Basic College Math

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Basic College Math

John [email protected]

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Tobey & Slater, Basic College Mathematics, 6e 3

Standard & Expanded Notation

Billions Millions Thousands Ones

Hundreds

Tens

Ones

Hundreds

Tens

Ones

Hundreds

Tens

Ones

Hundreds

Tens

Ones

3 57 015 4

3,000,000 + 500,000 + 70,000 + 5,000 + 100 + 4

Expanded Notation

3,575,104

Standard Notation


Properties of Addition

1. Associative Property of AdditionWhen 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 AdditionTwo numbers can be added in either order with the same result.

8 + 4 = 4 + 8

12 = 12

3. Identity Property of ZeroWhen zero is added to a number, the sum is that number.

7 + 0 = 7

0 + 12 = 12


Properties of Multiplication1. Associative Property of Multiplication

When we multiply three numbers, we cangroup them in any way.

(2 4) 3 = 2 (4 3)8 3 = 2 12

24 = 242. Commutative Property of Multiplication

Two numbers can be multiplied in either order withthe same result.

8 4 = 4 832 = 32

3. Identity Property of OneWhen one is multiplied by a number, the result is that number.

7 1 = 71 12 = 12

4. Distributive Property of MultiplicationMultiplication can be distributed over addition without changing the result.

3 (2 + 4) = (3 2) + (3 4)

18 = 183 6 = 6 + 12


Examples:

34

53

= 3 3 3 3 = 81

= 5 5 5 = 125

35 = 3 3 3 3 3 = 243

122 = 12 12 = 144

74 = 7 7 7 7 = 2401

Exponents


Order of Operations

Order of Operations1. Parentheses.2. Exponents.3. Multiply or Divide from left to right.4. Add or Subtract from left to right.

Example: 24 ÷ 2 – 4 2

Do first

Do last

= 12 – 8

= 4

24 ÷ 2 – 4 2


Order of Operations

Example:

Work inside the parentheses.

Evaluate the exponent.

Add or subtract.

Add or subtract.

= 4 + 34 – 3

= 4 + 81 – 3

= 85 – 3

= 82

= 4 + (16 – 13)4 – 3


Fractions

Chapter 2


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.


Multiplying Fractions

To multiply fractions, multiply numerators across and denominators across.

3 2 67 5 35


Dividing Fractions

When fractions are divided,invert the second fraction and multiply.

3 14 4

3 44 1

3 or 3.1


Adding Fractions

Fractions must have common denominators before they can be added or subtracted.

24

+ =

14

34

2 1 34 4 4


Creating Equivalent Fractions

Fractions with unlike denominators cannot be added.

In order to add fractions with different denominators: 1) Find a Common Denominator2) Build into equivalent fractions with the Common Denominator.

3 4 + 4 5

3 ?4 20

cc

4 ?5 20

cc

55 , 15

c

44 , 14

c



If fractions have different denominators, find the Common Denominator and build up each fraction so that its denominators are the same.

3 18 6

Example:

3 3 98 3 24

1 4 46 4 24

9 4 1324 24 24



5 712 30

Example:

5 5 2512 5 60

7 2 1430 2 60

25 14 1160 60 60


Decimals

Chapter 3


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 Ones position1.234567 Tenths1.234567 Hundredths1.234567 Thousandths1.234567 Ten thousandths1.234567 Hundred Thousandths1.234567 Millionths


Adding Decimals

718.97

Example: 718.97 + 496.5

+ 496.5

Line up decimal points

0Place holder1215.47


Subtracting Decimals

243.967

Example: 243.967 – 84.2

– 84.2

Line up decimal points

00Place holders 159.767

13 131


Multiplying Decimals

Example: Multiply 0.17 0.4

2 decimal places0.17

0.4 1 decimal place

3 decimal places in product (2 + 1 = 3) .068


Dividing by a Decimal

Example: Divide 4.209 ÷ 1.83

1.83 4.20 9

2.3

Move each decimal point to the right two places.. .

Mark the new position by a caret ().

1.83 4.20 9

Place the decimal point of the answer directly above the caret.

366

549549


Converting a Fraction to a Decimal

Example:Write as an equivalent decimal 5

18

5 18 518

repeating remainder

.277 .0 0 0

3 6140126

140


Ratios, Rates and Proportions

Chapter 4


Equality Test for Proportions

8 72n

A variable is a letter used to represent a number we do not yet know.

An equation has an equal sign. This indicates that the values on each side are equivalent.

8 728 8n

8 72n

8 98

n

1 9n


Solving for a Variable

11.4 = 57n

Example: Solve for n.

11.11.4 57=4 11.4

n

11.4 = 511.4

n

= 5nCheck: 5 11.4 = 57

Basic College Mathematics Chapter 5 – Percentages

John J. Moore

Drop the % symbol. Convert to a fraction. Change to decimal form. Move decimal point two places to the left.

Changing a Percent to a Decimal

2727%100

27 0.27100

• Example: Write 19% as a decimal.19% = 19. = 0.19

• Example: Write 2.67% as a decimal.2.67% = 2.67 = 0.0267

Writing Percents as Decimals

Add an extra zero to the left of the 2.

