1
1.1 Standard Notation and Place Value
digit – one of the numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
number – may have several digits, for example 367
1,234,567,890
1,234,567,890
Ex a For the number above what digit names the hundred thousands place?
Ex b What digit names the number of tens?
Ex c What does the digit “2” represent in the number above?
Ex d What does the digit “7” represent in the number above?
Ex e Write 5,620,487 in word form.
Ex f Write 5,620,478 in expanded form.
Ex g Write in standard notation: Thirty-two million
2
1.2 Addition
Most problems are added vertically, even if they are originally written horizontally.
Ex a 34 + 2413 + 222
Ex b 782 + 4365 + 28
The result of an addition problem is called a ___________
Perimeter – distance around the outside of an object.
Ex c Find the perimeter of each of the objects below:
Practice Problems
3
1.3 Subtraction
Subtraction is the inverse of addition.
Subtraction is also carried out vertically, even if the original problem is written horizontally.
For each example, subtract and check by adding:
Ex a 85 – 32
Ex b
425
- 86
Ex c 3000 - 1471
The result of a subtraction problem is called a _____________________
Practice Problems
1) 30,008 – 52
2) 5923
- 769
4
1.4 Multiplication
8 X 3 = 24
8 and 3 are called ____________, 24 is called the _______________
If you haven’t memorized the products of single digits (times tables), you should do so.
There are many ways to write products:
58
5X8
5(8)
(5)(8)
(5)8
(5)X8
When there is no operator shown, the operation which is understood is _____________
The purpose of parentheses is _______________________
(5) + (8) =
(5)(+8) =
5(+8) =
5 + (8) =
Special Products
Multiplying by zero: 0 X 17 =
29(0) =
Multiplying by one: 1 X 392 =
(53)(1) =
Multi-Digit Multiplication
Ex a Multiply 2 5 9
X 7
Ex b Multiply 4 5 2 7
X31
Ex c
Multiply
659
X 403
Sneaky Multiplication Tricks
Ex d Multiply 1000 X 7834
Ex e Multiply
6824
X300
5
Area
Ex f Find the area of the rectangle:
6
1.5 Division
Division is the inverse of multiplication:
12  6 
can be rewritten as
Ex a Divide and check:
45
5
21 3
6 30
Special Quotients
Dividing by 1:
19  1 
397
1
Dividing by itself:
19  19 
397
397
Dividing 0 by a number:
0  42 
0
20
Dividing a number by 0:
79  0 
11
0
Long Division
Ex b Divide 296  4
407
5
23 7 2 5 8
7
1.6 Rounding and Estimating
Ex a
Round 29 to the nearest 10
Round 22 to the nearest 10
Round 25 to the nearest 10
Rounding Whole Numbers Procedure – for a specific place
1. Find the digit in the specified place.
2. Look at the digit AFTER that place
3. If the digit ___________________________
If the digit ___________________________
4. Replace the rounded digits with __________
Ex b Round 3,682,357 to the nearest:
million
ten thousand
hundred
ten
Estimating
Ex c Estimate the following amounts for easier calculations:
Restaurant bill: $ 43.58
Truck: $27,875
House: $239,995
Ex d Estimate the sum by rounding each
number to the nearest ten: 58 + 91 + 37
Ex e Estimate the difference by rounding
each number to the nearest hundred:
564 – 238
Ex f Estimate the product by rounding to the
nearest hundred: 287 X 726
Ex g Estimate the quotient by rounding to the
nearest ten: 476  59
8
1.7 Solving Equations
Ex a My husband’s brother is 4 years older than he is. If his brother is 59, how old is he?
Solve by Trial (guess and check)
Ex b Solve: x – 9 = 33
Solve: 4x = 36
Solving by Opposite (Inverse) Operations
The opposite (inverse) of addition is ______________________
The opposite (inverse) of multiplication is __________________
We want to isolate the variable.
Ex c Solve each equation, then check your answer:
14 + x = 52
18 = 2  y
22 6 = p
9
1.8 Applications
Procedure
1. Familiarize – understand what is asked for, what numbers are important
2. Translate – make an equation
3. Solve
4. Check
5. State – Answer the question
Graph from Basic College Mathematics, 12/e, by Bittinger/Beecher/Johnson
Ex a (Problem 1) How much taller would the Aeropolis 2001 have been than the Nakeel
Tower?
Ex b (Problem 3) The Willis Tower (formerly Sears Tower) is the tallest building in Chicago.
If the Miglin-Beitler Skyneedle had been built, it would have been 551 ft. higher than the Willis
Tower. What is the height of the Willis Tower?
Ex c There are 520 seats in an auditorium. If all rows have the same number of seats, and
there are 20 rows, how many seats are in each row?
10
1.9 Exponential Notation and Order of Operations
24
33
52
10  10  10  10  10  10
Ex a Write in exponent form: 7  7  7
Ex b Evaluate: 7  7  7
10  10  10  10  10  10
Simplifying Expressions (Order of Operations for several operations)
1. Parentheses (and grouping symbols like { } or [ ])
2. Evaluate all exponential expressions
3. Multiplication and Division, in order from left to right
4. Addition and Subtraction, in order from left to right
Ex a 100 – (58 – 21)
(100 – 58) – 21
Ex b 5  22
(5  2)2
Ex c 60 - 36  3  4
(60 - 36)  3  4
Average - Add the numbers, divide by “how many”
Ex d Find the average of the test scores: 82, 72, 83
11
2.1 Factorizations
For the product a  b , a and b are called _____________________
Q
Dividing
, d is a factor of Q if the remainder is _____ .
d
If d is a factor of Q, Q is a _______________ of d, and Q is _________________ by d.
Ex a List all the factors of 24.
Ex b List all the factors of 23.
Ex c List the first 5 multiples of 13.
Ex d Show that 52 is divisible by 4.
Prime and Composite Numbers

