Basic College Math
John Moore
john_moore@heald.edu
916.414.2777
© 2010 Pearson Prentice Hall. All rights reserved
© 2010 Pearson Prentice Hall. All rights reserved
Standard & Expanded Notation
Billions
Millions
Thousands
Ones
5
7
5
1
0
4
Ones
Hundreds
Tens
Ones
Hundreds
Tens
Ones
Tens
Hundreds
Ones
Tens
Hundreds
3
3,575,104
Standard Notation
3,000,000 + 500,000 + 70,000 + 5,000 + 100 + 4
Expanded Notation
Tobey & Slater, Basic College Mathematics, 6e
3
Properties of Addition
1. Associative Property of Addition
When we add three numbers, we can
group them in any way.
(2 + 4) + 3 = 2 + (4 + 3)
6+3=2+7
9=9
2. Commutative Property of Addition
Two numbers can be added in either order
with the same result.
8+4=4+8
12 = 12
3. Identity Property of Zero
When zero is added to a number, the
sum is that number.
Tobey & Slater, Basic College Mathematics, 6e
7+0=7
0 + 12 = 12
4
Properties of Multiplication
1. Associative Property of Multiplication
(2  4)  3 = 2  (4  3)
8  3 = 2  12
24 = 24
When we multiply three numbers, we can
group them in any way.
2. Commutative Property of Multiplication
Two numbers can be multiplied in either order with
the same result.
3. Identity Property of One
When one is multiplied by a number, the
result is that number.
84=48
32 = 32
71=7
1  12 = 12
4. Distributive Property of Multiplication
Multiplication can be distributed over
addition without changing the result.
Tobey & Slater, Basic College Mathematics, 6e
3  (2 + 4) = (3  2) + (3  4)
3  6 = 6 + 12
18 = 18
5
Exponents
Examples:
34
=
3  3  3  3 = 81
53
=
5  5  5 = 125
35
=
3  3  3  3  3 = 243
122
=
12  12 = 144
74
=
7  7  7  7 = 2401
Tobey & Slater, Basic College Mathematics, 6e
6
Order of Operations
Order of Operations
Do first
1.
2.
3.
4.
Parentheses.
Exponents.
Multiply or Divide from left to right.
Add or Subtract from left to right.
Do last
Example:
24 ÷ 2 – 4  2
24 ÷ 2 – 4  2
= 12 – 8
=4
Tobey & Slater, Basic College Mathematics, 6e
7
Order of Operations
Example:
= 4 + (16 – 13)4 – 3
Work inside the parentheses.
= 4 + 34 – 3
Evaluate the exponent.
= 4 + 81 – 3
Add or subtract.
= 85 – 3
Add or subtract.
= 82
Tobey & Slater, Basic College Mathematics, 6e
8
Chapter 2
Fractions
© 2010 Pearson Prentice Hall. All rights reserved
Divisibility Rules
1. A number is divisible by 2 if the last digit is:
0, 2, 4, 6, or 8.
2. A number is divisible by 3 if:
the sum of the digits is divisible by 3.
3. A number is divisible by 5 if:
the last digit is 0 or 5.
Tobey & Slater, Basic College Mathematics, 6e
10
Multiplying Fractions
To multiply fractions, multiply numerators
across and denominators across.
3
2
6


7
5
35
Tobey & Slater, Basic College Mathematics, 6e
11
Dividing Fractions
When fractions are divided,
invert the second fraction and multiply.
3
1

4
4
3
4


4
1
3

or 3.
1
Tobey & Slater, Basic College Mathematics, 6e
12
Adding Fractions
Fractions must have common denominators before
they can be added or subtracted.
2
1
3


4
4
4
+
2
4
=
1
4
Tobey & Slater, Basic College Mathematics, 6e
3
4
13
Creating Equivalent Fractions
Fractions with unlike denominators cannot be added.
3
4
+
4
5
In order to add fractions with different denominators:
1) Find a Common Denominator
2) Build into equivalent fractions with the Common Denominator.
3
c
?


4
c
20
5
c  5,
 1
5
4
c
?


5
c
20
c  4,
4
 1
4
Tobey & Slater, Basic College Mathematics, 6e
14
Creating Equivalent Fractions
If fractions have different denominators, find the Common Denominator and
build up each fraction so that its denominators are the same.
Example:
3
1

8
6
3
3
9


8
3
24
1
4
4


6
4
24
9
4
13


24
24
24
Tobey & Slater, Basic College Mathematics, 6e
15
Creating Equivalent Fractions
Example:
5
7

12
30
5
5
25


12
5
60
7
2
14


30
2
60
25
14
11


60
60
60
Tobey & Slater, Basic College Mathematics, 6e
16
Chapter 3
Decimals
© 2010 Pearson Prentice Hall. All rights reserved
Place Values

Digits in a decimal number have a value dependant on the place
of the digits.

