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.
Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
39
3.5 Translate into an algebraic expression: the quotient of a number and five fewer
than that number.
If x represents the unknown number, then x – 5 is five fewer than x. To calculate
the quotient of two numbers, divide them.
Note that, like subtraction, division is not commutative—divide the first value
stated by the second value stated.
3.6 Translate into an algebraic expression and simplify the expression: one-third
of a number less half of the same number.
The word “of,” when used alone, usually indicates multiplication. Specifically, it
usually indicates multiplication by a rational number. Here, one-third of a
number is written or . Similarly, one-half of the same unknown num-
ber should be written or .
The word “less” indicates subtraction. Because that operation is not
commutative, ensure that the values are written in the correct order: .
Because both terms of the expression contain the same variable (x), they are
like terms and can be combined. To combine them, however, the fractions must
have a common denominator. Use the technique outlined in Problem 2.23 (or
the more advanced technique in Problem 2.30) to identify a least common
denominator and combine the coefficients of the terms.
You could also
write x as a
fraction
and
multiply:
.
When you
have like
terms, combine
the coefcients
(the numbers) and
attach the matching
variable. For example,
to add 9y and 3y, add
9 + 3 and attach the
y that both terms have
in common: 12y. Problem
3.6 is a little rougher
because you have to
add fractions, but
it’s the same
principle.
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