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The Humongous Book of Algebra Problems by W. Michael Kelley

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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,” youd
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
scientic notation
(like in Problems
3.19–3.22), × notation is
better because youre
dealing with decimals.
It’s easy to confuse
multiplication dots
and decimal points
if youre writing
fast.

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