1 is ___________________________

A _____________________has only 1 and itself (2 different factors) as factors

A _____________________ can be “broken down” into other factors besides 1 and itself
Ex e Which of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are prime?
Ex f Is 128 divisible by 7?
Ex g Is 128 divisible by 8?
12
2.2 Divisibility

A number is divisible by 2 if its ones digit is ___________________________
Ex a Which numbers are divisible by 2?
17

4,201,122
3801
50,000
A number is divisible by 3 if ____________________________ is divisible by 3
Ex b Which numbers are divisible by 3?
29

4,201,122
3801
50,000
A number is divisible by 6 if it is ____________________________
Ex c Which numbers are divisible by 6?
29

4,201,122
3801
50,000
A number is divisible by 9 if ____________________________ is divisible by 9
Ex d Which numbers are divisible by 9?
387
4,201,122

A number is divisible by 10 if ____________________________

A number is divisible by 5 if ____________________________
Ex f Which numbers are divisible by 10? Which are divisible by 5?
295
3,729,231
1620

A number is divisible by 4 if ____________________________

A number is divisible by 8 if ____________________________
Ex g Which numbers are divisible by 4? Which are divisible by 8?
9024
387,231
420
13
2.3 Fractions and Fraction Notation
Ex a Shade the portions that represent 2
3
8
Ex b What fraction is represented by the shaded portions?
Ex c Find 1
7
3
and 3
16
8
Ratio – a quotient of 2 quantities (can be written as a fraction)
The 2 quantities are often separated by “to”
Ex d A job opening has 97 applicants, and 4 people are hired.
1) Write the ratio of people hired to applicants.
2) Write the ratio of people hired to people not hired.
14
2.4 Multiplying Fractions
Ex a Multiply
Multiplying a Fraction by a Fraction
1. Multiply the 2 numerators 2. Multiply the 2 denominators –
Do the same rules work for addition (add 2 numerators & keep, add 2 denominators & keep)?
Multiply a Fraction by Whole Number
Ex b Multiply
Applications
Ex c For a training program, 20 out of 71 applicants are accepted. Of the accepted students,
4/5 of the students are hired. What fraction of all applicants are hired?
15
2.5 Simplifying (reducing)
Fractions that reduce to 1:
Multiplicative Identity (Multiplying identity) - Using it gives the same value (no change)
a  _____ = a
Equivalent fractions – have the same value:
We can change fractions to have a new denominator, but the same value
Ex a Find a name for
2
with a denominator of 21.
7
Ex b Create an equivalent fraction with the new denominator.
3
?

4 36
Simplifying Fraction Notation (Reducing)
Simplest fractions have NO COMMON FACTOR in the numerator and denominator. To get
simplest form, remove fractions that equal 1 (common factors)
Ex c Simplify:
18
24
16
Practice Problems
Simplify:
1.
12
30
2.
18
54
3.
1170
1200
17
2.6 Multiplying, Simplifying, and Applications
It’s important to simplify a product before actually multiplying out the numbers
Ex a Simplify and multiply
3 10

5 9
1.
2.
3.
4.
Procedure
Put numerator and denominator factors together in the num. & denom., but don’t actually
multiply out the numbers
Factor the numerator and denominator
Remove factor fractions that equal 1, if possible.
Multiply out the products to get a single number in numerator & denominator.
Ex b Multiply (reminder before multiplying:
Applications
1
inch (this is how far it moves with every full turn). How far
16
into a piece of wood will it go when makes 6 full turns?
Ex c The pitch of a screw is
Ex d Financial aid covers
covered by financial aid?
3
of a student’s expenses. If expenses are $4500, how much is
5
18
2.7 Division and Applications
Reciprocals - Pairs of fractions whose product = 1. We find a reciprocal by________
Ex a Find the reciprocal of
Dividing Fractions
2 8
Ex b Divide: 
3 9
Solving Equations
2
10
x
Ex c Solve
5
7
Solve
3
x  600
2
Applications
Ex b How many 3/4 ounce servings of chips can be made from a 12 ounce bag?
19
Practice Problems
 5 
1. 9   
 12 
4.
3 6

