Chapter Three — Basic Algebraic Expressions

The Humongous Book of Algebra Problems

38

Translating Expressions

The alchemy of turning words into math

3.1 Translate the following phrases into algebraic expressions and evaluate the

expressions: “the sum of three and six” and “the difference of three and six.”

Add two numbers to calculate the sum (3 + 6 = 9) and subtract two numbers to

calculate the difference (3 – 6 = –3). Because addition is commutative, the order

in which you add is inconsequential: 3 + 6 = 6 + 3 = 9. However, subtraction is

not commutative. Ensure that you subtract in the order stated. In this case, the

difference of three and six translates to the expression 3 – 6, not 6 – 3.

3.2 Translate into an algebraic expression: eight more than a number.

In this instance, “more than” indicates addition; if one number is 8 more than

another, then adding 8 to the smaller number produces the larger number.

Unlike Problem 3.1, neither number is stated explicitly. Rather than adding

known values like 3 and 6, you are asked to add 8 to an unknown number. Use

a variable to represent the unknown number: x + 8. Although it is common to

represent the unknown value with x, any variable might be used. Thus, y + 8,

k + 8, and u + 8 are equally valid answers.

3.3 Explain the difference between the following expressions: “10 less a number”

and “10 less than a number.”

To understand the subtle difference in meaning, replace “a number” with a

real number value, such as 7. “10 less 7” represents the subtraction problem

10 – 7, whereas “10 less than 7” represents “7 – 10.” Though both phrases

indicate subtraction, the order in which the operation is performed differs, and

the result differs as well. Algebraically, “10 less a number” is translated as 10 – x,

and “10 less than a number” is interpreted as “x – 10.”

3.4 Translate into an algebraic expression: the product of a number and 7.

The word “product” indicates multiplication. Thus, the product of an unknown

number x and seven is x · 7 or 7x. There are two important things of which

to take note. First, the notation 7x is preferred to x7; numbers are typically

written before variables in a product and are called coefﬁcients. In this case,

x has a coefﬁcient of 7. Second, dot notation ·(rather than the traditional

multiplication operator ×) is preferred to avoid confusing the operator × with

the variable x. For that reason, and with rare exception, the dot notation is used

to represent multiplication for the remainder of this book.

See Problem

1.34 for more

information.

If the

phrase had

been “Eight

IS more than

a number,” you’d

translate that into

8 > x. The difference?

The verb “is.” This

chapter deals with math

PHRASES, not math

SENTENCES that

include verbs—those

are covered in

Chapter 4.

Because

subtraction is

not commutative, as

Problem 3.1 indicated.

When it comes to

scientic notation

(like in Problems

3.19–3.22), × notation is

better because you’re

dealing with decimals.

It’s easy to confuse

multiplication dots

and decimal points

if you’re writing

fast.

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