Translating Words Into Algebra

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	[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.

The difference of [latex]20[/latex] and [latex]4[/latex]

[latex]20[/latex] minus [latex]4[/latex]

[latex]20-4[/latex]

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

The quotient of [latex]10x[/latex] and [latex]3[/latex]

divide [latex]10x[/latex] by [latex]3[/latex]

[latex]10x\div{3}[/latex]

This can also be written as [latex](10x)/3[/latex] or [latex]\frac{10x}{3}[/latex]

try it

example

Translate each word phrase into an algebraic expression:

How old will you be in eight years? Let’s say your current age is [latex]y[/latex].
How old were you seven years ago? This is seven years less than your age now. Let’s say your current age is [latex]9z[/latex].

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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]

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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 [latex]6[/latex] and [latex]9[/latex] is [latex]15[/latex].

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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example

Translate the sentence into an algebraic equation: Twice the difference of [latex]x[/latex] and [latex]3[/latex] gives [latex]18[/latex].

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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 [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.

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

example

Translate and solve: Three more than [latex]x[/latex] is equal to [latex]47[/latex].

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example

Translate and solve: The difference of [latex]y[/latex] and [latex]14[/latex] is [latex]18[/latex].

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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:

[latex]12+14[/latex]
[latex]\left(30\right)\left(5\right)[/latex]
[latex]64\div 8[/latex]
[latex]x-y[/latex]

Solution:

[latex]12+14[/latex]

[latex]12[/latex] plus [latex]14[/latex]

the sum of twelve and fourteen

[latex]\left(30\right)\left(5\right)[/latex]

[latex]30[/latex] times [latex]5[/latex]

the product of thirty and five

[latex]64\div 8[/latex]

[latex]64[/latex] divided by [latex]8[/latex]

the quotient of sixty-four and eight

[latex]x-y[/latex]

[latex]x[/latex] minus [latex]y[/latex]

the difference of [latex]x[/latex] and [latex]y[/latex]

TRY IT

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 [latex]b[/latex] is greater than [latex]a[/latex], it means that [latex]b[/latex] is to the right of [latex]a[/latex] on the number line. We use the symbols [latex]\text{<}[/latex] and [latex]\text{>}[/latex] for inequalities.

[latex]a

The figure shows a horizontal number line that begins with the letter a on the left then the letter b to its right.
[latex]a>b[/latex] is read [latex]a[/latex] is greater than [latex]b[/latex]
[latex]a[/latex] is to the right of [latex]b[/latex] on the number line

The figure shows a horizontal number line that begins with the letter b on the left then the letter a to its right.

The expressions [latex]ab[/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

[latex]\begin{array}{l}aa.\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}[/latex]

When we write an inequality symbol with a line under it, such as [latex]a\le b[/latex], it means [latex]a Algebraic Notation Say [latex]a=b[/latex] [latex]a[/latex] is equal to [latex]b[/latex] [latex]a\ne b[/latex] [latex]a[/latex] is not equal to [latex]b[/latex] [latex]a [latex]a[/latex] is less than [latex]b[/latex] [latex]a>b[/latex] [latex]a[/latex] is greater than [latex]b[/latex] [latex]a\le b[/latex] [latex]a[/latex] is less than or equal to [latex]b[/latex] [latex]a\ge b[/latex] [latex]a[/latex] is greater than or equal to [latex]b[/latex]

Symbols [latex]<[/latex] and [latex]>[/latex]

The symbols [latex]<[/latex] and [latex]>[/latex] each have a smaller side and a larger side.

smaller side [latex]<[/latex] larger side

larger side [latex]>[/latex] 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:

[latex]20\le 35[/latex]
[latex]11\ne 15 - 3[/latex]
[latex]9>10\div 2[/latex]
[latex]x+2<10[/latex]

Show Answer

TRY IT

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

Locate the “equals” word(s).
Recognize the key words: twice; difference of …. and ….	Twice means two times.
Translate.
	[latex]2(x-3)=18[/latex]

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]

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]

		Three more than x is equal to [latex]47[/latex].
Translate.		[latex]x+3=47[/latex]
Subtract [latex]3[/latex] from both sides of the equation.		[latex]x+3\color{red}{--3}=47\color{red}{--3}[/latex]
Simplify.		[latex]x=44[/latex]
We can check. Let [latex]x=44[/latex] .	[latex]x+3=47[/latex]
	[latex]44+3=47[/latex]
	[latex]47=47\quad\checkmark[/latex]

		The difference of [latex]y[/latex] and [latex]14[/latex] is [latex]18[/latex].
Translate.		[latex]y-14=18[/latex]
Add [latex]14[/latex] to both sides.		[latex]y-14\color{red}{+14}=18\color{red}{+14}[/latex]
Simplify.		[latex]y=32[/latex]
We can check. Let [latex]y=32[/latex] .	[latex]y--14=18[/latex]
	[latex]32--14=18[/latex]
	[latex]18=18\quad\checkmark[/latex]

Module 2: Calculations and Solving Equations

Learning outcome

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Exercises

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Symbols [latex]<[/latex] and [latex]>[/latex]

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