Fundamentals of Math
An Activities and Applications Approach
by Soodi Zamani, Jon Freedman, and Rick Hough
August 2010
1
Contents
1 Numbers
1.1 Defining Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Comparing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Place Value
2.1 Digits . . . . . . . . . . . . . . . . . . . .
2.2 Introducing exponents with base 10 . .
2.3 Making Numbers . . . . . . . . . . . . .
2.4 Place Value and Value: . . . . . . . . . .
2.5 Saying and Writing Numbers in Words
3 Rounding, Estimation, and Measurement
3.1 Rounding . . . . . . . . . . . . . . . . . .
3.2 Graphing Numbers on a Number Line
3.3 Rounding to a Particular Decimal Place:
3.4 Estimation and Measurement . . . . . .
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4 Addition of Whole Numbers, Decimals, and Fractions
4.1 Totaling Money: . . . . . . . . . . . . . . . . . . . .
4.2 The Addition Process . . . . . . . . . . . . . . . . .
4.3 Estimation of Sums . . . . . . . . . . . . . . . . . .
4.4 Addition in Real Life . . . . . . . . . . . . . . . . .
4.5 Introduction to Fractions . . . . . . . . . . . . . . .
4.6 Fractions . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Measuring Lengths with Inches . . . . . . . . . . .
4.8 Adding Fractions . . . . . . . . . . . . . . . . . . .
4.9 Equivalent Fractions . . . . . . . . . . . . . . . . .
4.10 Adding Mixed Numbers: . . . . . . . . . . . . . . .
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7
7
8
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11
11
12
19
20
21
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23
23
25
28
32
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39
39
46
49
57
65
68
74
80
84
88
5 Subtraction
5.1 Subtracting Money . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Process of Subtraction: . . . . . . . . . . . . . . . . . . . .
5.3 Alternate subtraction method: Subtraction by Counting Up .
5.4 Estimation or Approximation . . . . . . . . . . . . . . . . . .
5.5 Subtracting Fractions with Like Denominators . . . . . . . .
5.6 Comparing Fractions . . . . . . . . . . . . . . . . . . . . . . .
5.7 Subtraction in Real Life . . . . . . . . . . . . . . . . . . . . . .
5.8 Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . .
5.9 Signed Numbers on a Number Line . . . . . . . . . . . . . .
5.10 Order of Operations and Grouping Symbols . . . . . . . . .
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95
95
97
100
104
106
111
114
120
123
128
6 Multiplication
6.1 Introduction to Multiplication . . . .
6.2 Significant Digits . . . . . . . . . . .
6.3 Multiplication is Repeated Addition
6.4 Introduction to Area . . . . . . . . .
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131
131
137
144
145
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2
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6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
Area Formula For a Rectangle . . . . . . . . . . . . . . . . . . . .
Pythagorean Theorem and Square Roots . . . . . . . . . . . . . .
The Distributive Property . . . . . . . . . . . . . . . . . . . . . .
Traditional Process of Multiplication . . . . . . . . . . . . . . . .
Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prime Factorization . . . . . . . . . . . . . . . . . . . . . . . . . .
Equivalent Fractions Revisited . . . . . . . . . . . . . . . . . . .
Factorization and Equivalent Fractions . . . . . . . . . . . . . . .
Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Finding a Common Denominator to Add or Subtract Fractions.
The Golden Rule of Fractions . . . . . . . . . . . . . . . . . . . .
Ratio and Proportion . . . . . . . . . . . . . . . . . . . . . . . . .
Multiplication by Negative Numbers . . . . . . . . . . . . . . . .
Order of Operations with Multiplication and Addition . . . . .
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147
153
162
172
173
178
179
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184
186
189
192
194
196
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201
205
210
214
219
229
241
248
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251
251
252
253
254
257
260
9 Order of Operations
9.1 Order of Operations Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 The Full Order of Operations Agreement . . . . . . . . . . . . . . . . . . . . . . . .
267
267
269
270
10 Extra Exercises
273
7 Division of Whole Numbers and Decimals
7.1 The Process of Division . . . . . . . . . .
7.2 Divisibility . . . . . . . . . . . . . . . . .
7.3 Estimation Again . . . . . . . . . . . . .
7.4 Multiplying Fractions . . . . . . . . . . .
7.5 Fraction × Fraction . . . . . . . . . . . .
7.6 Decimal Fractions . . . . . . . . . . . . .
7.7 Dividing with Negative Numbers . . . .
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8 Ratio and Proportion
8.1 Ratio . . . . . . . . . . . . . . . . . . . . . .
8.2 Equivalent Fractions . . . . . . . . . . . . .
8.3 Proportion . . . . . . . . . . . . . . . . . . .
8.4 Percent . . . . . . . . . . . . . . . . . . . . .
8.5 Proportion and Percent in Problem Solving
8.6 Unit Conversion . . . . . . . . . . . . . . . .
3
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.
