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Solving Linear Equations and Inequalities: Application I - Translating from Verbal to Mathetical Expressions

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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 "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). Objectives of this module: be able to translate from verbal to mathematical expressions.

Overview

  • Translating from Verbal to Mathematical Expressions

Translating 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.

Table 1: 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 x x (or any symbol)

Sample Set A

Translate the following phrases or sentences into mathematical expressions or equations.

Example 1

six 6 morethan + anumber x 6+x . six 6 morethan + anumber x 6+x .

Example 2

Fifteen 15 minus - anumber x 15-x . Fifteen 15 minus - anumber x 15-x .

Example 3

Aquantity y less - eight 8 y-8 . Aquantity y less - eight 8 y-8 .

Example 4

Twice 2 anumber x is = ten. 10 2x=10 Twice 2 anumber x is = ten. 10 2x=10

Example 5

Onehalf 1 2 of anumber z is = twenty. 20 1 2 z=20 Onehalf 1 2 of anumber z is = twenty. 20 1 2 z=20

Example 6

Three 3 times anumber y is = five 5 morethan + twice 2 thesamenumber. y 3y=5+2y Three 3 times anumber y is = five 5 morethan + twice 2 thesamenumber. y 3y=5+2y

Practice Set A

Translate the following phrases or sentences into mathematical expressions or equations.

Exercise 1

Eleven more than a number.

Solution

11+x 11+x

Exercise 2

Nine minus a number.

Solution

9x 9x

Exercise 3

A quantity less twenty.

Solution

x20 x20

Exercise 4

Four times a number is thirty two.

Solution

4x=32 4x=32

Exercise 5

One third of a number is six.

Solution

x 3 =6 x 3 =6

Exercise 6

Ten times a number is eight more than five times the same number.

Solution

10x=8+5x 10x=8+5x

Sometimes the structure of the sentence indicates the use of grouping symbols.

Sample Set B

Translate the following phrases or sentences into mathematical expressions or equations.

Example 7

Anumberdividedbyfive, (  x  ÷ 5) minus ten, 10 is = fifteen. 15 x 5 10=15 Anumberdividedbyfive, (  x  ÷ 5) minus ten, 10 is = fifteen. 15 x 5 10=15

Commas set off terms.

Example 8

Eight 8 dividedby ÷ fivemorethananumber ( 5 + x ) is = ten 10 Thewordingindicatesthisistobeconsideredasa singlequantity. Eight 8 dividedby ÷ fivemorethananumber ( 5 + x ) is = ten 10 Thewordingindicatesthisistobeconsideredasa singlequantity.

8 5+x =10 8 5+x =10

Example 9

Anumber x multipliedby tenmorethanitself (10 + x) is = twenty. 20 x(10 + x) = 20 Anumber x multipliedby tenmorethanitself (10 + x) is = twenty. 20 x(10 + x) = 20

Example 10

A number plus one is divided by three times the number minus twelve and the result is four.
(x+1)÷(3x12) = 4 x+1 3x12 = 4 (x+1)÷(3x12) = 4 x+1 3x12 = 4
Notice that since the phrase "three times the number minus twelve" does not contain a comma, we get the expression 3x12 3x12 . If the phrase had appeared as "three times the number, minus twelve," the result would have been
x+1 3x 12=4 x+1 3x 12=4

Example 11

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 translation of a verbal expressions 'Fifteen is subtracted from some quantity' into mathematical expressions is x minus fifteen.

The word from indicates the motion (subtraction) is to begin at the point of "some quantity."

Example 12

Eight less than some quantity. Notice that less than could be replaced with from.
x8 x8

Practice Set B

Translate the following phrases and sentences into mathematical expressions or equations.

Exercise 7

A number divided by sixteen, plus one, is five.

Solution

x 16 +1=5 x 16 +1=5

Exercise 8

Seven times two more than a number is twenty-one.

Solution

7(2+x)=21 7(2+x)=21

Exercise 9

A number divided by two more than itself is zero.

Solution

x 2+x =0 x 2+x =0

Exercise 10

A number minus five is divided by twice the number plus three and the result is seventeen.

Solution

x5 2x+3 =17 x5 2x+3 =17

Exercise 11

Fifty-two is subtracted from some quantity.

Solution

x52 x52

Exercise 12

An unknown quantity is subtracted from eleven and the result is five less than the unknown quantity.

Solution

11x=x5 11x=x5

Exercises

For the following problems, translate the following phrases or sentences into mathematical expressions or equations.

Exercise 13

A quantity less four.

Solution

a4 a4

Exercise 14

Eight more than a number.

Exercise 15

A number plus seven.

