin{align*}\frac{1}{2} \times 3x\end{align*}

To see the Review answers, open this PDF file and look for section 2.10.

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Color Highlighted Text Notes

### 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

The multiplicative identity for multiplication of real numbers is one.

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

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.