HOW TO BE SUCCESSFUL IN A MATH CLASS
(regardless of your past experiences)
Success isn’t a spectator sport. You succeed because you commit to your decision to do
something and you act on that decision. People don’t succeed because they’re lucky, they
succeed because they do something about it. You may have done well in math classes in the
past or you may have had some difficulty but in the end the message is the same. If you want
to learn math, if you want to pass this class, then you have to make it a conscious decision and
follow a plan to see it though.
Although there are many reasons you may not have met your goals, there are also multiple
strategies for succeeding in your next math class, especially if you begin with some good habits
you maintain throughout the semester. From the beginning of the class, almost every day,
through to the end of the semester, a successful student will follow most or all of the following:
• Come to class. This seems obvious, but it’s critical. If you could do the work without
attending class, you wouldn’t be taking the class in the first place.
• Do all the assigned work. If you ever do get behind, get help immediately so that you
can catch up with the current topics of each class.
• If any of the material is unclear, get help from your instructor, a tutor, or a classmate.
The longer you wait to do this, the more the amount of unclear material builds up until
virtually nothing makes sense in class anymore.
• Study outside of class with classmates. You may pick up ideas from them, and its easier
to motivate for a study session if other people are counting on you.
• Visit the Learning Center to study in a structured environment with a tutor.
• Be honest with yourself. Only you know if you really understand a concept. Only you
know if you have truly been doing your best. It’s easy to make an excuse to yourself as to
why you can’t do something. Successful people persevere despite the obstacles in their
way.
One last thing before we begin. Many students who have started the semester hating math
have reported starting to enjoy it after working hard enough to start understanding the
material. They confess that once they began doing the math correctly with confidence, the work
actually became fun! So, I say to you, “Get busy, get help, and have fun!”
4
HOW TO USE THIS BOOK
The book in your hands is meant to be used. At the end of the semester it will be full of your
work, your notes, your sweat, and your tears. Whether your Fundamentals of Math class
allows the students to use their notes on exams or not, you need the work in the book to be
correct, clear to read, and organized so that you can easily refer to previous work to figure out
how to do a problem
Some of the exercises or activities in the book don’t leave enough space to show a clear solution.
Some of the exercises or activities in the book require that you work with physical objects,
reflect on what you have learned, then summarize the ideas in clear sentences.
Some of the exercises or activities in the book ask you to fill in a table.
Often you will be asked to explain how something works rather than being told in the text how
it works.
All of these require that you figure out how to show your work in such a way that both you and
your instructor can clearly understand what you have done. The following suggestions may
help you to find a way to do this for yourself.
• If there is very little space to show your work in the book, write your solution on a
separate piece of paper.
• Any work you write on a separate piece of paper, label carefully with the page number
and exercise number so that it is easy to see where it goes. Write on paper that is 3-hole
punched so that you can put it in your binder next to the page it refers to.
• Do your work and write your notes on scratch paper first, then when you have checked
that it is correct and checked it for clarity, re-write it as neatly as you can before putting it
together in the book.
5
Activity 0.1 Problem Solving
Objective: To find a solution to a problem by working in a group. To share problem solving
strategies with the class.
Materials: None.
Group size: 3 to 4.
Instructions: First solve the problem clearly by writing out all of your steps, then answer the
questions below.
You are invited to your boss’s house at 8:00 pm. At 6:30 pm you have just left the candy shop in
your neighborhood where you picked up a box of candy for a hostess gift. You notice that your
favorite store is having a sale. You have had your eye on a designer leather jacket for a while
now. There is a line outside the store and the manager says there cannot be more than 10 people
in the store at any time. You decide to wait in the line. There are 22 people ahead of you. People
are leaving the store at a rate of 2 people every 5 minutes and once you get in the store, it takes
you 10 minutes to find the jacket in the right size and 15 minutes to pay for it. Will you make it
to your boss’s house on time? Don’t forget you have a 12 minute drive from the shopping
center to her house.
