Translating Algebraic Expressions From Words

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]a[/latex] increased by [latex]b[/latex]

[latex]b[/latex] more than [latex]a[/latex]

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]b[/latex] subtracted from [latex]a[/latex]

[latex]a[/latex] decreased by [latex]b[/latex]

[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]

Look closely at these phrases using the four operations:

  • 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]

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

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]

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.
[latex]\begin{array}{}\\ \text{the difference of }20\text{ and }4\hfill \\ 20\text{ minus }4\hfill \\ 20 - 4\hfill \end{array}[/latex]

2. The key word is quotient, which tells us the operation is division.
[latex]\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]
This can also be written as [latex]\begin{array}{l}10x/3\text{ or}\frac{10x}{3}\hfill \end{array}[/latex]

 

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example

Translate each word phrase into an algebraic expression:

  1. 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.
  2. 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,a. Seven less than means seven subtracted from your present age,a.

 

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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]

 

try it

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

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

 

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.

 

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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.

 

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