14 7
2.
5 1

7 10
4
5. 6   
3
 4   15 
3.     
 25   16 
6. Solve:
5
3
x
4
8
20
3.1 Least Common Multiples
Ex a Find the least common multiple of 9 and 12 by making a list of multiples.
12:
9:
Some common multiples are:
The Least Common Multiple (LCM) is:
Finding LCM’s by Listing Multiples (Method 1)
a) Is the largest number a multiple of the other numbers?
b) If not, list multiples of the largest number until you find one that is a multiple of the
other numbers.
Ex b Ex c Find the LCM of 4, 10, and 20
Ex c Find the LCM of 4, 6, and 10:
10:
6:
4:
Prime Factorization - Breaking down numbers to the smallest possible factors
Tree method:
Divide Up Method:
21
Finding the LCM by Prime Factorization (Method 2)
a) Write the prime factorization of each number
b) Create a product – for each factor, use the greatest number of repeats in ONE
number
Ex c Find the LCM of 9 and 12
Ex d Find the LCM of 8, 18, and 12 using prime factorization and exponents
Practice Problem: Find the LCM of 25 and 35
We will skip Method 3 (p. 150).
22
3.2 Addition and Applications
Like Denominators
1. Add numerators
2. Keep same denominator
3. Reduce if possible
Ex a Add:
1
7

=
12 12
It doesn’t say reduce – should we?
Different Denominators
1. Get LCD (LCM of denominators)
2. Multiply top & bottom by the needed factor.
3. Add as above.
Ex b Add:
1 3
 =
4 8
Ex c Add:
5 2

=
9 15
Ex d Add:
1 1
 =
3 8
How is this different from multiplying?
23
Applications
Ex e A tile is
1
7
1
in. thick and is glued to a board
in. thick. The glue is
. How thick is the
4
8
16
result?
Practice Problems
1. Add:
3 2
 =
4 5
2. Add:
2 4 1
  =
15 9 6
24
3.3 Subtraction, Order, Applications
Like Denominators
1. Subtract numerators
2. Keep same denominator
3. Reduce if possible
Ex a Add:
9
3
=

10 10
Different Denominators
1. Get LCD
2. Multiply top & bottom by the needed factor.
3. Subtract as above.
Ex b Subtract:
5 1
 =
3 4
Ex c Subtract:
7 1