1. What is the answer to the problem?
2. Show clearly all of the steps used to find the answer.
3. What did you do to check whether the answer is correct? Does everyone in the group
agree that it is correct?
4. What pictures did you draw to help you solve the problem?
5. How did you organize the information on paper so that you could communicate your
ideas clearly to your group-mates?
6. How can you write your solution clearly enough so that you will be able to follow your
steps long after you’ve forgotten what the problem was about?
6
Notes:
7
1 Numbers
1.1 Defining Numbers
Numbers have two main uses in our world: to identify something the same way that a name
does, and to represent a value. Some examples of identification numbers are account numbers,
phone numbers, address numbers, social security and G-numbers, and titles or names like
Chapter 5 or R2D2. Numbers that represent value are much more common, and are the subject
of the study of mathematics. Some examples are anything to do with money like prices or
wages, scores in sports or on an exam, measurements like area, distance, weight or speed, a
time or date, and age. Often, value numbers are used to compare one object with another. For
example, a jacket which costs $45 with another that costs $100; a job that offers a salary of
$25,000 per year with a job that offers a salary of $40,000 per year; an apartment with an area of
500 square feet with one with an area of 800 square feet. People use value numbers to make
decisions, and mathematics helps people make informed decisions.
Activity 1.1 Introducing Units: How can the meaning of the same number be different?
Objective: To learn how the meaning of numbers change based on the units assigned to them.
Materials: None.
Group size: 3 to 4.
Instructions: Use the numbers 10, 16, 0, 21, and 5 to answer the first four questions, then work
on the last two.
1. Discuss together what each number in the list above reminds you of. Notice that numbers
just by themselves are a very abstract concept. When numbers quantify SOMETHING or
describe a quantity of SOMETHING, called a unit, they are more meaningful. Therefore
Units are associated with numbers. For example: 10 yards, is quite different from 10 trees.
2. By yourselves, assign different units to each of the numbers 10, 16, 0, 21, and 5.
3. When everyone is done, compare your answers.
(a) Which of the numbers did more than one person have the same units for?
(b) Explain why that might have happened. Did those numbers HAVE to have those
units to make sense, or was it just a common thing for that number to stand for?
4. Explain why it is important to include proper units with answers to word problems.
5. Tonight at home, search a newspaper for three examples of identification numbers and
three examples of value numbers. For the value numbers, write a sentence explaining
what object in real life is being represented with that value and include the units if there
are any.
6. Tomorrow at the beginning of class, find your group members and discuss what you
found.
8
1.2 Comparing Numbers
What determines if one number is greater than another?
We use the symbol “>” to indicate when one number represents more of something than
another number. For instance, since 8 pounds of something is more than 5 pounds, we would
write 8 lbs. > 5 lbs. We use “<” to indicate when one number represents less of something than
another number. From the same example above, we would write 5 lbs. < 8 lbs.
Exercise 1.2.1 To search for the answer to this question, let’s do the following exercise:
1. For each question insert either > , < , or a ? if it is unclear whether one quantity is more
or less than the other.
(a) 10 dogs
7 dogs
(b) 10 dogs
7 cats
(c) 10 dogs
7 buckets
(d) $5
34 c/
(e) 2 feet
(f) 12 miles
20 inches
3 gallons
(g) $42,000 per year
(h) $12.95 per hour
$5,000 per month
$2,500 per month
2. Which part(s) of question (1) could you clearly answer? Why?
3. What information do you need to answer the questions that were not clear? Why?
4. Explain your answers to parts (e) through (h).
5. Write a conclusion about what you need in order to compare two things using > or <.
9
Activity 1.2 Powers of 10
Objective: To investigate the pattern when multiplying by powers of ten.
Materials: Calculator.
Group size: 2 to 5.
1. Use your calculator to help you find the products below.
Number of 10’s Expression
Result
2
10 × 10 =
h
3
10 × 10 × 10 =
h
4
10 × 10 × 10 × 10 =
h
5
10 × 10 × 10 × 10 × 10 =
h
6
10 × 10 × 10 × 10 × 10 × 10 =
h
2. Without using your calculator, anticipate the result of the product:
10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 =
3. Describe a shortcut for multiplying tens.
4. How many 10’s are multiplied together to obtain the number 1, 000, 000, 000?
10
5. Find the results of the products below, then write the names of the results out in words.
(Hint: For the decimals, think of money. For example, 0.1 = 0.10 is a dime which is a tenth
of a dollar, and 0.01 is a penny which is a hundredth of a dollar!)
