Translating Words to Symbols
Practical problems seldom, if ever, come in equation form. The job of the problem solver is to translate the problem from phrases and statements into mathematical expressions and equations, and then to solve the equations.
As problem solvers, our job is made simpler if we are able to translate verbal phrases to mathematical expressions and if we follow the five-step method of solving applied problems. To help us translate from words to symbols, we can use the following Mathematics Dictionary.
| Word or Phrase | Mathematical Operation |
|---|---|
| Sum, sum of, added to, increased by, more than, and, plus | + |
| Difference, minus, subtracted from, decreased by, less, less than | - |
| Product, the product of, of, multiplied by, times, per | ⋅ |
| Quotient, divided by, ratio, per | ÷ |
| Equals, is equal to, is, the result is, becomes | = |
| A number, an unknown quantity, an unknown, a quantity | \(x\) (or any symbol) |
Sample Set A
Translate each phrase or sentence into a mathematical expression or equation.
\(\underbrace{\text{Nine}}_{9}\underbrace{\text{More than}}_{+}\underbrace{\text{some number}}_{x}\).
Solution
Translation: \(9 + x\).
Sample Set A
\(\underbrace{\text{Eighteen}}_{18}\underbrace{\text{minus}}_{-}\underbrace{\text{a number}}_{x}\).
Solution
Translation: \(18 - x\).
Sample Set A
\(\underbrace{\text{A quantity}}_{y}\underbrace{\text{less}}_{-}\underbrace{\text{five}}_{5}\).
Solution
Translation: \(y - 5\).
Sample Set A
\(\underbrace{\text{Four}}_{4}\underbrace{\text{times}}_{\cdot}\underbrace{\text{a number}}_{x}\underbrace{\text{is}}_{=}\underbrace{\text{sixteen}}_{16}\).
Solution
Translation: \(4x = 16\).
Sample Set A
\(\underbrace{\text{One fifth}}_{\dfrac{1}{5}}\underbrace{\text{of}}_{\cdot}\underbrace{\text{a number}}_{n}\underbrace{\text{is}}_{=}\underbrace{\text{thirty}}_{30}\).
Solution
Translation: \(\dfrac{1}{5} n = 30\), or \(\dfrac{n}{5} = 30\).
Sample Set A
\(\underbrace{\text{Five}}_{5}\underbrace{\text{times}}_{\cdot}\underbrace{\text{a number}}_{x}\underbrace{\text{is}}_{=}\underbrace{\text{two}}_{2}\underbrace{\text{more than}}_{+}\underbrace{\text{twice}}_{2}\underbrace{\text{the number}}_{x}\).
Solution
Translation: \(5x = 2 + 2x\).
Practice Set A
Translate each phrase or sentence into a mathematical expression or equation.
Twelve more than a number.
- Answer
-
\(12 + x\)
Practice Set A
Eight minus a number.
- Answer
-
\(8 - x\)
Practice Set A
An unknown quantity less fourteen.
- Answer
-
\(x - 14\)
Practice Set A
Six times a number is fifty-four.
- Answer
-
\(6x = 54\)
Practice Set A
Two ninths of a number is eleven.
- Answer
-
\(\dfrac{2}{9} x = 11\)
Practice Set A
Three more than seven times a number is nine more than five times the number.
- Answer
-
\(3 + 7x = 9 + 5x\)
Practice Set A
Twice a number less eight is equal to one more than three times the number.
- Answer
-
\(2x - 8 = 3x + 1\) or \(2x - 8 = 1 + 3x\)
Sample Set B
Sometimes the structure of the sentence indicates the use of grouping symbols. We’ll be alert for commas. They set off terms.
\(\underbrace{\text{A number}}_{(x}\underbrace{\text{divided by}}_{\div}\underbrace{\text{four}}_{4)}\underbrace{\text{minus}}_{-}\underbrace{\text{six}}_{6}\underbrace{\text{is}}_{=}\underbrace{\text{twelve}}_{12}\).
Solution
Translation: \(\dfrac{x}{4} - 6 = 12\).
Sample Set B
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.
Solution
Translation: \(x - 20\)
The word from indicated the motion (subtraction) is to begin at the point of “some number.”
Sample Set B
Ten less than some number. Notice that less than can be replaced by from.
Ten from some number.
Solution
Translation: \(x - 10\).
Practice Set B
Translate each phrase or sentence into a mathematical expression or equation.
A number divided by eight, plus seven, is fifty.
- Answer
-
\(\dfrac{x}{8} + 7 = 50\)
Practice Set B
A number divided by three, minus the same number multiplied by six, is one more than the number.
- Answer
-
\(\dfrac{2}{3} - 6x = x + 1\)
Practice Set B
Nine from some number is four.
- Answer
-
\(x - 9 = 4\)
Practice Set B
Five less than some quantity is eight.
- Answer
-
\(x - 5 = 8\)
Exercises
Translate each phrase or sentence to a mathematical expression or equation.
Exercise \(\PageIndex{1}\)
A quantity less twelve.
- Answer
-
\(x - 12\)
Exercise \(\PageIndex{2}\)
Six more than an unknown number.
Exercise \(\PageIndex{3}\)
A number minus four.
- Answer
-
\(x - 4\)
Exercise \(\PageIndex{4}\)
A number plus seven.
Exercise \(\PageIndex{5}\)
A number increased by one.