Adding extra zeros to the right of the last decimal digit does not
change the value of the number.
 1.2345670
1.234567
1.234567
1.234567
1.234567
1.234567
1.234567
1.234567
Ones position
Tenths
Hundredths
Thousandths
Ten thousandths
Hundred Thousandths
Millionths
Tobey & Slater, Basic College Mathematics, 6e
18
Adding Decimals
Example: 718.97 + 496.5
718.97
+ 496.5 0
1215.47
Place holder
Line up decimal points
Tobey & Slater, Basic College Mathematics, 6e
19
Subtracting Decimals
Example: 243.967 – 84.2
1 13 13
243.967
– 84.2 00
159.767
Place holders
Line up decimal points
Tobey & Slater, Basic College Mathematics, 6e
20
Multiplying Decimals
Example: Multiply 0.17  0.4
0.17
 0.4
.068
2 decimal places
1 decimal place
3 decimal places in product (2 + 1 = 3)
Tobey & Slater, Basic College Mathematics, 6e
21
Dividing by a Decimal
Example: Divide 4.209 ÷ 1.83
1.83. 4.20. 9
Place the decimal point
of the answer directly
above the caret.
Move each decimal
point to the right two
places.
2.3
1.83 4.20 9
366
549
549
Tobey & Slater, Basic College Mathematics, 6e
Mark the new position
by a caret ().
22
Converting a Fraction to a Decimal
Example:
Write 5 as an equivalent decimal
18
.277
5
 18 5 .000
18
36
140
126
140
repeating
remainder
Tobey & Slater, Basic College Mathematics, 6e
23
Chapter 4
Ratios, Rates and Proportions
© 2010 Pearson Prentice Hall. All rights reserved
Equality Test for Proportions
A variable is a letter used to represent a number we
do not yet know.
8  n  72
An equation has an equal sign. This indicates that
the values on each side are equivalent.
8  n  72
8  n
72

8
8
8
 n  9
8
1 n  9
Tobey & Slater, Basic College Mathematics, 6e
25
Solving for a Variable
Example: Solve for n.
n  11.4 = 57
n  11.4
57
=
11.4
11.4
11.4
n 
= 5
11.4
n = 5
Tobey & Slater, Basic College Mathematics, 6e
Check: 5  11.4 = 57
26
Basic College Mathematics
Chapter 5 – Percentages
John J. Moore
Changing a Percent to a Decimal




Drop the % symbol.
Convert to a fraction.
Change to decimal form.
Move decimal point two places to the left.
27
27% 
100
27
 0.27
100
Writing Percents as Decimals
• Example: Write 19% as a decimal.
19% = 19. = 0.19
• Example: Write 2.67% as a decimal.
2.67% = 2.67 = 0.0267
Add an extra zero to the left of the 2.
Fractions to Percents
• Write the fraction as a decimal then convert to a percent.
Example: Write
as a percent.
7
8
7
 7  8
8
 0.875
 87.5%
Divide.
Write as a decimal.
Convert to a percent.
Solving Percent Problems
Using Equations
• Use the following table to translate from a written
problem to a mathematical equation.
Word
Mathematical Step
of
Multiplication
is
Equal
what
Any letter: n
find
Any letter: n =
Percent Problems into Equations
 Example: Translate into an equation.
What is 9% of 65?
n = 9% x 65
 Example: 24 is what percent of 144?
24 =
n
x 144
Percent Proportion
amount
percent number
=
base
100
p is the
percent
number.
10% of 500 is 50
The base, b, is the
entire quantity
(usually follows the
word of ).
50
10

500
100
The amount, a,
is the part
compared to
the whole.
Markup Problems
 Example:
o Mark and Peggy are out to dinner. They have $66 to spend. They want to tip
the server 20%, how much can they afford to spend on the meal?
• n = the cost of the meal
Cost of meal n
100% of n
+
tip at 20% of meal cost
=
$66
+
20% of n
=
$66
120% of n
=
$66
1.2n  66
1.2n
66

