# 2.18: Translating Words Into Algebra

- Page ID
- 45778

### Learning outcome

- Translate simple word phrases into math notation

We know there are many operation symbols that are used in algebra. Now, we’ll translate word phrases into algebraic expressions and equations. The symbols and variables we’ve talked about will help us do that. They are summarized below.

Operation | Phrase | Expression |
---|---|---|

Addition | plus the sum of and increased by more than the total of and added to | |

Subtraction | minus the difference of and subtracted from decreased by less than | |

Multiplication | times the product of and | , , , |

Division | divided by the quotient of and the ratio of and divided into | , , , |

Look closely at these phrases using the four operations:

- the sum
*of**and* - the difference
*of**and* - the product
*of**and* - the quotient
*of**and*

Each phrase tells you to operate on two numbers. Look for the words ** of** and

**to find the numbers.**

*and*### example

Translate each word phrase into an algebraic expression:

1. The difference of and

2. The quotient of and

Solution

1. The key word is *difference*, which tells us the operation is subtraction. Look for the words *of* and *and* to find the numbers to subtract.

The difference of and

minus

2. The key word is *quotient*, which tells us the operation is division.

The quotient of and

divide by

This can also be written as or

### try it

[ohm_question]146541[/ohm_question]

[ohm_question]143240[/ohm_question]

[ohm_question]143207[/ohm_question]

[ohm_question]146542[/ohm_question]

### example

Translate each word phrase into an algebraic expression:

- How old will you be in eight years? Let’s say your current age is .
- How old were you seven years ago? This is seven years less than your age now. Let’s say your current age is .

[reveal-answer q=”879313″]Show Answer[/reveal-answer]

[hidden-answer a=”879313″]

Solution:

1. Eight more than

The key words are *more than*. They tell us the operation is addition. *More than* means “added to”.

Eight more than

Eight added to

2. Seven less than .

The key words are *less than*. They tell us the operation is subtraction. *Less than* means “subtracted from”.

Seven less than

Seven subtracted from

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[ohm_question]144907[/ohm_question]

### example

Translate each word phrase into an algebraic expression:

1. Five times the sum of and

2. The sum of five times and

[reveal-answer q=”888823″]Show Answer[/reveal-answer]

[hidden-answer a=”888823″]

Solution

1. There are two operation words: *times* tells us to multiply and *sum* tells us to add. Because we are multiplying times the sum, we need parentheses around the sum of and .

five times the sum of and

2. To take a sum, we look for the words *of* and *and* to see what is being added. Here we are taking the sum *of* five times and .

the sum of five times and

Notice how the use of parentheses changes the result. In part 1, we add first and in part 2, we multiply first.

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Watch the video below to better understand how to write algebraic expressions from statements.

We’ll eventually apply our skills in algebra to solving equations in complex word problems. Usually start by translating a word phrase to an algebraic equation. Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean *equals*. Some words that translate to the equal sign are:

- is equal to
- is the same as
- is
- gives
- was
- will be

It may be helpful to put a box around the *equals* word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

### example

Translate the sentence into an algebraic equation: The sum of and is .

Solution

The word *is* tells us the equal sign goes between 9 and 15.

Locate the “equals” word(s). | |

Write the = sign. | |

Translate the words to the left of the equals word into an algebraic expression. | |

Translate the words to the right of the equals word into an algebraic expression. |

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[ohm_question]146546[/ohm_question]

### example

Translate the sentence into an algebraic equation: Twice the difference of and gives .

[reveal-answer q=”79895″]Show Answer[/reveal-answer]

[hidden-answer a=”79895″]

Solution

Locate the “equals” word(s). | |

Recognize the key words: twice; difference of …. and …. | Twice means two times. |

Translate. | |

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Now let’s apply our understanding of translating words to algebra in a real world scenario.

### example

The height of a rectangular window is inches less than the width. Let represent the width of the window. Write an expression for the height of the window.

