 Translating from Verbal to Mathematical Expressions
Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "FiveStep Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). Objectives of this module: be able to translate from verbal to mathematical expressions.
To solve a problem using algebra, we must first express the problem algebraically. To express a problem algebraically, we must scrutinize the wording of the problem to determine the variables and constants that are present and the relationships among them. Then we must translate the verbal phrases and statements to algebraic expressions and equations.
To help us translate verbal expressions to mathematics, we can use the following table as a mathematics dictionary.
Word or Phrase  Mathematical Operation 
Sum, sum of, added to, increased by, more than, plus, and  
Difference, minus, subtracted from, decreased by, less, less than 

Product, the product of, of, muitiplied by, times 

Quotient, divided by, ratio 

Equals, is equal to, is, the result is, becomes 

A number, an unknown quantity, an unknown, a quantity 

Translate the following phrases or sentences into mathematical expressions or equations.
Translate the following phrases or sentences into mathematical expressions or equations.
Eleven more than a number.
Nine minus a number.
A quantity less twenty.
Four times a number is thirty two.
One third of a number is six.
Ten times a number is eight more than five times the same number.
Sometimes the structure of the sentence indicates the use of grouping symbols.
Translate the following phrases or sentences into mathematical expressions or equations.
Commas set off terms.
A number plus one is divided by three times the number minus twelve and the result is four.
Notice that since the phrase "three times the number minus twelve" does not contain a comma, we get the expression
Some phrases and sentences do not translate directly. We must be careful to read them properly. The word from often appears in such phrases and sentences. The word from means "a point of departure for motion." The following translation will illustrate this use.
The word from indicates the motion (subtraction) is to begin at the point of "some quantity."
Eight less than some quantity. Notice that less than could be replaced with from.
Translate the following phrases and sentences into mathematical expressions or equations.
A number divided by sixteen, plus one, is five.
Seven times two more than a number is twentyone.
A number divided by two more than itself is zero.
A number minus five is divided by twice the number plus three and the result is seventeen.
Fiftytwo is subtracted from some quantity.
An unknown quantity is subtracted from eleven and the result is five less than the unknown quantity.
For the following problems, translate the following phrases or sentences into mathematical expressions or equations.
A quantity less four.
Eight more than a number.
A number plus seven.
A number minus three.
Negative five plus an unknown quantity.
Negative sixteen minus some quantity.
Fourteen added to twice a number.
Ten added to three times some number.
One third minus an unknown quantity.
Twice a number is eleven.
Four ninths of a number is twentyone.
One third of a number is two fifths.
Three times a number is nine more than twice the number.
Five times a number is that number minus two.
Twice a number added to six results in thirty.
Ten times a number less four results in sixtysix.
A number less twentyfive is equal to
Seven more than some number is five more than twice the number.
When a number is divided by four, the result is sixtyeight.
Eleven fifteenths of two more than a number is eight.
One tenth of a number is that number less one.
Two more than twice a number is one half the number less three.
A number is equal to itself plus four times itself.
Three fifths of a quantity added to the quantity itself is thirtynine.
A number plus seven is divided by two and the result is twentytwo.
Ten times a number minus one is divided by fourteen and the result is one.
A number is added to itself then divided by three. This result is then divided by three. The entire result is fifteen.
Ten divided by two more than a number is twentyone.
Five divided by a number plus six is fourteen.
Twelve divided by twice a number is fiftyfive.
Twenty divided by eight times a number added to one is nine.
A number divided by itself, plus one, results in seven.
A number divided by ten, plus four, results in twentyfour.
A number plus six, divided by two, is seventyone.
A number plus six, divided by two, plus five, is fortythree.
A number multiplied by itself added to five is thirtyone.
A quantity multiplied by seven plus twice itself is ninety.
A number is increased by one and then multiplied by five times itself. The result is eightyfour.
A number is added to six and that result is multiplied by thirteen. This result is then divided by six times the number. The entire result is equal to fiftynine.
A number is subtracted from ten and that result is multiplied by four. This result is then divided by three more than the number. The entire result is equal to six.
An unknown quantity is decreased by eleven. This result is then divided by fifteen. Now, one is subtracted from this result and five is obtained.
Ten less than some number.
Five less than some unknown number.
Twelve less than a number.
One less than an unknown quantity.
Sixteen less than some number is fortytwo.
Eight less than some unknown number is three.
Seven is added to ten less than some number. The result is one.
Twentythree is divided by two less than twice some number and the result is thirtyfour.
One less than some number is multiplied by three less than five times the number and the entire result is divided by six less than the number. The result is twentyseven less than eleven times the number.
((Reference)) Supply the missing word. The point on a line that is associated with a particular number is called the
of that number.
graph
((Reference)) Supply the missing word. An exponent records the number of identical
in a multiplication.
((Reference)) Write the algebraic definition of the absolute value of the number
((Reference)) Solve the equation
((Reference)) Solve the equation
"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"