### Learning Outcomes

- Translate word phrases into algebraic expressions
- Write an algebraic expression that represents the relationship between two measurements such as length and width or the amount of different types of coins

## Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized below.Operation | Phrase | Expression |
---|---|---|

Addition | [latex]a[/latex] plus [latex]b[/latex]the sum of [latex]a[/latex] and [latex]b[/latex][latex-display]a[/latex] increased by [latex]b[/latex-display][latex-display]b[/latex] more than [latex]a[/latex-display]the total of [latex]a[/latex] and [latex]b[/latex][latex]b[/latex] added to [latex]a[/latex] | [latex]a+b[/latex] |

Subtraction | [latex]a[/latex] minus [latex]b[/latex]the difference of [latex]a[/latex] and [latex]b[/latex][latex-display]b[/latex] subtracted from [latex]a[/latex-display][latex-display]a[/latex] decreased by [latex]b[/latex-display][latex]b[/latex] less than [latex]a[/latex] | [latex]a-b[/latex] |

Multiplication | [latex]a[/latex] times [latex]b[/latex]the product of [latex]a[/latex] and [latex]b[/latex] | [latex]a\cdot b[/latex] , [latex]ab[/latex] , [latex]a\left(b\right)[/latex] , [latex]\left(a\right)\left(b\right)[/latex] |

Division | [latex]a[/latex] divided by [latex]b[/latex]the quotient of [latex]a[/latex] and [latex]b[/latex]the ratio of [latex]a[/latex] and [latex]b[/latex][latex]b[/latex] divided into [latex]a[/latex] | [latex]a\div b[/latex] , [latex]a/b[/latex] , [latex]\frac{a}{b}[/latex] , [latex]b\overline{)a}[/latex] |

- the sum
*of*[latex]a[/latex]*and*[latex]b[/latex] - the difference
*of*[latex]a[/latex]*and*[latex]b[/latex] - the product
*of*[latex]a[/latex]*and*[latex]b[/latex] - the quotient
*of*[latex]a[/latex]*and*[latex]b[/latex]

**and**

*of***to find the numbers.**

*and*### example

Translate each word phrase into an algebraic expression:1. The difference of [latex]20[/latex] and [latex]4[/latex]2. The quotient of [latex]10x[/latex] and [latex]3[/latex]Solution1. 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.[latex-display]\begin{array}{}\\ \text{the difference of }20\text{ and }4\hfill \\ 20\text{ minus }4\hfill \\ 20 - 4\hfill \end{array}[/latex-display]2. The key word is *quotient*, which tells us the operation is division.[latex-display]\begin{array}{}\\ \text{the quotient of }10x\text{ and }3\hfill \\ \text{divide }10x\text{ by }3\hfill \\ 10x\div 3\hfill \end{array}[/latex-display]This can also be written as [latex]\begin{array}{l}10x/3\text{ or}\frac{10x}{3}\hfill \end{array}[/latex]

### 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? What age is eight more years than your age now? Did you add [latex]8[/latex] to your present age? Eight
*more than*means eight added to your present age. - How old were you seven years ago? This is seven years less than your age now. You subtract [latex]7[/latex] from your present age. Seven
*less than*means seven subtracted from your present age.

Answer:Solution:1. Eight more than [latex]y[/latex]2. Seven less than [latex]9z[/latex]1. The key words are *more than*. They tell us the operation is addition. *More than* means "added to".[latex-display]\begin{array}{l}\text{Eight more than }y\\ \text{Eight added to }y\\ y+8\end{array}[/latex-display]2. The key words are *less than*. They tell us the operation is subtraction. *Less than* means "subtracted from".[latex-display]\begin{array}{l}\text{Seven less than }9z\\ \text{Seven subtracted from }9z\\ 9z - 7\end{array}[/latex-display]

### try it

[ohm_question]144907[/ohm_question]

### example

Translate each word phrase into an algebraic expression:1. five times the sum of [latex]m[/latex] and [latex]n[/latex]2. the sum of five times [latex]m[/latex] and [latex]n[/latex]

Answer:Solution1. There are two operation words: *times* tells us to multiply and *sum* tells us to add. Because we are multiplying [latex]5[/latex] times the sum, we need parentheses around the sum of [latex]m[/latex] and [latex]n[/latex].five times the sum of [latex]m[/latex] and [latex]n[/latex][latex-display]\begin{array}{}\\ \\ 5\left(m+n\right)\hfill \end{array}[/latex-display]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 [latex]m[/latex] and [latex]n[/latex].the sum of five times [latex]m[/latex] and [latex]n[/latex][latex-display]\begin{array}{}\\ \\ 5m+n\hfill \end{array}[/latex-display]Notice how the use of parentheses changes the result. In part 1, we add first and in part 2, we multiply first.

### try it

[ohm_question]144916[/ohm_question]

### example

The height of a rectangular window is [latex]6[/latex] inches less than the width. Let [latex]w[/latex] represent the width of the window. Write an expression for the height of the window.

Answer:Solution

Write a phrase about the height. | [latex]6[/latex] less than the width |

Substitute [latex]w[/latex] for the width. | [latex]6[/latex] less than [latex]w[/latex] |

Rewrite 'less than' as 'subtracted from'. | [latex]6[/latex] subtracted from [latex]w[/latex] |

Translate the phrase into algebra. | [latex]w - 6[/latex] |

### try it

[ohm_question]144917[/ohm_question]

### example

Blanca has dimes and quarters in her purse. The number of dimes is [latex]2[/latex] less than [latex]5[/latex] times the number of quarters. Let [latex]q[/latex] represent the number of quarters. Write an expression for the number of dimes.

Answer:Solution

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

Substitute [latex]q[/latex] for the number of quarters. | [latex]2[/latex] less than five times [latex]q[/latex] |

Translate [latex]5[/latex] times [latex]q[/latex] . | [latex]2[/latex] less than [latex]5q[/latex] |

Translate the phrase into algebra. | [latex]5q - 2[/latex] |

### try it

[ohm_question]144918[/ohm_question]

## Licenses & Attributions

### CC licensed content, Original

- Write Algebraic Expressions from Statements: Form ax+b and a(x+b).
**Authored by:**Sousa, James (mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Write Basic Expressions from Words Modeling Situations.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Question ID: 144907, 144916, 144917, 144918,146542,146541 .
**Authored by:**Alyson Day.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.

### CC licensed content, Specific attribution

- Prealgebra.
**Provided by:**OpenStax**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[emailprotected].