=
5 14
Order (which is bigger?) - Use < or >
Ex e Use < or > to write a true statement.
3
5
2
5
3
4
Ex f I have ½ lb. butter, and use 1/3 lb. How much is left?
5
7
25
3.4 Mixed Numerals (also called mixed numbers)
What fraction is represented below?
Ex a Write as a mixed numeral: 11 +
3
=
4
9+
3
=
5
Ex b Convert to fraction notation (commonly called _________________________ )
4
3
5
5
2
3
Fraction notation (improper fractions) and mixed numbers.
Ex c Convert the improper fractions to mixed numbers:
95
7
158
3
26
3.5 Add & Subtract Using Mixed Numerals; Applications
Addition
Ex a
1
4
3
 11
8
3
3
5
2
4
3
Ex b 1
Subtraction
Ex c
5
6
1
-2
12
8
1
7
5
7
7
Ex d 13
5
3
Ex e (# 38) A plumber uses 2 pipes, each of length 51 , and one pipe of length 34 . How
16
4
much pipe was used in all?
27
3.6 Multiplication and Division with Mixed Numbers
Convert mixed numbers to fraction notation (improper fractions)
3
1
Ex a ( 1 )( 2 )
5
4
Ex b (12)( 3
Ex c 2
5
)
6
1
3
1
2
5
Ex d 8  1
1
3
1
Ex e A space shuttle orbits the earth in 1 hr. How many orbits are made in 24 hours?
2
28
3.7 Order of Operations/Complex Fractions/Estimation
2
1  1
2
Ex a Simplify      2 
3  2
3
3
Ex b Simplify: 10
6
35
Ex c Find the average of
3
5
and
4
8
Ex d Estimate, rounding to the nearest whole number
Compare the numerator to ______________________________
5
8
1
3
4
3
11
12
7
8
Ex d Estimate each term to the nearest whole number, then perform the operations:
7
3
4
31
5 2
19
5
37
29
4.1 Decimal Notation, Order, and Rounding
Decimal Values
Ex a Write the value of $178.95 in expanded form.
Place Value Chart (for 1.73205)
Decimal Notation and Word Names – decimal words are similar to the fraction they represent
1. Number left of decimal point:
2. Point:
3. Number right of point:
Ex b Give the word name of 1.73205
Ex c The median age in CA is 35.2 -- write the word name.
30
Converting Decimals to Fractions:
1. Count the number of decimal places
2. Write that number of zeroes in the denominator, with 1 in front
3. Write the digits in the numerator
Ex
Convert 0.357
0.0182
23.41
Note: Whole number parts on the left of the decimal point make ______________________
Converting Fractions to Decimals (“simple” denominators with powers of 10)
1. Count the number of zeroes
2. Use that number of place values to make the numerator smaller (move left).
Ex
Order (state which is larger using > or <)
How to make equivalent decimals:
31
Comparing numbers in decimal notation
1. If needed, tack on zeroes to make the decimals equal length
2. Compare digits beginning starting immediately after the point (if needed, tack on zeroes to
make numbers the same length)
3. When digits differ, the larger number gives the larger amount
Ex
Rounding
1. Look at the specified digit
2. Look at the next place value (immediately after the specified one)
3. If the next digit is
0 – 4, keep the desired digit
5 – 9, round up
Ex
Round
to the nearest
a) thousandth
b) hundredth
c) whole number (unit)
d) ten
e) hundred
Ex
Round
a) hundredth
b) tenth
c) whole number (unit)
d) ten
e) hundred
to the nearest
32
4.2 Addition & Subtraction
Procedure:
1. Line up decimal points! (most important)
2. Fill in zeroes at the end of decimals if needed
Ex a Add: 2.68 + 11.3 + 0.009
Ex b Subtract: 6 – 4.27
Ex c Solve: x + 3.7 = 9.431
Practice Problems - Perform the operations or solve
1) 7 – 2.381
2) 14.843 + 0.34 + 1.9 + 10
3)
Solve: x – 42.87 = 19.4
33
4.3 Multiplication
Decimals have fractional equivalents:
2.35 X 0.4
Procedure:
1. Multiply digits as if they were whole numbers
2. Move the point the number of decimal places after all points
3. Fill in zeroes if needed
Ex a Multiply: (6.7)(0.038)
Multiply by 0.1, 0.01, 0.001, etc. (small numbers)
Ex b Multiply 18.47 X 0.001
Multiply by 10, 100, 1000, etc. (large numbers
Ex c Multiply 18.47 X 1000
Large Number Names
Ex d Convert $14.5 million to digits
Dollars and Cents $1 = 100¢
Ex e Convert 89 cents to dollars
and 1¢ = 0.01$
Ex f Convert $22.51 to cents
34
Practice Problems
1) 4.6 X 0.9
2) 0.01 X 821.37
3) Convert 530,792¢ to dollars
4) Convert 192.5 thousand to standard form
4.4 Division
Divide decimals by whole numbers – similar to whole number long division, but put the
decimal point in the quotient
Ex a
15 25.5
Decimal divisors (denominators) - make fraction and move point to get a whole number in
denominator.
Ex b 2.732  0.04
35
Divide by 10, 100, 1000, etc. (large numbers)
Ex c Divide 128.54  1000
Divide by 0.1, 0.01, 0.001, etc. (small numbers)
Ex d Divide
0.063
0.001
Solving
Ex e Solve: 2.5t = 300
Practice Problems
1)
14.31
0.01
2) 11.2  4
3) Solve: 0.3y = 1.38
36
4.5 Converting Fractions to Decimals - Use Long Division
Write in decimal notation:
Ex a
5
8
Ex b
1
2
3
Ex c
3
11
For each of the decimals above, round to the nearest tenth, hundredth, and thousandth.
tenth:
hundredth
thousandth
Practice Problems
Write in decimal notation and round to the nearest tenth
1)
13
4
2)
5
6
37
4.6 – 4.7 Estimating and Applications
Estimating Sums and Differences – operations are
- Round to the same place value(s) then add or subtract.
Ex a On a shopping trip, Mia buys items
costing $38.95, $129.99 and $9.77. Estimate
the cost by rounding to the nearest ten.
Ex b A $491.79 tablet is
discounted by $109.21. Estimate
the final price.
Estimating Products and Quotients – operations are
- Round to one non-zero digit OR round to “easy” digits.
Ex b Dan is paid $892.12 for 11 days. Estimate his daily pay, then calculate the exact
amount to the nearest cent.
Practice Problems - Perform the operations or solve
1) Estimate, rounding to the nearest tenth:
1.4368 + 0.1724 – 0.0913
2) Coffee costs $3.61 (including tax).
How much is spent in a 30-day month?
3) Cole earned $620.80 working 40 hours
in a week. What is his hourly wage?
4) A shipment of seafood costs $88.65,
and there are 6.245 lb. Estimate each
number, then divide the estimates to
approximate the cost/lb.
38
5.1 Intro to Ratios
ratio – a ___________________ of 2 quantities
There are several ways to write ratios. For example for a TV screen 16 inches wide and 9
inches tall, the width to height ratio can be written as:
Ex a Write the ratios in 2 other formats (without reducing)
3 to 5
14.7:100
8½ to 11
Ex b Find the ratio of length to width:
Ex c For the triangle, find the ratios listed and reduce.
height to base ratio
ratio
hypotenuse to base ratio
height to hypotenuse
39
5.2 Rates and Unit Prices
rate – a ratio whose numerator and denominator have different units.
Ex a My car travels 500 miles on 15 gallons of gas. What is the rate of miles per gallon (also
known as gas mileage)?
Ex b Al earns $30,000 in a year. What is his rate of pay in dollars per month?
Unit rate – ratio where the denominator number is 1
Unit price – ratio of price to number of units, where the number of units is reduced to
____________
To find unit rates (including unit price), use ______________________________
Ex c Find the unit cost of the following jars of peanut butter. Which is the better buy?
Brand A is 40 oz. and costs $5.00
Brand B is 28 oz. and costs $3.00
Ex d I drive 390 miles in 6 hours. What is the unit rate in miles/hour?
40
5.3 Proportions
Proportion – 2 ratios that equal each other:
The pairs 1, 2 and 3, 6 can be used to form a ratio:
We can test if 2 proportions are equal if their cross products are _____________________
Ex a Are the pairs or ratios proportional?
1)
4
7
3)
1
20
5
11
2) 3,5 and 21, 35
0.04
0.8
4) 2½ , 4½ and 10, 18
Solving Proportions – set cross products ____________________________, then solve for x
Ex b Solve:
x 9