1.2
1.2
n  55
Simple Interest Problems
 Interest is money paid for borrowing money.
 Principal is the amount deposited or borrowed.
 Interest rate is per year, unless otherwise stated.
o If the interest rate is in years, the time is also in years.
Interest = principal  rate  time
I=PRT
Example:
Find the simple interest on a loan of $3600 borrowed at 6% for 8 years.
= 3600  0.06  8
= $1728
Chapter 9
Signed Numbers
© 2010 Pearson Prentice Hall. All rights reserved
The Number Line
A number line is a line on which each point is
associated with a number.
–5 –4 –3 –2 –1
– 4.8
0
1
2
3
4
5
1.5
Negative numbers
Positive numbers
The set of positive numbers, negative numbers,
and zero make up all Signed Numbers.
Tobey & Slater, Basic College Mathematics, 6e
37
Ordering Numbers
Signed numbers are listed in order on the number line.
–5 –4 –3 –2 –1
–4<–1
0
1
2
3
4
5
2>1
“greater
than”
“less than”
Tobey & Slater, Basic College Mathematics, 6e
38
Absolute Value
The absolute value of a number is the distance
between that number and zero on a number line.
| – 4| = 4
|5| = 5
Distance of 4
–5 –4 –3 –2 –1
Distance of 5
0
1
2
3
4
Tobey & Slater, Basic College Mathematics, 6e
5
39
Adding Two Numbers with Same Signs
1.
2.
Add the absolute value of the numbers.
Use the common sign in the answer.
Example:
Add (– 3) + (– 11)
–3
+ –11
Add the absolute values of
the numbers 3 and 11.
– 14
Tobey & Slater, Basic College Mathematics, 6e
40
Adding Two Numbers with Different Signs
1.
2.
Subtract the absolute value of the numbers.
Use the sign of the number with the larger absolute
value.
Example:
Add 5 + (– 9)
5
+ –9
Subtract the absolute values of the
numbers 5 and 9.
–4
Tobey & Slater, Basic College Mathematics, 6e
41
Adding Two Numbers with Different Signs
Example:
Add (–24) + (38)
– 24
+ 38
Subtract the absolute values of
the numbers 24 and 38.
14
Example: (– 36) + 4
– 36
+ 4
– 32
Tobey & Slater, Basic College Mathematics, 6e
42
Adding Two Numbers with Different Signs
Commutative Property of Addition
a + b = b + a.
Tobey & Slater, Basic College Mathematics, 6e
43
Adding Three or More Signed Numbers
Example:
(–56) + 6 + (–14)
Because addition is commutative, the numbers can be added in any way.
(–56) + 6 + (–14)
– 50
+ (–14)
(–56) + (–14) + 6
or
– 64
– 70
+6
– 64
Tobey & Slater, Basic College Mathematics, 6e
44
Opposite Numbers
The opposite of a positive number is a negative number
with the same absolute value.
The opposite of 4 is – 4.
–5 –4 –3 –2 –1
0
1
4 + (– 4) = 0
10 – 4 = 6
10 + (–4) = 6
15 – 8 = 7
15 + (–8) = 7
12 – 2 = 10
12 + (–2) = 10
2
3
4
5
The sum of a number and
its opposite is zero.
Subtracting is the
same as adding the
opposite.
Tobey & Slater, Basic College Mathematics, 6e
46
Subtraction of Signed Numbers
To subtract signed numbers, add the opposite of
the second number to the first number.
Example: Subtract – 6 – 14
The opposite of 14 is –14.
–6 + (–14)
–20
Change the subtraction to
addition.
Perform the addition of the
two negative numbers.
Tobey & Slater, Basic College Mathematics, 6e
47
Subtraction of Signed Numbers
Example: Subtract –21 – (–13)
The opposite of –13 is 13.
–21 + (13)
–8
Change the subtraction to
addition.
Perform the
addition.
When subtracting two signed numbers:
1.
The first number does not change.
2.
The subtraction sign is changed to addition.
3.
Write the opposite of the second number.
4.
Find the result of the addition problem.
Tobey & Slater, Basic College Mathematics, 6e
48
Multiplying and Dividing
Signed Numbers
© 2010 Pearson Prentice Hall. All rights reserved
Multiplication with Different Signs
Notice the following pattern when multiplying
numbers with different signs.
3(4) =
2(4) =
12
8
1(4) =
0(4) =
–1(4) =
4
0
–4
–2(4) =
–3(4) =
–8
–12
Note that when we multiply a
positive number by a negative
number, we get a negative number.
Tobey & Slater, Basic College Mathematics, 6e
51
Multiplication with Different Signs
To multiply two numbers with different signs, multiply the absolute value.
The result is negative.
Example:
Multiply –6 (4)
–6 (4) = –24
Example:
Multiply 12 (–9)
The result will always be negative.
12 (–9) = –108
Tobey & Slater, Basic College Mathematics, 6e
52
Division with Different Signs
To divide numbers with different signs, divide the absolute value. The
result is negative.
Example:
Divide. –36 ÷ 4
–36 ÷ 4 = – 9
Example:
Divide. 100 ÷ (–20)
The result will always be negative.
100 ÷ (–20) = –5
Tobey & Slater, Basic College Mathematics, 6e
53
Multiplication with Same Signs
Notice the following pattern when multiplying negative numbers. (The same is
true for division of signed numbers with the same sign.)
3(–4) = –12
2(–4) = –8
1(–4) = –4
0(–4) =
–1(–4) =
–2(–4) =
0
4
8
–3(–4) =
12
When multiplying two
negative numbers, we
get a positive number.
Tobey & Slater, Basic College Mathematics, 6e
54
Multi & Div with Same Signs
To multiply or divide two numbers with the same sign, multiply or divide the
absolute values. The result is positive.
Example:
Divide. –75 ÷ (– 3)
–75 ÷ (– 3) = 25
Example:
Multiply.
The result will always be positive.
 5  2 
 12  3 

 
1
 5  2   5
 12  3  18
 6  
Tobey & Slater, Basic College Mathematics, 6e
55
Multiplying Three or More Signed Numbers
Example:
Multiply –8(–3)(2)
–8(–3)(2) = 24(2)
= 48
First multiply –8(–3) = 24.
Then multiply 24(2) = 48.
Example:
Perform the operations
 2    2  3    2    3  3
 3   3  5   3   2  5 
         
3
 1  
 5
3
  
 5
To divide the first two fractions,
invert and multiply the second
fraction.
Tobey & Slater, Basic College Mathematics, 6e
56