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4.13: Simplify Variable Expressions Involving Integer Division

Difficulty Level: At Grade Created by: CK-12

Let’s Think About It

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 \begin{align*}b\end{align*}b 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 \begin{align*}b\end{align*}b?

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:

\begin{align*}\frac{24x}{3} + 18xy\end{align*}24x3+18xy

In this variable expression there are two terms. The first term is \begin{align*}\frac{24x}{3}\end{align*}24x3 and the second term is \begin{align*}18xy\end{align*}18xy.

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 \begin{align*}\frac{24x}{3}\end{align*}24x3 can be simplified. Let's look at how you would simplify it.

Remember that a fraction bar is the same as division. So \begin{align*}\frac{24x}{3}\end{align*}24x3 is the same as \begin{align*}24x \div 3\end{align*}24x÷3. 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.

\begin{align*}\frac{24x}{3}\end{align*}24x3

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

\begin{align*}\frac{24x}{3} = \frac{24 \cdot x}{3}\end{align*}24x3=24x3

Now, divide the integers within the term.

\begin{align*}\frac{24}{3}=8\end{align*}243=8

There is only one \begin{align*}x\end{align*} in the expression, so it will not change.

The answer is \begin{align*}\frac{24x}{3}=8x\end{align*}.

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

Simplify \begin{align*}-24y \div 2y\end{align*}.

First, rewrite using a fraction bar.

\begin{align*}-24y \div 2y = \frac{-24y}{2y}\end{align*}

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

\begin{align*}\frac{-24y}{2y} = \frac{-24 \cdot y}{2 \cdot y}\end{align*}

Now, focus on the integers. Divide \begin{align*}\frac{-24}{2}\end{align*}. You know that \begin{align*}\frac{24}{2} = 12\end{align*} and a negative divided by a positive equals a negative.

\begin{align*}\frac{-24}{2} = -12\end{align*}

Next, look at the variables. There is a \begin{align*}y\end{align*} in both the numerator and the denominator. Any number divided by itself is equal to 1, so \begin{align*}\frac{y}{y}\end{align*} is equal to 1. You might say that the \begin{align*}y\end{align*}'s “cancel out”.

\begin{align*}\frac{-24 \cdot y}{2 \cdot y} = -12 \cdot 1= -12\end{align*}

The answer is \begin{align*}-24y \div 2y=-12\end{align*}.

Guided Practice

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

First, rewrite using a fraction bar.

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

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

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

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

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

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{-18 \cdot a \cdot b}{9 \cdot b}=-2 \cdot a \cdot 1=-2a\end{align*}

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

Examples

Example 1

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

First, rewrite using a fraction bar.

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

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

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

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

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

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

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

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

Example 2

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

First, rewrite using a fraction bar.

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

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

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

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

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

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

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.

  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*}

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Difficulty Level:
At Grade
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Date Created:
Dec 02, 2015
Last Modified:
Mar 23, 2016
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