4 6
x 0

5 21
0.5 1.5

y
3.5
41
Practice Problems:
1) Write fraction notation and reduce:
8 to 12
2.4 to 6
2) Find each rate:
65 meters to 5 seconds
243 miles per 4 hours
3) Are the pairs proportional:
3, 7 and 15, 45
2.4, 1.5 and 0.16, 0.1
4) Solve
2 10

7 x
p
1.1

1.2 0.6
42
5.4 Applications of Proportions
When to use proportions? You have 2 quantities that are related. One quantity changes,
and you want to find the changed value of the second quantity.
Ex a Ravi makes $315 working 21 hours. How much would he make if he works 40 hours?
Ex b Two cities on a map are 2½ inches apart, which represents 300 miles. How far apart
are two cities if they are 6¼ inches apart on the map?
Ex c In 2015, 1 US dollar is worth 16.8 pesos. How many dollars is 400 pesos worth, to the
nearest dollar?
43
6.1 Percent Notation
The Butte fire was 15% contained (as of Sept. 12, 2015). What does that mean?
Percent notation:
Fraction notation:
or ratio:
Decimal notation:
Percent of a Quantity
What is
3
of 40?
5
What is 60% of 40?
Proper fractions (less than 1) and “common” percentages (less than 100%) are similar
Amount = Percent of Base
(If you have “Percent of”, the next is always base)
Ex b A discount is 20% of the original price. If the item is marked $30, what is the discount?
44
Converting percent to decimal – Replace % with ___________ or ______________
This causes you to remove _______, make number ____________
Ex a Convert to decimal:
58%
7.2%
150%
0.03%
2
Ex b Convert 5 % to decimal:
3
Ex c Write decimal values for each of the percentages listed.
Monthly Expenses
If monthly income is $1000, how much spent on transportation?
Convert decimal to percent – multiply by ___________
Does this change the value?
Ex d Write percent notation for
0.27
0.735
2.7
0.0009
0.4
45
6.2 Percents and Fractions
Converting a fraction to percent
1. First, convert fraction to decimal (From 4.5, use ________________________
2. Next, convert decimal to percent (From 6.1, multiply by ______________________
Ex a Convert 7/8 to a percent
Ex b Convert 7
2
to a percent
3
Shortcut - Only works when denominator is a factor of 100
1. Multiply top and bottom to build the denominator to 100.
2. Change /100 to %
Ex c Convert to a percent:
13
20
7
25
3
10
47
50
Converting Percent to a Fraction (postpone repeating decimal to Math 20)
1. Replace % with __________________
2. Reduce
Ex d Convert to a fraction and simplify
70%
12.5%
0.4%
2
16 %
3
46
Practice Problems
1. A lawn requires 300 gallons of water for every 500 square feet. How much water does a
lawn which is 1800 square feet require?
2. Sal burned 200 calories in ¾ hour of walking. How many calories would be burned in 1 ¾
hours of walking?
3 Find percent notation for:
0.7
0.3891
5
12
7
25
4. Find decimal notation for:
57%
1.5%
22 ½ %
240%
22 ½ %
240%
5. Find fraction notation for:
57%
1.5%
47
6.3 Solving Percent Problems - Percent Equations
Translating to Equations
of  multiply
is  equals
%  convert number to decimal or a fraction using 1/100
What 
Ex a What is 7% of 45?
Ex b
28% of 30 is what?
Ex c 15 is what percent of 75?
Practice Problems
1. What percent of 42 is 7?
2. 9 is 25% of what?
3. 70% of what is 35?
48
6.5 Applications
Ex a From the pie chart below:
Monthly Expenses
1) If a person makes $3000/month, how much is
spent on housing?
2) If a person is spending $600/month on
transportation, what is their total income?
Ex b A test has 60 questions, and Jan gets 49 correct. What percent are correct (to the
nearest whole number percent)?
Percent Increase & Decrease
Ex c Rent was $750/month last month and increased to $800 this month. What is the
percent increase?
Ex d A TV cost $400 last year but costs $320 this year. What is the percent decrease?
49
6.6 Sales Tax, Commission, and Discount
Sales tax and commission increase an original price.
Discount decreases the original price.
Rate of discount or increase is the same as percent of discount or increase.
Ex a Sales tax adds $12.74 to the price of a fire pit. If the sales tax rate is 8%, find the
original price.
Ex b A real estate agent earns a 6% commission on a house valued at $240,000. How
much commission does he earn?
Ex c A backpack is discounted and sells for $25. If the amount of discount is $15, find the
original price and rate of discount.
Practice Problems
1. A $60 meal is charged 7.35% sales tax. How much tax is charged, and what is the final
price?
2. A sweater originally costing $75 is marked down to $45. What is the rate of discount?
50
7.1 Average, Median and Mode
Mean (Average) – Procedure for calculating
1. Find the sum of values
2. Divide by how many values
Ex a Trey’s test scores are 88, 92, 79, and 84. What is the average?
Weighted Average - If a class has a higher number of units, it “counts more”
GPA = total grade points/# of units
Ex b A student takes earns an “A” in a 3-unit English course and a “C” in a 4-unit math