Solution

b+7 b+7

Exercise 16

A number minus three.

Exercise 17

Negative five plus an unknown quantity.

Solution

5+c 5+c

Exercise 18

Negative sixteen minus some quantity.

Exercise 19

Fourteen added to twice a number.

Solution

2d+14 2d+14

Exercise 20

Ten added to three times some number.

Exercise 21

One third minus an unknown quantity.

Solution

1 3 e 1 3 e

Exercise 22

Twice a number is eleven.

Exercise 23

Four ninths of a number is twenty-one.

Solution

4 9 f=21 4 9 f=21

Exercise 24

One third of a number is two fifths.

Exercise 25

Three times a number is nine more than twice the number.

Solution

3g=2g+9 3g=2g+9

Exercise 26

Five times a number is that number minus two.

Exercise 27

Twice a number added to six results in thirty.

Solution

2h+6=30 2h+6=30

Exercise 28

Ten times a number less four results in sixty-six.

Exercise 29

A number less twenty-five is equal to 3.019 3.019 .

Solution

k25=3.019 k25=3.019

Exercise 30

Seven more than some number is five more than twice the number.

Exercise 31

When a number is divided by four, the result is sixty-eight.

Solution

m 4 =68 m 4 =68

Exercise 32

Eleven fifteenths of two more than a number is eight.

Exercise 33

One tenth of a number is that number less one.

Solution

n 10 =n1 n 10 =n1

Exercise 34

Two more than twice a number is one half the number less three.

Exercise 35

A number is equal to itself plus four times itself.

Solution

x=x+4x x=x+4x

Exercise 36

Three fifths of a quantity added to the quantity itself is thirty-nine.

Exercise 37

A number plus seven is divided by two and the result is twenty-two.

Solution

Q+7 2 =22 Q+7 2 =22

Exercise 38

Ten times a number minus one is divided by fourteen and the result is one.

Exercise 39

A number is added to itself then divided by three. This result is then divided by three. The entire result is fifteen.

Solution

r+r 3 3 =15 r+r 3 3 =15

Exercise 40

Ten divided by two more than a number is twenty-one.

Exercise 41

Five divided by a number plus six is fourteen.

Solution

5 s+6 =14 5 s+6 =14

Exercise 42

Twelve divided by twice a number is fifty-five.

Exercise 43

Twenty divided by eight times a number added to one is nine.

Solution

20 8x +1=9 20 8x +1=9

Exercise 44

A number divided by itself, plus one, results in seven.

Exercise 45

A number divided by ten, plus four, results in twenty-four.

Solution

v 10 +4=24 v 10 +4=24

Exercise 46

A number plus six, divided by two, is seventy-one.

Exercise 47

A number plus six, divided by two, plus five, is forty-three.

Solution

w+6 2 +5=43 w+6 2 +5=43

Exercise 48

A number multiplied by itself added to five is thirty-one.

Exercise 49

A quantity multiplied by seven plus twice itself is ninety.

Solution

7y+2y=90 7y+2y=90

Exercise 50

A number is increased by one and then multiplied by five times itself. The result is eighty-four.

Exercise 51

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 fifty-nine.

Solution

( z+16 )13 6z =59 ( z+16 )13 6z =59

Exercise 52

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.

Exercise 53

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.

Solution

x11 15 1=5 x11 15 1=5

Exercise 54

Ten less than some number.

Exercise 55

Five less than some unknown number.

Solution

n5 n5

Exercise 56

Twelve less than a number.

Exercise 57

One less than an unknown quantity.

Solution

m1 m1

Exercise 58

Sixteen less than some number is forty-two.

Exercise 59

Eight less than some unknown number is three.

Solution

p8=3 p8=3

Exercise 60

Seven is added to ten less than some number. The result is one.

Exercise 61

Twenty-three is divided by two less than twice some number and the result is thirty-four.

Solution

23 2n2 =34 23 2n2 =34

Exercise 62

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 twenty-seven less than eleven times the number.

Exercises for Review

Exercise 63

((Reference)) Supply the missing word. The point on a line that is associated with a particular number is called the

          
of that number.

Solution

graph

Exercise 64

((Reference)) Supply the missing word. An exponent records the number of identical

          
in a multiplication.

Exercise 65

((Reference)) Write the algebraic definition of the absolute value of the number a a .

Solution

| a |={ a,ifa0 a,ifa<0 | a |={ a,ifa0 a,ifa<0

Exercise 66

((Reference)) Solve the equation 4y+5=3 4y+5=3 .

Exercise 67

((Reference)) Solve the equation 2(3x+1)5x=4(x6)+17 2(3x+1)5x=4(x6)+17 .

Solution

x=3 x=3

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