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Applications I: Translating Words to Mathematical Symbols

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to translate word to mathematical symbols. By the end of the module students should be able to translate phrases and statements to mathematical expressions and equations.

Section Overview

  • Translating Words to Symbols

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.

Table 1
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 xx size 12{x} {} (or any symbol)

Sample Set A

Translate each phrase or sentence into a mathematical expression or equation.

Example 1

Nine ︸ 9 more than ︸ + some number. ︸ x Nine ︸ 9 more than ︸ + some number. ︸ x

Translation: 9+x9+x size 12{9+x} {}.

Example 2

Eighteen ︸ 18 minus ︸ - a number. ︸ x Eighteen ︸ 18 minus ︸ - a number. ︸ x

Translation: 18−x18−x size 12{"18" - x} {}.

Example 3

A quantity ︸ y less ︸ - five. ︸ 5 A quantity ︸ y less ︸ - five. ︸ 5

Translation: y−5y−5 size 12{y - 5} {}.

Example 4

Four ︸ 4 times ︸ ⋅ a number ︸ x is ︸ = sixteen. ︸ 16 Four ︸ 4 times ︸ ⋅ a number ︸ x is ︸ = sixteen. ︸ 16

Translation: 4 x=164 x=16 size 12{4x="16"} {}.

Example 5

One fifth ︸ 15  of ︸ ⋅  a number ︸ n  is ︸ =  thirty. ︸ 30 One fifth ︸ 15  of ︸ ⋅  a number ︸ n  is ︸ =  thirty. ︸ 30

Translation: 15n=3015n=30 size 12{ { {1} over {5} } n="30"} {}, or n5=30n5=30 size 12{ { {n} over {5} } ="30"} {}.

Example 6

 Five ︸ 5  times ︸ ⋅  a number ︸ x  is ︸ =  two ︸ 2  more than ︸ +  twice ︸ 2⋅  the number. ︸ x  Five ︸ 5  times ︸ ⋅  a number ︸ x  is ︸ =  two ︸ 2  more than ︸ +  twice ︸ 2⋅  the number. ︸ x

Translation: 5 x=2+2x5 x=2+2x size 12{5x=2+2x} {}.

Practice Set A

Translate each phrase or sentence into a mathematical expression or equation.

Exercise 1

Twelve more than a number.

Solution

12+x12+x size 12{"12"+x} {}

Exercise 2

Eight minus a number.

Solution

8−x8−x size 12{8 - x} {}

Exercise 3

An unknown quantity less fourteen.

Solution

x−14x−14 size 12{x - "14"} {}

Exercise 4

Six times a number is fifty-four.

Solution

6 x=546 x=54 size 12{6x="54"} {}

Exercise 5

Two ninths of a number is eleven.

Solution

29x=1129x=11 size 12{ { {2} over {9} } x="11"} {}

Exercise 6

Three more than seven times a number is nine more than five times the number.

Solution

3+7 x=9+5 x3+7 x=9+5 x size 12{3+7x=9+5x} {}

Exercise 7

Twice a number less eight is equal to one more than three times the number.

Solution

2 x−8=3 x+12 x−8=3 x+1 size 12{2x - 8=3x+1} {} or 2 x−8=1+3 x2 x−8=1+3 x size 12{2x - 8=1+3x} {}

Sample Set B

Example 7

Sometimes the structure of the sentence indicates the use of grouping symbols. We’ll be alert for commas. They set off terms.

A number ︸ (x divided by ︸ ÷ four, ︸ 4) minus ︸ - six, ︸ 6 is ︸ = twelve ︸ 12 A number ︸ (x divided by ︸ ÷ four, ︸ 4) minus ︸ - six, ︸ 6 is ︸ = twelve ︸ 12

Translation: x4−6=12x4−6=12 size 12{ { {x} over {4} } - 6="12"} {}.

Example 8

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 statement twenty is subtracted from some number can be broken into three parts and converted into a mathematical expression. Some number becomes x, the phrase, is subtracted from, becomes a minus symbol, and twenty becomes the number 20. This translates to the expression x - 20.

Translation: x−20x−20 size 12{x - "20"} {}.

The word from indicated the motion (subtraction) is to begin at the point of “some number.”

Example 9

Ten less than some number. Notice that less than can be replaced by from.

Ten from some number.

Translation: x−10x−10 size 12{x - "10"} {}.

Practice Set B

Translate each phrase or sentence into a mathematical expression or equation.

Exercise 8

A number divided by eight, plus seven, is fifty.

Solution

x8+7=50x8+7=50 size 12{ { {x} over {8} } +7="50"} {}

Exercise 9

A number divided by three, minus the same number multiplied by six, is one more than the number.

Solution

23−6 x=x+123−6 x=x+1 size 12{ { {2} over {3} } - 6x=x+1} {}

Exercise 10

Nine from some number is four.

Solution

x−9=4x−9=4 size 12{x - 9=4} {}

Exercise 11

Five less than some quantity is eight.

Solution

x−5=8x−5=8 size 12{x - 5=8} {}

Exercises

Translate each phrase or sentence to a mathematical expression or equation.

