# 4.13: Simplify Variable Expressions Involving Integer Division

**At Grade**Created by: CK-12

### Let’s Think About It

**Credit**: Sonny Abesamis

**Source**: https://www.flickr.com/photos/enerva/12097480044/

**License**: CC BY-NC 3.0

Jake and his 3 friends are working together to sell bird houses that they've made. They are selling the bird houses for $20 each and plan to divide up the money they make equally. If

In this concept, you will learn how to simplify terms within variable expressions using integer division.

### Guidance

Recall that a **variable expression** is a math phrase that has numbers, variables, and operations in it. Variable expressions are made up of **terms** that are separated by addition or subtraction.

Here is an example:

In this variable expression there are two terms. The first term is

Sometimes individual terms can be simplified if they contain more than one number or the same variable more than once.

For example, in the variable expression above, the term

Remember that a fraction bar is the same as division. So

Your next step is to separate out the integers and variable in the numerator and the denominator.

Now, divide the integers within the term.

There is only one

The answer is

Let's look at another example where there is the same variable more than once.

Simplify

First, rewrite using a fraction bar.

Next, separate out the integers and variables in the numerator and the denominator.

Now, focus on the integers. Divide

Next, look at the variables. There is a

The answer is

### Guided Practice

Simplify

First, rewrite using a fraction bar.

Next, separate out the integers and variables in the numerator and the denominator.

Now, focus on the integers. Divide

Next, look at the variables. There is a

The answer is

### Examples

#### Example 1

Simplify

First, rewrite using a fraction bar.

Next, separate out the integers and variable in the numerator and the denominator.

Now, focus on the integers. Divide

Next, look at the variable. There is only one

The answer is

#### Example 2

Simplify

First, rewrite using a fraction bar.

Next, separate out the integers and variables in the numerator and the denominator.

Now, focus on the integers. Divide

Next, look at the variables. There is a \begin{align*}b\end{align*} in both the numerator and the denominator. \begin{align*}\frac{b}{b}\end{align*} is equal to 1. There is only one \begin{align*}a\end{align*}, so it will not change.

\begin{align*}\frac{28 \cdot a \cdot b}{-7 \cdot b} = -4 \cdot a \cdot 1=-4a\end{align*}

The answer is \begin{align*}28ab \div -7b=-4a\end{align*} .

#### Example 3

Simplify \begin{align*}-6x \div -2y\end{align*}.

First, rewrite using a fraction bar.

\begin{align*}-6x \div -2y= \frac{-6x}{-2y}\end{align*}

Next, separate out the integers and variables in the numerator and the denominator.

\begin{align*}\frac{-6x}{-2y} = \frac{-6 \cdot x}{-2 \cdot y}\end{align*}

Now, focus on the integers. Divide \begin{align*}\frac{-6}{-2}\end{align*}. You know that \begin{align*}\frac{6}{2}=3\end{align*} and a negative divided by a negative equals a positive.

\begin{align*}\frac{-6}{-2}=3\end{align*}

Next, look at the variables. There is only one \begin{align*}x\end{align*}, so it will not change. There is only one \begin{align*}y\end{align*}, so it will not change.

\begin{align*}\frac{-6 \cdot x}{-2 \cdot y}= \frac{3x}{y}\end{align*}

Notice that the \begin{align*}y\end{align*} must stay in the denominator of the fraction!

The answer is \begin{align*}-6x \div -2y= \frac{3x}{y}\end{align*}.

### Follow Up

**Credit**: USFWS Mountain-Prairie

**Source**: https://www.flickr.com/photos/usfwsmtnprairie/8803101648/

**License**: CC BY-NC 3.0

Remember Jake and his bird houses? Jake and his three friends will sell \begin{align*}b\end{align*} bird houses for $20 each and Jake wonders how much money he will make if he and his friends divide the money equally.

If Jake and his friends sell \begin{align*}b\end{align*} bird houses at $20 each, they will make 20b dollars total. They will divide up that money four ways amongst the four friends. Each person will get:

\begin{align*}\frac{20b}{4}\end{align*}

You can simplify this expression by dividing the integers. First, separate the integers from the variable:

\begin{align*}\frac{20 \cdot b}{4}\end{align*}

Now, focus on the integers. Divide \begin{align*}\frac{20}{4}\end{align*}.

\begin{align*}\frac{20}{4} = 5\end{align*}

Next, look at the variable. There is only one \begin{align*}b\end{align*}, so it will not change.

\begin{align*}\frac{20 \cdot b}{4} = 5b\end{align*}

The answer is \begin{align*}\frac{20b}{4} = 5b\end{align*}.

Jake will make \begin{align*}5b\end{align*} dollars from selling the bird houses.

### Explore More

Simplify each variable expression.

- \begin{align*}36t \div (-9)\end{align*}
- \begin{align*}-56n \div (-7)\end{align*}
- \begin{align*}-22n \div -11n\end{align*}
- \begin{align*}-28n \div 7\end{align*}
- \begin{align*}18xy \div 2x\end{align*}
- \begin{align*}72t \div (-9t)\end{align*}
- \begin{align*}48xy \div (-8y)\end{align*}
- \begin{align*}54xy \div (-9xy)\end{align*}
- \begin{align*}16a \div (4a)\end{align*}
- \begin{align*}-16ab \div (-4b)\end{align*}
- \begin{align*}-99xy \div (-9x)\end{align*}
- \begin{align*}121a \div (11b)\end{align*}
- \begin{align*}-144xy \div (-12)\end{align*}
- \begin{align*}-169y \div (-13x)\end{align*}
- \begin{align*}-225xy÷ (5z)\end{align*}

### Image Attributions

**[1]****^**Credit: Sonny Abesamis; Source: https://www.flickr.com/photos/enerva/12097480044/; License: CC BY-NC 3.0**[2]****^**Credit: USFWS Mountain-Prairie; Source: https://www.flickr.com/photos/usfwsmtnprairie/8803101648/; License: CC BY-NC 3.0

## Description

## Learning Objectives

In this concept, you will learn how to simplify terms within variable expressions using integer division.

## Related Materials

## Difficulty Level:

At Grade## Subjects:

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## Date Created:

Aug 10, 2015## Last Modified:

Sep 23, 2015## Vocabulary

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