course. What is the GPA?
Class
Units
Grade
English
3
A (4.0)
Math
4
C (2.0)
Ex b My syllabus has the following weights:
Category
Weight
Your score
Exams
50%
Homework
20%
Participation
5%
Final Exam
25%
51
Median – for an odd number of values
1. Rewrite the list of values in order (smallest to largest or largest to smallest)
2. Choose the middle value (the median)
Ex
The list prices of neighborhood houses is below. Find the median.
180,000
250,000
176,000
220,000
1,000,000
206,000 240,000 192,000 220,000
Median – a more sophisticated approach for an even number of values
1. Rewrite the list in order
2. Look at the middle 2 values.
3. Calculate the average of the middle 2 values.
Ex Find the median:
180,000
250,000
176,000
220,000
1,000,000
206,000
240,000 192,000
Mode
The most common value is the mode. If there are 2 (or more) most common values, there
are 2 (or more) modes. If no value is more common than any other, there is no mode.
Ex
Find the mode of the ages of students:
22, 20, 19, 20, 18, 35, 19, 58, 21, 19, 28
Find the mode of these ages:
9, 9, 9, 12, 15
Find the mode:
9, 9, 12, 12, 15, 15, 17, 17
52
7.2 Interpreting Data from Tables and Graphs
Ex a From the table in Example 1, page 419
1) Which country has the smallest land area? Which has the largest land area?
2) Which country or countries had a population decrease from 2008 to 2012
3) Find the average population density of China, Japan, and India in 2012.
4) Estimate the population in the United States in 2000 and 2012 to the nearest million.
Calculate the percent increase to the nearest whole number percent.
Ex b From the pictograph in Example 2, page 421
1) Which continent has the greatest number of roller coasters?
2) About how many roller coasters are there in Asia?
3) About how many more roller coasters are there in South American than in Australia?
53
7.3 Bar Graphs and Line Graphs
Ex a From the bar graph in Example 1, page 430
1) Which country has the highest per capital tea consumption?
2) What is the coffee consumption in Brazil?
3) In what country do people drink almost no coffee?
4) What is the difference in coffee consumption between the 2 highest coffee consumers?
5) What is the percent decrease from the largest consumer to the second largest?
Ex c Make a bar graph of class data showing favorite technology applications
App
Number of users
54
Line Graphs
Ex c From the line graph, Example 3 on page 432,
1) What was the average price of gold in 2010?
2) During which 5-year period did the price of gold remain essentially unchanged?
3) In what year was the average gold price about $600?
4) What was the change in average price from 2010 – 2012?
5) What was the percent change during that period?
Ex b Make a line graph of class data showing the number of siblings of the students in the
class.
How many
siblings?
0
1
2
3
4
5
6
7 or more
Number of
responses
55
7.4 Circle Graphs (aka Pie Graphs)
Ex a From Example 1, page 439
1) What 2 species have the largest populations?
2) What species accounts for 15% of the population of endangered whales?
3) What is the percent of right whales (from both North Atlantic and North Pacific)?
4) If there are about 300,000 endangered whales of these species, how many endangered
right whales are there?
Ex b Draw a circle graph for the following data:
Age of student
< 18
2%
18-24
61%
25 – 34
20%
35 -44
9%
45+
8%
56
8.1 Linear Measures: American Units
Length
Area
Volume (capacity) are 3 different kinds of quantities.
Units:
Ex a Convert: 5 ½ yards = ___________ inches
Note: When converting from larger units to smaller ones:
Converting between units: Multiply by fractions that equal 1
Ex b 64 inches = _________ feet.
Ex c Convert 0.6 miles to feet
Ex d Convert 11.37 ft to yards
57
8.2 Linear Measures: Metric
Examples of metric length:
Recall: Converting large units to small units means
Ex a 13.4 kilometers = ____________________ meters
Ex b
2.98 m = _________________cm
Ex c Convert 681 mm to meters
Ex d
57.7 mm = _____________________cm
Using mental conversion
Practice Problems (8.1 and 8.2)
1. 927.1 dm = _________________km
2. 69 inches = _____________ft
3. 0.3 miles = ____________ ft
4. 578 mm = _______________ m
58
8.3 Converting Between American and Metric Units
Ex a Convert 10 cm to inches
Ex b 400 m = _______________ ft.
Ex c How many miles is a 10K run (10 km)?
Ex d The Horseshoe Falls of Niagara Falls is 173 ft tall. How many meters is it (to the
nearest meter)?
Practice Problems
1. 12 inches = _____________ cm.
2. 1000 meters = _______________ miles
3. Convert 40 ft. to meters.
59
8.4 Weight & Mass/ Medical Applications
Ex a Convert 100 ounces to pounds.
Ex b 3.4 tons = _____________ pounds
Ex c 56 kg = ______________pounds
Ex d 325 mg = __________ g
Medical – Micrograms
1 micrograms = 1 mcg =
1
grams;
1,000,000
1,000,000 mcg = 1 g
1
mg
1000
1000 mcg = 1 mg
1 mcg =
