11.5: Applications I- Translating Words to Mathematical Symbols
( \newcommand{\kernel}{\mathrm{null}\,}\)
- be able to translate phrases and statements to mathematical expressions and equations
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) |
Translate each phrase or sentence into a mathematical expression or equation.
Nine⏟9More than⏟+some number⏟x.
Solution
Translation: 9+x.
Eighteen⏟18minus⏟−a number⏟x.
Solution
Translation: 18−x.
A quantity⏟yless⏟−five⏟5.
Solution
Translation: y−5.
Four⏟4times⏟⋅a number⏟xis⏟=sixteen⏟16.
Solution
Translation: 4x=16.
One fifth⏟15of⏟⋅a number⏟nis⏟=thirty⏟30.
Solution
Translation: 15n=30, or n5=30.
Five⏟5times⏟⋅a number⏟xis⏟=two⏟2more than⏟+twice⏟2the 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
-
29x=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
Sometimes the structure of the sentence indicates the use of grouping symbols. We’ll be alert for commas. They set off terms.
A number⏟(xdivided by⏟÷four⏟4)minus⏟−six⏟6is⏟=twelve⏟12.
Solution
Translation: x4−6=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.
Solution
Translation: x−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.
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
-
x8+7=50
Practice Set B
A number divided by three, minus the same number multiplied by six, is one more than the number.
- Answer
-
23−6x=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 11.5.1
A quantity less twelve.
- Answer
-
x−12
Exercise 11.5.2
Six more than an unknown number.
Exercise 11.5.3
A number minus four.
- Answer
-
x−4
Exercise 11.5.4
A number plus seven.
Exercise 11.5.5
A number increased by one.
- Answer
-
x+1
Exercise 11.5.6
A number decreased by ten.
Exercise 11.5.7
Negative seven added to some number.
- Answer
-
−7+x
Exercise 11.5.8
Negative nine added to a number.
Exercise 11.5.9
A number plus the opposite of six.
- Answer
-
x+(−6)
Exercise 11.5.10
A number minus the opposite of five.
Exercise 11.5.11
A number minus the opposite of negative one.
- Answer
-
x−[−(−1)]
Exercise 11.5.12
A number minus the opposite of negative twelve.
Exercise 11.5.13
Eleven added to three times a number.
- Answer
-
3x+11
Exercise 11.5.14
Six plus five times an unknown number.
Exercise 11.5.15
Twice a number minus seven equals four.
- Answer
-
2x−7=4
Exercise 11.5.16
Ten times a quantity increased by two is nine.
Exercise 11.5.17
When fourteen is added to two times a number the result is six.
- Answer
-
14+2x=6
Exercise 11.5.18
Four times a number minus twenty-nine is eleven.
Exercise 11.5.19
Three fifths of a number plus eight is fifty.
- Answer
-
35x+8=50
Exercise 11.5.20
Two ninths of a number plus one fifth is forty-one.
Exercise 11.5.21
When four thirds of a number is increased by twelve, the result is five.
- Answer
-
43x+12=5
Exercise 11.5.22
When seven times a number is decreased by two times the number, the result is negative one.
Exercise 11.5.23
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 11.5.24
Five more than some number is three more than four times the number.
Exercise 11.5.25
When a number divided by six is increased by nine, the result is one.
- Answer
-
x6+9=1
Exercise 11.5.26
A number is equal to itself minus three times itself.
Exercise 11.5.27
A number divided by seven, plus two, is seventeen.
- Answer
-
x7+2=17
Exercise 11.5.28
A number divided by nine, minus five times the number, is equal to one more than the number.
Exercise 11.5.29
When two is subtracted from some number, the result is ten.
- Answer
-
x−2=10
Exercise 11.5.30
When four is subtracted from some number, the result is thirty-one.
Exercise 11.5.31
Three less than some number is equal to twice the number minus six.
- Answer
-
x−3=2x−6
Exercise 11.5.32
Thirteen less than some number is equal to three times the number added to eight.
Exercise 11.5.33
When twelve is subtracted from five times some number, the result is two less than the original number.
- Answer
-
5x−12=x−2
Exercise 11.5.34
When one is subtracted from three times a number, the result is eight less than six times the original number.
Exercise 11.5.35
When a number is subtracted from six, the result is four more than the original number.
- Answer
-
6−x=x+4
Exercise 11.5.36
When a number is subtracted from twenty-four, the result is six less than twice the number.
Exercise 11.5.37
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 11.5.38
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 11.5.39
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 11.5.40
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 11.5.41
89 of what number is 23?
- Answer
-
34
Exercise 11.5.42
Find the value of 2140+1730.
Exercise 11.5.43
Find the value of 3112+413+114
- Answer
-
823
Exercise 11.5.44
Convert 6.1115 to a fraction.
Exercise 11.5.45
Solve the equation 3x4+1=−5.
- Answer
-
x=−8