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# 4.13: Simplify Variable Expressions Involving Integer Division

Difficulty Level: At Grade Created by: CK-12

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 b\begin{align*}b\end{align*} represents the number of bird houses they sell, how could Jake write and simplify a variable expression that represents how much money he will make in terms of b\begin{align*}b\end{align*}? In this concept, you will learn how to simplify terms within variable expressions using integer division. ### Simplifying Variable Expressions 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: 24x3+18xy\begin{align*}\frac{24x}{3} + 18xy\end{align*} In this variable expression there are two terms. The first term is 24x3\begin{align*}\frac{24x}{3}\end{align*} and the second term is 18xy\begin{align*}18xy\end{align*}. 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 24x3\begin{align*}\frac{24x}{3}\end{align*} can be simplified. Let's look at how you would simplify it. Remember that a fraction bar is the same as division. So 24x3\begin{align*}\frac{24x}{3}\end{align*} is the same as 24x÷3\begin{align*}24x \div 3\end{align*}. You will see expressions written in both ways. If the expression is not already in fraction form, it helps to rewrite it in fraction form. 24x3\begin{align*}\frac{24x}{3}\end{align*} Your next step is to separate out the integers and variable in the numerator and the denominator. 24x3=24x3\begin{align*}\frac{24x}{3} = \frac{24 \cdot x}{3}\end{align*} Now, divide the integers within the term. 243=8\begin{align*}\frac{24}{3}=8\end{align*} There is only one x\begin{align*}x\end{align*} in the expression, so it will not change. The answer is 24x3=8x\begin{align*}\frac{24x}{3}=8x\end{align*}. Let's look at another example where there is the same variable more than once. Simplify 24y÷2y\begin{align*}-24y \div 2y\end{align*}. First, rewrite using a fraction bar. 24y÷2y=24y2y\begin{align*}-24y \div 2y = \frac{-24y}{2y}\end{align*} Next, separate out the integers and variables in the numerator and the denominator. 24y2y=24y2y\begin{align*}\frac{-24y}{2y} = \frac{-24 \cdot y}{2 \cdot y}\end{align*} Now, focus on the integers. Divide 242\begin{align*}\frac{-24}{2}\end{align*}. You know that 242=12\begin{align*}\frac{24}{2} = 12\end{align*} and a negative divided by a positive equals a negative. 242=12\begin{align*}\frac{-24}{2} = -12\end{align*} Next, look at the variables. There is a y\begin{align*}y\end{align*} in both the numerator and the denominator. Any number divided by itself is equal to 1, so yy\begin{align*}\frac{y}{y}\end{align*} is equal to 1. You might say that the y\begin{align*}y\end{align*}'s “cancel out”. 24y2y=121=12\begin{align*}\frac{-24 \cdot y}{2 \cdot y} = -12 \cdot 1= -12\end{align*} The answer is 24y÷2y=12\begin{align*}-24y \div 2y=-12\end{align*}. ### Examples #### Example 1 Earlier, you were given a problem about Jake and his bird houses. Jake and his three friends will sell b\begin{align*}b\end{align*} bird houses for20 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 b\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:

20b4\begin{align*}\frac{20b}{4}\end{align*}

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

20b4\begin{align*}\frac{20 \cdot b}{4}\end{align*}

Now, focus on the integers. Divide 204\begin{align*}\frac{20}{4}\end{align*}.

204=5\begin{align*}\frac{20}{4} = 5\end{align*}

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

20b4=5b\begin{align*}\frac{20 \cdot b}{4} = 5b\end{align*}

The answer is 20b4=5b\begin{align*}\frac{20b}{4} = 5b\end{align*}.

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

#### Example 2

Simplify 18ab÷9b\begin{align*}-18ab \div 9b\end{align*}.

First, rewrite using a fraction bar.

18ab÷9b=18ab9b\begin{align*}-18ab \div 9b = \frac{-18ab}{9b}\end{align*}

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

18ab9b=18ab9b\begin{align*}\frac{-18ab}{9b} = \frac{-18 \cdot a\cdot b}{9 \cdot b}\end{align*}

Now, focus on the integers. Divide 189\begin{align*} \frac{-18}{9}\end{align*}. You know that 189=2\begin{align*}\frac{18}{9}=2\end{align*} and a negative divided by a positive equals a negative.

189=2\begin{align*}\frac{-18}{9}=-2\end{align*}

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

18ab9b=2a1=2a\begin{align*}\frac{-18 \cdot a \cdot b}{9 \cdot b}=-2 \cdot a \cdot 1=-2a\end{align*}

The answer is 18ab÷9b=2a\begin{align*}-18ab \div 9b=-2a\end{align*}.

#### Example 3

Simplify 14a÷7\begin{align*}-14a \div -7\end{align*}.

First, rewrite using a fraction bar.

14a÷7=14a7\begin{align*}-14a \div -7= \frac{-14a}{-7}\end{align*}

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

14a7=14a7\begin{align*}\frac{-14a}{-7} = \frac{-14 \cdot a}{-7}\end{align*}

Now, focus on the integers. Divide 147\begin{align*}\frac{-14}{-7}\end{align*}. You know that 147=2\begin{align*}\frac{14}{7}=2\end{align*} and a negative divided by a negative equals a positive.

147=2\begin{align*}\frac{-14}{-7}=2\end{align*}

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

14a7=2a\begin{align*}\frac{-14 \cdot a}{-7}=2a\end{align*}

The answer is 14a÷7=2a\begin{align*}-14a \div -7=2a\end{align*}.

#### Example 4

Simplify 28ab÷7b\begin{align*}28ab \div -7b\end{align*}.

First, rewrite using a fraction bar.

28ab÷7b=28ab7b\begin{align*}28ab \div -7b = \frac{28ab}{-7b}\end{align*}

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

28ab7b=28ab7b\begin{align*}\frac{28ab}{-7b} = \frac{28 \cdot a \cdot b}{-7 \cdot b}\end{align*}

Now, focus on the integers. Divide 287\begin{align*}\frac{28}{-7}\end{align*}. You know that 287=4\begin{align*}\frac{28}{7}=4\end{align*} and a positive divided by a negative equals a negative.

287=4\begin{align*}\frac{28}{-7}=-4\end{align*}

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

28ab7b=4a1=4a\begin{align*}\frac{28 \cdot a \cdot b}{-7 \cdot b} = -4 \cdot a \cdot 1=-4a\end{align*}

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

#### Example 5

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

First, rewrite using a fraction bar.

6x÷2y=6x2y\begin{align*}-6x \div -2y= \frac{-6x}{-2y}\end{align*}

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

6x2y=6x2y\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*}.

### Review

Simplify each variable expression.

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

To see the Review answers, open this PDF file and look for section 4.13.

1. [1]^ License: CC BY-NC 3.0

## Date Created:

Dec 02, 2015

Dec 02, 2015
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