Exercise 12

A quantity less twelve.

Solution

x−12x−12 size 12{x - "12"} {}

Exercise 13

Six more than an unknown number.

Exercise 14

A number minus four.

Solution

x−4x−4 size 12{x - 4} {}

Exercise 15

A number plus seven.

Exercise 16

A number increased by one.

Solution

x+1x+1 size 12{x+1} {}

Exercise 17

A number decreased by ten.

Exercise 18

Negative seven added to some number.

Solution

−7+x−7+x size 12{ - 7+x} {}

Exercise 19

Negative nine added to a number.

Exercise 20

A number plus the opposite of six.

Solution

x+−6x+−6 size 12{x+ left ( - 6 right )} {}

Exercise 21

A number minus the opposite of five.

Exercise 22

A number minus the opposite of negative one.

Solution

x−−−1x−−−1 size 12{x - left [ - left ( - 1 right ) right ]} {}

Exercise 23

A number minus the opposite of negative twelve.

Exercise 24

Eleven added to three times a number.

Solution

3 x+113 x+11 size 12{3x+"11"} {}

Exercise 25

Six plus five times an unknown number.

Exercise 26

Twice a number minus seven equals four.

Solution

2 x−7=42 x−7=4 size 12{2x - 7=4} {}

Exercise 27

Ten times a quantity increased by two is nine.

Exercise 28

When fourteen is added to two times a number the result is six.

Solution

14+2 x=614+2 x=6 size 12{"14"+2x=6} {}

Exercise 29

Four times a number minus twenty-nine is eleven.

Exercise 30

Three fifths of a number plus eight is fifty.

Solution

35x+8=5035x+8=50 size 12{ { {3} over {5} } x+8="50"} {}

Exercise 31

Two ninths of a number plus one fifth is forty-one.

Exercise 32

When four thirds of a number is increased by twelve, the result is five.

Solution

43x+12=543x+12=5 size 12{ { {4} over {3} } x+"12"=5} {}

Exercise 33

When seven times a number is decreased by two times the number, the result is negative one.

Exercise 34

When eight times a number is increased by five, the result is equal to the original number plus twenty-six.

Solution

8 x+5=x+268 x+5=x+26 size 12{8x+5=x+"26"} {}

Exercise 35

Five more than some number is three more than four times the number.

Exercise 36

When a number divided by six is increased by nine, the result is one.

Solution

x6+9=1x6+9=1 size 12{ { {x} over {6} } +9=1} {}

Exercise 37

A number is equal to itself minus three times itself.

Exercise 38

A number divided by seven, plus two, is seven­teen.

Solution

x7+2=17x7+2=17

Exercise 39

A number divided by nine, minus five times the number, is equal to one more than the number.

Exercise 40

When two is subtracted from some number, the result is ten.

Solution

x−2=10x−2=10 size 12{x - 2="10"} {}

Exercise 41

When four is subtracted from some number, the result is thirty-one.

Exercise 42

Three less than some number is equal to twice the number minus six.

Solution

x−3=2 x−6x−3=2 x−6 size 12{x - 3=2x - 6} {}

Exercise 43

Thirteen less than some number is equal to three times the number added to eight.

Exercise 44

When twelve is subtracted from five times some number, the result is two less than the original number.

Solution

5x−12=x−25x−12=x−2 size 12{5x - "12"=x - 2} {}

Exercise 45

When one is subtracted from three times a num­ber, the result is eight less than six times the original number.

Exercise 46

When a number is subtracted from six, the result is four more than the original number.

Solution

6−x=x+46−x=x+4 size 12{6 - x=x+4} {}

Exercise 47

When a number is subtracted from twenty-four, the result is six less than twice the number.

Exercise 48

A number is subtracted from nine. This result is then increased by one. The result is eight more than three times the number.

Solution

9−x+1=3 x+89−x+1=3 x+8 size 12{9 - x+1=3x+8} {}

Exercise 49

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 50

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.

Solution

2 x−7−4 x=−x−62 x−7−4 x=−x−6 size 12{2x - 7 - 4x= - x - 6} {}

Exercise 51

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 52

((Reference)) 8989 size 12{ { {8} over {9} } } {} of what number is 2323 size 12{ { {2} over {3} } } {}?

Solution

3434 size 12{ { {3} over {4} } } {}

Exercise 53

((Reference)) Find the value of 2140+17302140+1730 size 12{ { {"21"} over {"40"} } + { {"17"} over {"30"} } } {}.

Exercise 54

((Reference)) Find the value of 3112+413+1143112+413+114 size 12{3 { {1} over {"12"} } +4 { {1} over {3} } +1 { {1} over {4} } } {}.

Solution

823823 size 12{8 { {2} over {3} } } {}

Exercise 55

((Reference)) Convert 6.11156.1115 size 12{6 "." "11" { {1} over {5} } } {} to a fraction.

Exercise 56

((Reference)) Solve the equation 3x4+1=−53x4+1=−5 size 12{ { {3x} over {4} } +1= - 5} {}.

Solution

x=−8x=−8 size 12{x= - 8} {}

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