In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we'll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we've talked about will help us do that. They are summarized in Table 2.7.

Operation Phrase Expression
Addition a plus b
the sum of a and b
a increased by b
b more than a
the total of a and b
b added to a
 a + b
Subtraction a minus b
the difference of a and b
b subtracted from a
a decreased by b
b less than a
 a - b
Multiplication a times b
the product of a and b		 a \cdot b, a b, a(b),(a)(b)
Division a divided by b
the quotient of a and b
the ratio of a and b
b divided into a
 a \div b, a / b, \frac{a}{b}, b \text{⟌} { a }


Table 2.7

Look closely at these phrases using the four operations:

  • the sum of a and b
  • the difference of a and b
  • the product of a and b
  • the quotient of a and b

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.


Example 2.23

Translate each word phrase into an algebraic expression:
    ⓐ the difference of 20 and 4
    ⓑ the quotient of 10x and 3

Solution

ⓐ The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

\text{the difference of 20 and 4}
20 \text{minus} 4
20−4


ⓑ The key word is quotient, which tells us the operation is division.

\text{the quotient of 10x and 3}
\text{divide 10x by 3}
10x÷3
This can also be written as 10x/3or \frac{10x}{3}


Try It 2.45

Translate the given word phrase into an algebraic expression:
ⓐ the difference of 47 and 41
ⓑ the quotient of 5x and 2

Try It 2.46

Translate the given word phrase into an algebraic expression:
ⓐ the sum of 17 and 19
ⓑ the product of 7 and x

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract 7 from your present age. Seven less than means seven subtracted from your present age.


Example 2.24

Translate each word phrase into an algebraic expression:

ⓐ Eight more than y
ⓑ Seven less than 9z

Solution

ⓐ The key words are more than. They tell us the operation is addition. More than means "added to".

Eight more than y
Eight added to y
y+8


ⓑ The key words are less than. They tell us the operation is subtraction. Less than means "subtracted from".

Seven less than 9z
Seven subtracted from 9z
9z−7


Try It 2.47

Translate each word phrase into an algebraic expression:
ⓐ Eleven more than x
ⓑ Fourteen less than 11a

Try It 2.48

Translate each word phrase into an algebraic expression:
ⓐ 19 more than j
ⓑ 21 less than 2x

Example 2.25

Translate each word phrase into an algebraic expression:

ⓐ five times the sum of m and n

ⓑ the sum of five times m and n

Solution

ⓐ There are two operation words: times tells us to multiply and sum tells us to add. Because we are multiplying 5 times the sum, we need parentheses around the sum of m and n.

five times the sum of m and n

5(m+n)

ⓑ To take a sum, we look for the words of and and to see what is being added. Here we are taking the sum of five times m and n.

the sum of five times m and n

5m+n

Notice how the use of parentheses changes the result. In part ⓐ, we add first and in part ⓑ, we multiply first.


Try It 2.49

Translate the word phrase into an algebraic expression:

ⓐ four times the sum of p and q
ⓑ the sum of four times p and q

Try It 2.50

Translate the word phrase into an algebraic expression:

ⓐ the difference of two times x and 8
ⓑ two times the difference of x and 8

Later in this course, we'll apply our skills in algebra to solving equations. We'll usually start by translating a word phrase to an algebraic expression. We'll need to be clear about what the expression will represent. We'll see how to do this in the next two examples.


Example 2.26

The height of a rectangular window is 6 inches less than the width. Let w represent the width of the window. Write an expression for the height of the window.

Solution
Write a phrase about the height. 6 less than the width
Substitute w for the width.     6 less than w
Rewrite 'less than' as 'subtracted from'.     6 subtracted from w
Translate the phrase into algebra.     w−6

Try It 2.51

The length of a rectangle is 5 inches less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.


Try It 2.52

The width of a rectangle is 2 meters greater than the length. Let l represent the length of the rectangle. Write an expression for the width of the rectangle.


Example 2.27

Blanca has dimes and quarters in her purse. The number of dimes is 2
less than 5 times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

Solution

Write a phrase about the number of dimes.

two less than five times the number of quarters
Substitute q for the number of quarters.     2 less than five times q
Translate 5 times q.     2 less than 5q
Translate the phrase into algebra. 5q−2

Try It 2.53

Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.


Try It 2.54

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let n represent the number of nickels. Write an expression for the number of dimes.

Source: Rice University, https://openstax.org/books/prealgebra-2e/pages/2-2-evaluate-simplify-and-translate-expressions
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Last modified: Thursday, 24 March 2022, 12:08 PM