Ex e Convert 0.3 mg to mcg
Practice Problems
1. 2 kg = ___________ grams
2. 2 lb = ______________ oz.
3. 475 mcg = _____________ mg
4. 2 lb = ______________ g
60
8.5 Capacity/Medical Applications
Capacity (volume) -- American Units
Ex a How many gallons is 32 quarts?
Ex b Convert 3 quarts to fluid ounces (more than 1 step)
Metric Units
The same prefixes used for length are used for capacity.
Abbreviations for liters:
Also: 1 ml = 1 centimeter cubed (cc) = 1 cm3
Ex c
0.287 L = ____________________mL
Ex d How many liters is 4380 mL?
Medical Applications
Ex e A bottle of medicine contains 300 mL. If 1 tsp = 5 mL, how many tsp. is this?
If each dose is 2 tsp., how many doses are in the bottle?
61
Ex f A doctor orders 40 mL per hour of saline ordered intravenously. How many mL should
be given in a day?
If a bag contains 500 mL saline, how long will it take to empty the bag?
Practice Problems
1. How many mL is 3.75 L?
2. A punch bowl contains 3 gallons of punch. How many cups is this?
3. A patient needs to receive 2.0 L of saline over a 24 hour period.
a) How many liters should be given in 1 hour?
b) How many mL is this equivalent to?
62
8.6 Time and Temperature
Your book states that 1 year = 365 ¼ days, so we use this conversion. More precisely, 1
year = 365.24 days, which is why we have a leap year every 4 years, but we skip the leap
year at the turn of the century (years ending in 00).
Ex a 4 minutes = ______________ seconds
Ex b A movie has a run time of 160 minutes. How many hours is this?
Ex c Convert 7200 seconds to hours
Ex d Convert 68o F to Celsius
Ex e A patient has a fever of 40o C. What is this in Fahrenheit degrees?
Practice Problems
1. How many hours are in a week?
2. 900 seconds = ____________ hours.
3. Convert 68o F to Celsius
4. 100 o C = ______________ o F
63
9.1 Perimeter and Applications
Perimeter – the sum of the lengths of the sides:
Ex a Find the perimeter of each object:
Rectangle: has 4 angles which are 90o (right angles)
Formula for Perimeter of a rectangle:
P=
Question:
Formula for Perimeter of a square:
P=
Ex b A living room is 18 ft X 12 ft. The doorway into the living room is 6 ft wide.
a) If baseboard costs $2.25/foot, what is the cost of installing baseboard?
b) If baseboard is only sold in 8-ft segments for $12 each, what is the cost?
64
Practice Problems
1. Find the perimeter of a square picture frame with 9.5 inches on each side.
2. A 9’ X 10’ room is decorated with border paper. If each roll is 12 ft, how many rolls are
needed?
3. A yard is enclosed with chicken wire fencing. If each roll of 50 ft. costs $26, how much
does it cost to enclose a 60 ft X 30 ft back yard?
65
9.2 Area
Formula for Area of a rectangle:
A=
Formula for Area of a square:
A=
Ex a Find the area of a rectangle that is 4 yards X 5 yards.
Ex b Convert 4 yards and 5 yards to feet, and find the area using these units.
Triangles
Formula for Area of a Triangle:
Ex c Find the areas:
66
Formula for Area of a Parallelogram:
Ex d Find the area of a parallelogram whose base is 4½ inches and height is 5¾ inches.
Formula for Area of a Trapezoid:
Practice Problems - Identify the shapes, then find the areas:
67
9.3 Circles
Formulas:
Ex a Find the radius and the diameter of each circle
Circumference – the distance around the _____________________________
Formula for Circumference: C =
Also: C =
Estimates for 
Ex b Find the circumference of a circle whose radius is 10 inches. Use 3.14 as an estimate
for .
68
Formula for Area of a Circle
A=
Ex d Find the area of a circle with radius = 3 ft. Use 22/7 as an estimate for 
Practice Problems:
1. Which has more pizza: a 12-inch square pizza, or a 14-inch round pizza?
2. Find the perimeter of this shape.
3. Find the area of the shape above
69
9.4 Volume
Rectangular Solids (fill with cubes)
Formula for Volume: V =
Ex a Find the volume of a box whose dimensions are 8” X 10” X 3”
Cylinders
Formula for Volume: Volume = Area X height
V=
Ex b Find the volume of a can whose diameter is 14 cm and whose height is 10 cm.
Practice Problems:
1. How many cubic feet is a refrigerator that is 2½ ft wide, 1½ ft deep, and 5 ft tall?
2. A tower in Germany has a height of 110 m and radius of 21 m. Find the volume, using
3.14 for .
70
10.1 The Real Numbers
Integers – positive and negative whole numbers.
Ex a May’s bank balance is $20 overdrawn. Write the balance as an integer
Ex b Represent the following numbers on the number line: 3, -1, 0, -4
Rational Numbers - Include integers, fractions, and decimals that can be written as
fractions.
Ex c Graph the numbers:
Decimal Notation for Rational Numbers (any fraction can be converted to decimal)
3
Ex d Write  as a decimal
5
Ex e Write 
5
as a decimal
11
Alternate formats:
0
=
3
–
31
100
- 1¾
Real Numbers
Irrational numbers - the decimals do not stop or repeat
–
3
2
71
Ex f Use < or > to write a true statement for each pair
3
-8
0
-3
- 12

