# 2.1.4: Multiplication and Division Phrases as Equations

- Page ID
- 4348

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Multiplication and Division Phrases as Equations

There is a bunny population behind Ishmael’s house. Ishmael is helping the local park ranger track the population numbers. At the start of July there are 127 females. By the end of the month, Ishmael and the ranger count 415 new bunnies. What is the average number of bunnies that each female had during that time? How can Ishmael write an **equation** that shows this situation?

In this concept, you will learn how to write multiplication and division phrases as single **variable** equations.

**Writing Multiplication and Division Phrases as Equations**

Just as you can write addition and subtraction expressions from words or phrases, you can also write multiplication and division expressions. You can use key words to help you with this. The more familiar you become with the key words that identify a multiplication or division **expression**, the better you will become at writing expressions.

Here are some phrases that can be translated into multiplication or division expressions.

Multiplication Phrases |
Division Phrases |
||

9 times k |
9×k or 9k | 8 divided into n groups |
8÷n or 8/n |

twice as much as m |
2×m or 2m | q shared equally by 3 people |
q÷3 or q/3 |

half of r |
r÷2 or r/2 | ||

one-third of p |
p÷3 or p/3 |

Remember, these words are only a helpful guide. You should always think about which operation makes sense for a particular situation.

Let’s look at an example.

Write an **algebraic expression** to represent the phrase “three times a number, t.”

Use a number, an operation sign, or a variable to represent each part of the phrase.

First, consider the phrase and look for key terms.

three times a number, t

Next, rewrite the phrase using numbers and operations.

3×t

Then, simplify the expression.

3t

The phrase is represented by the expression 3t.

Here is another example.

Mr. Warren separated 30 students into n equal groups. Write an algebraic expression to represent the number of students in each group.

First, consider the phrase and look for key terms.

30 students into n equal groups

Separating 30 students into n equal groups means dividing 30 students into n equal groups.

Next, write a division expression.

30/n

The phrase is represented by the expression 30/n.

When an expression equals something it is an equation. If the number of students in each group needed to be five, for instance, you could write the following equation.

n=5

Then, you could evaluate the expression 30/n using the given variable 5.

**Examples**

Example \(\PageIndex{1}\)

Earlier, you were given a problem about Ishmael’s local bunnies.

There are 127 females and 415 bunnies. Ishmael needs to write an equation that represents the average number of bunnies each female had.

**Solution**

First, let n be the average number of bunnies per female.

127n

This is the total number of bunnies in the population.

Then, because you know there are 415 bunnies, you can write an equation.

127n=415

Example \(\PageIndex{1}\)

Write a single variable equation for the following phrase.

Keith bought tickets to the movies. The tickets were $8.50 each. Keith spent a total of $34.00. How many tickets did Keith buy?

**Solution**

First, let t represent the number of tickets that Keith bought.

The cost of each ticket is $8.50. The total price of the tickets will be the number of tickets, represented by t, times the price of each ticket.

Next, write an expression that represents this.

8.5t

Then, because you know the total cost of the tickets was $34.00, write an equation.

8.5t=34.00

**Write a multiplication or division equation for the following phrases.**

Example \(\PageIndex{1}\)

Four times a number is eight.

**Solution**

Let x be “a number.”

“Times” means multiplication and “is” means equals.

The answer is 4x=8.

Example \(\PageIndex{1}\)

Sixteen candles divided into a number of piles is two candles in each pile. Let x be the number of piles.

**Solution**

“Divided into” means division but be careful about the order, and “is” means equals.

The answer is 162=2.

Example \(\PageIndex{1}\)

The product of five and a number is fifteen.

**Solution**

Let x be “a number.”

“Product” means you are multiplying, and “is” means equals.

The answer is 5x=15.

**Review**

Write an equation for each phrase.

- The product of four and a number is twelve.
- Six times a number is thirty.
- Twelve times a number is forty-eight.
- Fourteen times a number is twenty-eight.
- The product of five and a number is thirty.
- Eight times a number is sixty-four.
- Twenty divided by a number is four.
- Eighty divided by a number is four.
- Nineteen times an unknown number is ninety-five.
- Thirteen times an unknown number is thirty-nine.
- Twelve divided into groups is six.
- An unknown number divided by two is eight.
- An unknown number divided by seven is fourteen.
- An unknown number times five is thirty-five.
- An unknown number divided by twelve is twelve.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.2.

### Vocabulary

Term | Definition |
---|---|

Algebraic Expression |
An expression that has numbers, operations and variables, but no equals sign. |

Equation |
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs. |

Expression |
An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols. |

Variable |
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n. |

### Additional Resources

Video:

Practice: **Multiplication and Division Phrases as Equations**