(a)
(b)
Expression
4 × 100, 000 =
6 × 0.01 =
Result
400,000
Names in Words
four hundred thousand
0.06
six hundredths
(c)
7 × 1000 =
h
(d)
2 × 100 =
h
h
(e)
3 × 1000 =
h
h
(f)
5 × 10, 000 =
h
h
(g)
9 × 10 =
h
h
(h)
2 × 0.1 =
h
h
(i)
5 × 0.1 =
h
h
(j)
3 × 0.01 =
h
h
h
6. What is the name of the number 407,200?
Notice how the name of a number combines its digits with the power of ten
implied by the position of the digit. The 4 represents 4 × 100, 000 = 400, 000, the
7 represents 7 × 1000 = 7, 000 and the 2 represents 2 × 100 = 200.
7. What is the name of the number 53,090?
8. How much does the 5 represent in the number 53,090?
9. How much is 10 quarters worth?
10. How much is 10 dimes worth?
11. How much is 100 pennies worth?
12. How much is 1000 pennies worth?
13. How much is 1000 dimes worth?
11
2 Place Value
2.1 Digits
There are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Numbers are represented by placing digits
together in a string. The digits in a number have different values associated with them
depending on their relative position in the number.
For example:
276 has three digits and 7 has a value of seventy or 7 × 10 = 70. As you read the
number, the place values are spoken as, two hundred seventy-six.
A decimal point separates whole number values from fractional values.
24.89 has 4 digits and the digit 8 has the value: 8 × 0.1 = 0.8. For example, if it was
$24.89, the 8 would represent 8 dimes, or 8×$0.1=$0.80.
Exercise 2.1.1 Associating digits with money. $1 bill, $10 bill, $100 bill, $1000 bill
Leticia wants to buy a new motorcycle that costs $6,327.95
1. How many digits are in 6,327.95?
2. Which is the largest digit?
3. Which digit has the largest place value?
4. What is the value of the digit with the largest value?
5. Which is the smallest digit?
6. Which digit has the smallest value and what is its value?
7. Leticia pays cash for the bike and the bills she can get from the bank are in the
denominations $10,000, $1000, $100, $10, $1, as well as dimes and pennies. How many of
each denomination will she need to cover the exact price if she uses the minimum number
of bills?
8. The value of each denomination of the bills Leticia used in question (7) is the place value
of each digit within the number 6,327.95. Name the place value of each digit.
Note: As you see in exercise 2.1.1, there is a monetary value associated with each digit of a
number according to the placement of the number. For example, the value of the digit 6 is 6
thousand since you need 6 one thousand dollar bills and the value of the 5 is 5 hundredths
because you need 5 pennies and each penny is worth one hundredth of a dollar.
12
2.2 Introducing exponents with base 10
The following chart shows some of the place values in the number system:
one billion
one hundred million
ten million
one million
one hundred thousand
ten thousand
one thousand
one hundred
ten
one
one tenth
one hundredth
one thousandth
1, 000, 000, 000 = 1 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1 × 109
100, 000, 000 = 1 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1 × 108
10, 000, 000 = 1 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1 × 107
1, 000, 000 = 1 × 10 × 10 × 10 × 10 × 10 × 10 = 1 × 106
100, 000 = 1 × 10 × 10 × 10 × 10 × 10 = 1 × 105
10, 000 = 1 × 10 × 10 × 10 × 10 = 1 × 104
1, 000 = 1 × 10 × 10 × 10 = 1 × 103
100 = 1 × 10 × 10 = 1 × 102
10 = 1 × 10 = 1 × 101
1 = 1 × 1 = 1 × 100
1
0.1 = 10
= 1 ÷ 10
1
0.01 = 100
= 1 ÷ 100 = 1 ÷ 10 ÷ 10
1
0.001 = 1000 = 1 ÷ 1000 = 1 ÷ 10 ÷ 10 ÷ 10
onetent
h
onehun
dred
th
onethou
sand
th
,
one
one
hun
dred
thou
ten
sand
thou
sand
one
thou
sand
,
,
one
hun
dred
ten
one
hun
dred
mill
ten
ion
mill
ion
one
mill
ion
one
billi
on
The following chart illustrates the place value of the digits:
.
one
4
6
5
1
7
5
,
2
8
,
13
1
onetent
h
onehun
dred
th
onethou
sand
th
one
hun
dred
ten
,
one
hun
dred
thou
ten
sand
thou
sand
one
thou
sand
3
one
hun
dred
mill
ten
ion
mill
ion
one
mill
ion
one
billi
on
Therefore, the number three billion, four hundred seventy five million, six hundred twenty
eight thousand, five hundred eleven and nine-tenths (3,475,628,511.9) would look like this in
the chart:
.
