Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
3.1 Mathematical Expressions
Variable
A variable is a symbol (usually a letter) that stands for a value or number.
Examples of Mathematical Expressions:
2a
x5
y2
2 a  b
2x  2 y  3
Review from Chapter 1: Translating Words to Math
Key words that translate to or indicate ADDITION:
Add
Plus
Sum
Increased by
Gain
Total
More than
Combined
Key words that translate to or indicate SUBTRACTION:
Minus
Take away
Difference
Decreased by
Subtracted from
Reduced by
Less (fewer) than
Fewer
Exceeds
Loss
Debit
Note: Remember subtraction is not commutative so the order of the subtraction will change your
answer. Most subtractions are written in the order they appear. However, there are two
phrases that are “backwards phrases”, meaning the second quantity goes before the first
quantity. The two phrases are “less than” and “subtracted from”.
They are denoted with a
.
EXs: “The difference between x and 7” translates to “ x  7 ” (the quantities are written in the order they appear)
“x less than 7” translates to “ 7  x ”
(the quantities are “backwards”, the 1st quantity appears last)
“x subtracted from 7” translates to “ 7  x ”
(the quantities are “backwards”, the 1st quantity appears last)
Note: “is” usually translates to the equal symbol, =.
Key words that translate to or indicate MULTIPLICATION:
Multiply
Times
Product
Of
Triple
Twice
Key words that translate to or indicate DIVISION:
Divide
Goes into
Quotient
Ratio
Double
Shared equally
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2015 Worrel
Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
Example 1: Translate the following phrases into mathematical expression.
a) “12 more than x”
b) “11 less than y”
c) “r decreased by 9”
Solution: a) “more than” translates to addition
So, “12 more than x” becomes 12  x
(Note: x  12 is also an acceptable answer because of the commutative property of addition)
b) “less than” translates to subtraction, but remember it is a “backwards” subtraction , hence the
last quantity has to go first
So, “11 less than y” becomes y 11
(Note: this is the only acceptable answer since there is no commutative property of subtraction, y  11  11  y )
c) “decreased by” translates to subtraction (it is a regular subtraction)
So, “r decreased by 9” becomes r  9
You Try It 1: Translate the following phrases into mathematical expression.
a) “13 more than x”
b) “12 fewer than y”
Example 2: Let W represent the width of the rectangle. The length of a rectangle is 4 feet longer than its width.
Express the length of the rectangle in terms of its width, W.
Solution:
The length of a rectangle is 4 feet longer than its width .
Length

4

W
Hence, Length  4  W
You Try It 2: The width of a rectangle is 5 inches shorter than its length, L. Express the width of the rectangle
in terms of its length, L.
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2015 Worrel
Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
Example 3: A string that measures 15 inches is cut into two pieces. Let x represent the length of one of the
resulting pieces. Express the length of the second piece in terms of the length, x, of the first piece.
Solution:
15 inches
Let’s say this green line is the piece of string.
Length of first piece = x
Length of second piece
If the entire length of the piece is 15 inches and the first piece of string has a length of x, if we
subtract the length of the first piece from the entire length, it will give us the remainder which is the
length of the second piece. In other words, the entire length of the string minus the length of the
first piece will equal the length of the second piece.
The entire length minus the length of the the 1st piece equals the length of the 2nd piece.
15


x
length of the second piece
So, the length of the second piece  15  x .
You can check that this is true by adding the length of the first piece with the length of the second
piece and making sure they add up to 15 inches.
The length of the 1st piece + the length of the 2nd piece = 15
x
x


15  x
15  x
 15
x 's cancel and we do get 15  15.
You Try It 3: A string is cut into two pieces, the first of which measures 12 inches. Express the total length of
the string as a function of x, where x represents the length of the second piece of string.
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2015 Worrel
Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
Example 4: Translate the following phrases into mathematical expressions:
a) “11 times x”
b) “the quotient of y and 4”
c) “twice a”
Solution:
a) 11 times x  11 x or 11x

11
x
b) the quotient of
y

y (goes first)
and
4
4 (goes last)
 y  4 or
y
4
c) twice x  2 x or 2 x
2
x
You Try It 4: Translate the following phrases into mathematical expressions:
a) “the product of 5 and x”
b) “12 divided by y”
Example 5: A plumber has a pipe of unknown length, x. He cuts it into 4 equal pieces. Find the length of each
piece in terms of the unknown length, x.
Solution:
You can rephrase the information to say, “the plumber divides the pipe into 4 equal pieces”
Divides the pipe into

x (goes first)
4
4 (goes last)
equal pieces  x  4 or
x
4
You Try It 5: A carpenter cuts a board of unknown length, L, into three equal pieces. Express the length of
each piece in terms of L.
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2015 Worrel
Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
Example 6: Mary invests A dollars in a savings account paying 2% interest per year. She invests five times this
amount in a certificate of deposit paying 5% per year. How much does she invest in the certificate
of deposit, in terms of the amount A in the savings account?
Solution:
Mary invests five times the savings account amount in a certificate of deposit
5


A
certificate of deposit
If you flip the equation around, you get certificate of deposit  5 A or 5 A
You Try It 6: David invests K dollars in a savings account paying 3% per year. He invests half this amount in a
mutual fund paying 4% a year. Express the amount invested in the mutual fund in terms of K, the
amount invested in the savings account.
Example 7: Let the first number equal x. The second number is 3 more than twice the first number. Express
the second number in terms of the first number, x.
Solution:
the second number is 3 more than twice the first number
the second number

3

2
x
Hence, we get the second number  3  2 x
You Try It 7: The second number is 4 less than 3 times the first number. Express the second number in terms
of the first number, y.
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2015 Worrel
Math 40
Prealgebra
Section 3.1 – Mathematical Expressions
Example 8: The length of a rectangle is L. The width is 15 feet less than 3 times the length. What is the width
of the rectangle in terms of the length, L?
Solution:
the width is 15 feet
the width

15
less than
 (backwards minus)
3 times the length
3
L
Hence, we get the width  3L  15
You Try It 8: The width of a rectangle is W. The length is 7 inches longer than twice its width. Express the
length of the rectangle in terms of its width, W.
Example 9: Translate the phrase, “5 times the sum of the number, n, and 8.”
Solution:
5 times the sum of the number and 8
5


n
8
It says “times the sum”, there are two operation terms next to each other. This tells you that you
will need to use parentheses, ( ), to separate the times and the sum.
Hence, we get 5  n  8  or 5  n  8 
You Try It 9: Translate the phrase, “21 times the difference of the number, x, and 17.”
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