- Answer
-
\(x + 1\)
Exercise \(\PageIndex{6}\)
A number decreased by ten.
Exercise \(\PageIndex{7}\)
Negative seven added to some number.
- Answer
-
\(-7 + x\)
Exercise \(\PageIndex{8}\)
Negative nine added to a number.
Exercise \(\PageIndex{9}\)
A number plus the opposite of six.
- Answer
-
\(x + (-6)\)
Exercise \(\PageIndex{10}\)
A number minus the opposite of five.
Exercise \(\PageIndex{11}\)
A number minus the opposite of negative one.
- Answer
-
\(x -[-(-1)]\)
Exercise \(\PageIndex{12}\)
A number minus the opposite of negative twelve.
Exercise \(\PageIndex{13}\)
Eleven added to three times a number.
- Answer
-
\(3x + 11\)
Exercise \(\PageIndex{14}\)
Six plus five times an unknown number.
Exercise \(\PageIndex{15}\)
Twice a number minus seven equals four.
- Answer
-
\(2x - 7 = 4\)
Exercise \(\PageIndex{16}\)
Ten times a quantity increased by two is nine.
Exercise \(\PageIndex{17}\)
When fourteen is added to two times a number the result is six.
- Answer
-
\(14 + 2x = 6\)
Exercise \(\PageIndex{18}\)
Four times a number minus twenty-nine is eleven.
Exercise \(\PageIndex{19}\)
Three fifths of a number plus eight is fifty.
- Answer
-
\(\dfrac{3}{5} x + 8 = 50\)
Exercise \(\PageIndex{20}\)
Two ninths of a number plus one fifth is forty-one.
Exercise \(\PageIndex{21}\)
When four thirds of a number is increased by twelve, the result is five.
- Answer
-
\(\dfrac{4}{3} x + 12 = 5\)
Exercise \(\PageIndex{22}\)
When seven times a number is decreased by two times the number, the result is negative one.
Exercise \(\PageIndex{23}\)
When eight times a number is increased by five, the result is equal to the original number plus twenty-six.
- Answer
-
\(8x + 5 = x + 26\)
Exercise \(\PageIndex{24}\)
Five more than some number is three more than four times the number.
Exercise \(\PageIndex{25}\)
When a number divided by six is increased by nine, the result is one.
- Answer
-
\(\dfrac{x}{6} + 9 = 1\)
Exercise \(\PageIndex{26}\)
A number is equal to itself minus three times itself.
Exercise \(\PageIndex{27}\)
A number divided by seven, plus two, is seventeen.
- Answer
-
\(\dfrac{x}{7} + 2 = 17\)
Exercise \(\PageIndex{28}\)
A number divided by nine, minus five times the number, is equal to one more than the number.
Exercise \(\PageIndex{29}\)
When two is subtracted from some number, the result is ten.
- Answer
-
\(x - 2 = 10\)
Exercise \(\PageIndex{30}\)
When four is subtracted from some number, the result is thirty-one.
Exercise \(\PageIndex{31}\)
Three less than some number is equal to twice the number minus six.
- Answer
-
\(x - 3 = 2x - 6\)
Exercise \(\PageIndex{32}\)
Thirteen less than some number is equal to three times the number added to eight.
Exercise \(\PageIndex{33}\)
When twelve is subtracted from five times some number, the result is two less than the original number.
- Answer
-
\(5x - 12 = x - 2\)
Exercise \(\PageIndex{34}\)
When one is subtracted from three times a number, the result is eight less than six times the original number.
Exercise \(\PageIndex{35}\)
When a number is subtracted from six, the result is four more than the original number.
- Answer
-
\(6 - x = x + 4\)
Exercise \(\PageIndex{36}\)
When a number is subtracted from twenty-four, the result is six less than twice the number.
Exercise \(\PageIndex{37}\)
A number is subtracted from nine. This result is then increased by one. The result is eight more than three times the number.
- Answer
-
\(9 - x + 1 = 3x + 8\)
Exercise \(\PageIndex{38}\)
Five times a number is increased by two. This result is then decreased by three times the number. The result is three more than three times the number.
Exercise \(\PageIndex{39}\)
Twice a number is decreased by seven. This result is decreased by four times the number. The result is negative the original number, minus six.
- Answer
-
\(2x - 7 - 4x = -x - 6\)
Exercise \(\PageIndex{40}\)
Fifteen times a number is decreased by fifteen. This result is then increased by two times the number. The result is negative five times the original number minus the opposite of ten.
Exercises for Review
Exercise \(\PageIndex{41}\)
\(\dfrac{8}{9}\) of what number is \(\dfrac{2}{3}\)?
- Answer
-
\(\dfrac{3}{4}\)
Exercise \(\PageIndex{42}\)
Find the value of \(\dfrac{21}{40} + \dfrac{17}{30}\).
Exercise \(\PageIndex{43}\)
Find the value of \(3\dfrac{1}{12} + 4 \dfrac{1}{3} + 1 \dfrac{1}{4}\)
- Answer
-
\(8\dfrac{2}{3}\)
Exercise \(\PageIndex{44}\)
Convert \(6.11\dfrac{1}{5}\) to a fraction.
Exercise \(\PageIndex{45}\)
Solve the equation \(\dfrac{3x}{4} + 1 = -5\).
- Answer
-
\(x = -8\)
