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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.
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) |
Translate each phrase or sentence into a mathematical expression or equation.
$\underset{9}{\underbrace{\text{Nine}}}\underset{+}{\underbrace{\text{more than}}}\underset{x}{\underbrace{\text{some number.}}}$
Translation: $9+x$ .
$\underset{18}{\underbrace{\text{Eighteen}}}\underset{-}{\underbrace{\text{minus}}}\underset{x}{\underbrace{\text{a number.}}}$
Translation: $\text{18}-x$ .
$\underset{\mathrm{y}}{\underbrace{\text{A quantity}}}\underset{-}{\underbrace{\text{less}}}\underset{5}{\underbrace{\text{five.}}}$
Translation: $y-5$ .
$\underset{4}{\underbrace{\text{Four}}}\underset{\cdot}{\underbrace{\text{times}}}\underset{x}{\underbrace{\text{a number}}}\underset{=}{\underbrace{\text{is}}}\underset{\mathrm{16}}{\underbrace{\text{sixteen.}}}$
Translation: $4x=\text{16}$ .
$\underset{\frac{1}{5}}{\underbrace{\text{One fifth}}}\underset{\cdot}{\underbrace{\text{of}}}\underset{n}{\underbrace{\text{a number}}}\underset{=}{\underbrace{\text{is}}}\underset{\mathrm{30}}{\underbrace{\text{thirty.}}}$
Translation: $\frac{1}{5}n=\text{30}$ , or $\frac{n}{5}=\text{30}$ .
$\underset{5}{\underbrace{\text{Five}}}\underset{\cdot}{\underbrace{\text{times}}}\underset{x}{\underbrace{\text{a number}}}\underset{=}{\underbrace{\text{is}}}\underset{2}{\underbrace{\text{two}}}\underset{+}{\underbrace{\text{more than}}}\underset{2\cdot}{\underbrace{\text{twice}}}\underset{x}{\underbrace{\text{the number.}}}$
Translation: $5x=2+2x$ .
Translate each phrase or sentence into a mathematical expression or equation.
Twelve more than a number.
$\text{12}+x$
Eight minus a number.
$8-x$
An unknown quantity less fourteen.
$x-\text{14}$
Six times a number is fifty-four.
$6x=\text{54}$
Two ninths of a number is eleven.
$\frac{2}{9}x=\text{11}$
Three more than seven times a number is nine more than five times the number.
$3+7x=9+5x$
Twice a number less eight is equal to one more than three times the number.
$2x-8=3x+1$ or $2x-8=1+3x$
Sometimes the structure of the sentence indicates the use of grouping symbols. We’ll be alert for commas . They set off terms.
$\underset{(x}{\underbrace{\text{A number}}}\underset{\xf7}{\underbrace{\text{divided by}}}\underset{4)}{\underbrace{\text{four,}}}\underset{-}{\underbrace{\text{minus}}}\underset{6}{\underbrace{\text{six,}}}\underset{=}{\underbrace{\text{is}}}\underset{\mathrm{12}}{\underbrace{\text{twelve}}}$
Translation: $\frac{x}{4}-6=\text{12}$ .
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.
Translation: $x-\text{20}$ .
The word from indicated the motion (subtraction) is to begin at the point of “some number.”
Ten less than some number. Notice that less than can be replaced by from .
Ten from some number.
Translation: $x-\text{10}$ .
Translate each phrase or sentence into a mathematical expression or equation.
A number divided by eight, plus seven, is fifty.
$\frac{x}{8}+7=\text{50}$
A number divided by three, minus the same number multiplied by six, is one more than the number.
$\frac{2}{3}-6x=x+1$
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