-5
2
3
1.5
-5
-4
- 2.7
½
Absolute Value – always positive
If number is positive, the absolute value is the same as the number
If the number is negative, the abs. value has the opposite sign
5 
-7
-3
4
7
0
2.58
72
10.2 Additing Real Numbers (esp. negative numbers)
Ex a 4 + ( - 7)
Ex b (-3) + 2
Ex c - 3 + (- 6)
Ex d - 3.5 + 0
Adding without number lines:
1. Add positive numbers - add as before
2. Add negative numbers - add absolute values; the final sign is negative
3. Add a positive and a negative
a) Subtract the amounts
b) Take the sign of the “dominant” number (greater absolute value)
Ex e
3.2 + (-7.8)
- 30 + (-14)

2
1
+
6
3
- 11.2 + 11.2
Opposites (Additive Inverses)
Ex f Find the opposite of:
17
The sum of a number and its opposite Is
The opposite of the opposite:
- 4.1
1
6
0
73
10.3 Subtracting Real Numbers
Subtraction – same as adding the opposite of the number
Ex a 2 – 6
Ex b 3 – (-1)
Ex c - 2 – 3
Ex d - 4 – (-3)
Practice Problems
1. 6 – 11
2. -8 – 5
3. - 7.2 – (- 3.1)
4. 1.8 - 4
5. 0 – 22.9
6. 
1 5

6 6
8. 
5
5

11 11
7.
3  2
  
5  3
74
10.4 Multiplication of Real Numbers
Product of a positive and a negative number – the result is __________________
Tip: Determine the sign and set it aside, then multiply the absolute values separately.
Ex a
7(-6)=
(-3.61)(5)=
 3   11
   =
7  6 
Product of 2 negative numbers – the result is __________________
Ex b
(-5)(-3) =
-2(-7.95) =
 25   14 


 =
 7   15 
Multiplying More than 2 Numbers
Every pair of multipled negative numbers produces a positive number.
For more than 2 negatives:
____________________________________________ produces a positive number.
_____________________________________________ produces a negative number.
Ex c
(- 4)(- 3)(2)(-1) =
Ex d (- 5)(4)(- 3)(- 2)(- 1) =
 1   2  3 
Ex e          =
 6   3  5 
75
10.5 Division of Real Numbers
The rules for signs in division are the same as for multiplication.
Quotient of a positive and a negative number – the result is __________________
Ex a
- 30
=
5
27
-9
Quotient of 2 negative numbers – the result is __________________
Ex b
- 42
=
-6
 4.8  (1.2) =
Division and Zero
Related equations:
- 24
 8
3
- 24
x
0
0
x
- 24
Ex c
0
- 19
- 52
0
Reciprocals - the product of 2 reciprocals is ____________
Ex d Find the reciprocal for:
number:
2
3
4

1
2
reciprocal:
The signs of a number and its reciprocal are:
-5
76
Dividing Fractions – Recall “Keep, Change, Flip”
2  4
Ex e     
3  9
Ex f Solve:
5
3
x
4
2
Dividing Decimals
Ex g – 6.48  (– 4)
Ex h 3.51  (–0.3)
Sign Placement – equivalent forms:
4 -4 4
 

7 7 -7
The sign cannot be “moved” for a mixed number:  2
Practice Problems:
 7 
1.  4   
 12 
1

4. 5    2 
2

1
3
- 3.4
- 20
3
3.  4  ( 10)
5
-77
5. 

 3 
6. (2.1)(-0.4)
2.