[reveal-answer q=”704093″]Show Answer[/reveal-answer]

[hidden-answer a=”704093″]

Solution

Write a phrase about the height. | less than the width |

Substitute for the width. | less than |

Rewrite ‘less than’ as ‘subtracted from’. | subtracted from |

Translate the phrase into algebra. |

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[ohm_question]144917[/ohm_question]

### example

Blanca has dimes and quarters in her purse. The number of dimes is less than times the number of quarters. Let represent the number of quarters. Write an expression for the number of dimes.

[reveal-answer q=”365611″]Show Answer[/reveal-answer]

[hidden-answer a=”365611″]

Solution

Write a phrase about the number of dimes. | two less than five times the number of quarters |

Substitute for the number of quarters. | less than five times |

Translate times . | less than |

Translate the phrase into algebra. |

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[ohm_question]144918[/ohm_question]

In the following video we show more examples of how to write basic algebraic expressions from words, and simplify.

Let’s practice translating sentences into algebraic equations and then solving them.

### example

Translate and solve: Three more than is equal to .

[reveal-answer q=”989526″]Show Answer[/reveal-answer]

[hidden-answer a=”989526″]

Solution

Three more than x is equal to . | ||

Translate. | ||

Subtract from both sides of the equation. | ||

Simplify. | ||

We can check. Let . | ||

So is the solution.

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[ohm_question]146551[/ohm_question]

### example

Translate and solve: The difference of and is .

[reveal-answer q=”763260″]Show Answer[/reveal-answer]

[hidden-answer a=”763260″]

Solution

The difference of and is . | ||

Translate. | ||

Add to both sides. | ||

Simplify. | ||

We can check. Let . | ||

So is the solution.

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[ohm_question]146552[/ohm_question]

[ohm_question]146386[/ohm_question]

In the following video we show more examples of how to translate an equation into words and solve. Note that this is different from the written examples on this page because we start with the mathematical equation then translate it into words.

### Exercises

Translate from algebra to words:

Solution:

1. |

plus |

the sum of twelve and fourteen |

2. |

times |

the product of thirty and five |

3. |

divided by |

the quotient of sixty-four and eight |

4. |

minus |

the difference of and |

### TRY IT

An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=100

An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=100

An interactive or media element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=100

What if we are working with expressions that are not equal? An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that is greater than , it means that is to the right of on the number line. We use the symbols and }" title="\text{>}" class="latex mathjax"> for inequalities.

is read is less than

is to the left of on the number line

b" title="a>b" class="latex mathjax"> is read is greater than

is to the right of on the number line

The expressions b" title="a**b" class="latex mathjax"> can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,**

a.\text{ For example, }7<11\text{ is equivalent to }11>7.\hfill \\ a>b\text{ is equivalent to }b4\text{ is equivalent to }4<17.\hfill \end{array}" title="\begin{array}{l}a<b\text{ is equivalent to }b>a.\text{ For example, }7<11\text{ is equivalent to }11>7.\hfill \\ a>b\text{ is equivalent to }b4\text{ is equivalent to }4<17.\hfill \end{array}" class="latex mathjax">

When we write an inequality symbol with a line under it, such as , it means or . We read this is less than or equal to . Also, if we put a slash through an equal sign, , it means not equal.

We summarize the symbols of equality and inequality in the table below.

Algebraic Notation | Say |
---|---|

is equal to | |

is not equal to | |

is less than | |

b" title="a>b" class="latex mathjax"> | is greater than |

is less than or equal to | |

is greater than or equal to |

### Symbols and

The symbols and each have a smaller side and a larger side.

smaller side larger side

larger side smaller side

The smaller side of the symbol faces the smaller number and the larger faces the larger number.

### Exercises

Translate from algebra to words:

- 10\div 2" title="9>10\div 2" class="latex mathjax">

[reveal-answer q=”346424″]Show Answer[/reveal-answer]

[hidden-answer a=”346424″]

Solution:

1. |

is less than or equal to |

2. |

is not equal to minus |

3. |

10\div 2" title="9>10\div 2" class="latex mathjax"> |

is greater than divided by |

4. |

plus is less than |

[/hidden-answer]

### TRY IT

In the following video we show more examples of how to write inequalities as words.

- Prealgebra.
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