9
Exercise 2.2.1
1. Find the place value and value of each digit underlined in following chart:
Number
Place Value
Value
(a)
237, 669
(b)
9, 932, 210
h
h
(c)
102
h
h
(d)
865, 106
h
h
(e)
65.293
h
h
(f)
10.226
h
h
(g)
0.226
h
h
(h)
650, 900, 563, 115.07
h
h
(i)
1127.08335
h
h
thousands
7,000
2. What number is written as forty two thousand, three hundred seventy five and twenty
three hundredths?
3. What number is written as eighty billion nine hundred seven thousand four?
4. A “googol” is defined as 10100 , that is, 10 multiplied by itself 100 times. Describe how you
could write this number on paper in standard notation.
14
Activity 2.1 Group Game: Spin the Digits
Objective: To introduce concepts of place value by strategizing digit placement.
Materials: Paper clip, paper, pencil.
Group size: 3 to 4.
1. Each group needs a paper clip and a separate piece of paper for each person.
2. Open the paper clip from one side and use it as a spinner.
3. Each member draws 3 lines on his or her paper.
4. Each time one of the members spins the paper clip and each member records the number
on one of the 3 lines that he or she has drawn.
5. After 3 spins compare your 3-digit numbers.
6. The largest 3-digit number wins the turn.
9
0
1
8
2
7
3
6
5
7. What is a good strategy for winning this game?
15
4
Activity 2.2 Learning Place Value with Money
Objective: To solidify concepts of place value by using money.
Materials: Bills of different denominations.
Group size: 3 to 4.
Notice that $235 could come in 235 $1 bills, or 2 $100 bills, 3 $10 bills and 5 $1 bills , or 23 $10
bills and 5 $1 bills.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
1. Which combination has the fewest number of bills totaling $235? (Write how many of
each denomination in the appropriate boxes.)
.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
2. Find three different combinations of bills/coins to make $400. (Write how many of each
denomination in the appropriate boxes.)
.
$0.1
0
$0.0
1
$0.0
1
,
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
16
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
3. Find three different combinations of bills/coins to make $1. (Write how many of each
denomination in the appropriate boxes.)
.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
4. Find three different combinations of bills/coins to make $7,000. (Write how many of each
denomination in the appropriate boxes.)
.
$0.1
0
$0.0
1
$0.0
1
,
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
17
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
5. Find three different combinations of bills/coins to make $5,628. (Write how many of each
denomination in the appropriate boxes.)
.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
6. Find three different combinations of bills/coins to make $63,921.42. (Write how many of
each denomination in the appropriate boxes.)
.
$0.1
0
$0.0
1
$0.0
1
,
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
18
,
$0.0
1
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
7. Find three different combinations of bills/coins to make $307,995. (Write how many of
each denomination in the appropriate boxes.)
.
$0.1
0
$0.0
1
$0.0
1
,
$0.1
0
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(a)
.
,
$1
$10
$100
$1,0
00
$10,
000
$100
,000
(b)
.
(c)
8. Raymond had 21 - $1000 bills, no $100 bills, no $10 bills, 26 - $1 bills, no dimes and 13
pennies. He went to the bank and exchanged them for the fewest number of bills/coins
that represented the same amount. How much of each kind of bill/coin did he get? In the
bank record below, record the value of Raymond’s money and the quantity of each type of
bill/coin he would have when using the fewest number of each to represent the amount.
Type of Bill/Coin
$10,000s
$1,000s
$100s
Count of Bill/Coin
How much would his money be worth?
9. What part (fraction) of $1 is a dime?
10. What part (fraction) of $1 is a penny?
11. What part (fraction) of $1 is a quarter?
19
$10s
$1s
$0.1s
(dimes)
$0.01s
(pennies)
2.3 Making Numbers
Exercise 2.3.1
1. When writing money in dollars and cents format, where do you put the decimal place?
(between the count of which of the bills/coins?)
2. Using each digit exactly once, what would be the largest number you could make with
the digits 0, 1, 2, 5, 8, 9 (with no decimal points)?