• Write the fraction as a decimal then convert to a percent.

Example: Write as a percent.

Fractions to Percents

78

7 7 88

Divide.

0.875 Write as a decimal.

87.5% Convert to a percent.

• Use the following table to translate from a written problem to a mathematical equation.

Solving Percent ProblemsUsing Equations

Word Mathematical Step

of Multiplication

is Equal

what Any letter: n

find Any letter: n =

Example: Translate into an equation.

Example: 24 is what percent of 144?

24 = n x 144

Percent Problems into Equations

What is 9% of 65?

n = 9% x 65

10% of 500 is 50

Percent Proportion

amount percent number = base 100

p is the percent number.

The base, b, is the entire quantity (usually follows the word of ).

The amount, a, is the part compared to the whole.

50 10 500 100

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

Markup Problems

Cost of meal n tip at 20% of meal cost $66+ =

100% of n 20% of n $66

$66

+ =

=120% of n

1.2 66n 1.2 66 1.2 1.2n

55n

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 timeI = P R T

Example:Find the simple interest on a loan of $3600 borrowed at 6% for 8 years.

= 3600 0.06 8

= $1728


Signed Numbers

Chapter 9


The Number Line

A number line is a line on which each point is associated with a number.

2– 2 0 1 3 4 5– 1– 3– 4– 5

Negative numbers Positive numbers

– 4.8 1.5

The set of positive numbers, negative numbers, and zero make up all Signed Numbers.


Ordering Numbers

Signed numbers are listed in order on the number line.

2– 2 0 1 3 4 5– 1– 3– 4– 5

– 4 < – 1

“less than”

2 > 1

“greater than”


Absolute Value

The absolute value of a number is the distance between that number and zero on a number line.

2– 2 0 1 3 4 5– 1– 3– 4– 5

| – 4| = 4

Distance of 4

|5| = 5

Distance of 5


Adding Two Numbers with Same Signs

1. Add the absolute value of the numbers.2. Use the common sign in the answer.

Example:Add (– 3) + (– 11)

–3 + –11

14

Add the absolute values of the numbers 3 and 11.

–


Adding Two Numbers with Different Signs

1. Subtract the absolute value of the numbers.2. Use the sign of the number with the larger absolute

value.

Example:Add 5 + (– 9)

5 + – 9

4

Subtract the absolute values of the numbers 5 and 9.

–



Example:Add (–24) + (38)

– 24 + 38

14

Subtract the absolute values of the numbers 24 and 38.

Example: (– 36) + 4

– 36 + 4

– 32



Commutative Property of Additiona + b = b + a.


Adding Three or More Signed Numbers

Because addition is commutative, the numbers can be added in any way.

Example: (–56) + 6 + (–14)

(–56) + 6 + (–14)

– 50 + (–14)

– 64

or(–56) + (–14) + 6

– 70 + 6

– 64


Opposite Numbers

The opposite of a positive number is a negative number with the same absolute value.

2– 2 0 1 3 4 5– 1– 3– 4– 5

The opposite of 4 is – 4.

4 + (– 4) = 0The sum of a number and its opposite is zero.

10 – 4 = 6 10 + (–4) = 6

15 – 8 = 7 15 + (–8) = 7

12 – 2 = 10 12 + (–2) = 10

Subtracting is the same as adding the opposite.


Subtraction of Signed Numbers

To subtract signed numbers, add the opposite of the second number to the first number.

Example: Subtract – 6 – 14

–6 + (–14)

–20

The opposite of 14 is –14.

Change the subtraction to addition.Perform the addition of the two negative numbers.


Subtraction of Signed NumbersExample: Subtract –21 – (–13)

–21 + (13)

–8

The opposite of –13 is 13.

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.


Multiplying and Dividing Signed Numbers


Multiplication with Different Signs

3(4) = 12

2(4) = 8

1(4) = 4

0(4) = 0

–1(4) = –4

–2(4) = –8

–3(4) = –12

Notice the following pattern when multiplying numbers with different signs.

Note that when we multiply a positive number by a negative number, we get a negative number.


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

The result will always be negative.Example:Multiply 12 (–9)

12 (–9) = –108


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

The result will always be negative.Example:Divide. 100 ÷ (–20)

100 ÷ (–20) = –5


Multiplication with Same Signs

3(–4) = –12

2(–4) = –8

1(–4) = –4

0(–4) = 0

–1(–4) = 4

–2(–4) = 8

–3(–4) = 12

Notice the following pattern when multiplying negative numbers. (The same is true for division of signed numbers with the same sign.)

When multiplying two negative numbers, we get a positive number.


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

The result will always be positive.Example:Multiply. 5 2

12 3

5 212 3

1

6

518


Multiplying Three or More Signed Numbers

Example:Multiply –8(–3)(2)

Example:Perform the operations

–8(–3)(2) = 24(2) First multiply –8(–3) = 24.

= 48 Then multiply 24(2) = 48.

2 2 3 2 3 33 3 5 3 2 5

To divide the first two fractions, invert and multiply the second fraction.

315

35

Basic College Math

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