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Simplify Variable Expressions Involving Integer Division

Divide integers with and without variables to create a new term.

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Simplify Variable Expressions Involving Integer Division
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 \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 \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:

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

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

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

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*}

Now, divide the integers within the term.

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

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*}.

Examples

Example 1

Earlier, you were given a problem about 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.

Example 2

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*}.

Example 3

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 4

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 5

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*}.

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*}

Answers for Review Problems

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

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Vocabulary

Quotient

The quotient is the result after two amounts have been divided.

Undefined

An expression in mathematics is undefined when it has no meaning. Division by zero is undefined.

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

Image Attributions

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

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