3. Why did you put the left-most digit in that spot?
4. Using each digit exactly once, what would be the smallest number you could make with
the digits 0, 1, 2, 5, 8, 9 (with no decimal points)?
5. Why can’t you put the smallest digit in the left-most place?
6. Using each digit exactly once, what would be the smallest 6-digit number with 2 digits
after the decimal point you could make with the digits 0, 1, 2, 5, 8, 9?
7. Using each digit exactly once, what would be the largest 6-digit number with 2 digits after
the decimal point you could make with the digits 0, 1, 2, 5, 8, 9?
8. If you had the digits 0, 1, 2, 5, 8, 9 to make a 6-digit number, using each digit exactly once,
what would be the smallest number you could make with the 6 digits? You choose the
number of digits after the decimal place.
9. What is the largest 5-digit number?
10. What is the smallest 5-digit number (without the decimal point)?
11. Using the digits 6, 4, 1, 0, and 8, exactly once each, what is the largest five digit number
you can make?
12. Using the digits 6, 4, 1, 0, and 8, exactly once each, what is the largest five digit number
you can make? You may include a decimal point.
20
2.4 Place Value and Value:
Exercise 2.4.1
1. In the number $3,461.85, which digit has the highest place value (i.e., accounts for the
most money?)
2. In the number $2.08, which digit has the highest place value?
3. In the number $2.08, which is the largest digit? What is its value?
4. In the number $2.08, which digit has the highest value? What is its value?
5. In the number $4501.32, what is the place value of the digit 0?
6. In the number $673.47, what is the place value of the digit 4?
7. In the number $86, 793.64, what digit is in the ten-thousands place?
8. In the number $86, 793.64, what digit is in the hundreds place?
9. In the number $86, 793.64, what digit is in the tenths place?
10. In the number $86, 793.64, what is the place value of the digit 9?
11. In the number $86, 793.64, what is the value of the digit 8?
12. In the number $86, 793.64, what is the place value of the digit 4?
13. In the number $86, 793.64, what is the value of the digit 4?
14. Create a number with a 9 in the thousands place.
15. Create a number with a 3 in the tens place and a 5 in the tenths place.
16. Create a number with a 2 in the hundreds place, a 1 in the hundredths place, and the digit
in the ones place is twice the digit in the tens place.
21
2.5 Saying and Writing Numbers in Words
To read and write whole numbers: The whole numbers are the numbers in the sequence 0, 1, 2,
3, 4, 5, 6, 7, 9, 10, 11, continuing indefinitely. When we write whole numbers in standard form
we separate a group of three digits by a comma. Each group of three digits forms what is called
a period. Each period has a particular name. We say the name of a period after we read all the
numbers in that period, except the last period before the decimal point.
Example 1: Read 8,412,769,215.
Solution: Eight billion, four hundred twelve million, seven hundred sixty nine thousand, two
hundred fifteen. (The name of each period is in italics. Notice that the last period doesn’t have a
name.)
To read and write decimal numbers: We read the whole number part exactly the same way as
before, then read the decimal as “and” followed by the name of the number to the right of the
decimal point using the place value of the last digit as the number’s value.
Example 2: Read 65,389,237,542.029.
Solution: Sixty five billion, three hundred eighty nine million, two hundred thirty seven thousand, five hundred forty two, and twenty nine thousandths. (Notice that the name of the decimal
part is thousandths since the last digit, the 9, is in the thousandths place.)
Exercise 2.5.1 Place commas to separate the periods, then write out the number in words:
1. 2863
2. 76.218
3. 28990.02
4. 3400651899
5. 23678.93
6. 1000267
7. 64822.2
Exercise 2.5.2
1. What number is 10 more than 5,687.13?
2. What number is 400 more than 38,265.79?
3. What number is 0.3 more than 21,975.52?
4. What number is 0.05 more than 5,687.13?
22
Notes:
23
3 Rounding, Estimation, and Measurement
3.1 Rounding
Some quantities are exact, like the number of people in a room, while others are approximations.
An approximation is a number that is close to the exact value, but has fewer non-zero digits.
More zeros makes it easier to work with.
We round a number to a less accurate but simpler approximate value to make it easier to work
with, while keeping most of the important information about the number.
For example, if we learned that the U.S. trade deficit in June 2005 was $58,753,421,989, we
probably could analyze how this was important to our lives just as well if it was rounded to
$59,000,000,000! In fact, without all of the non-zero digits, we could even write this number as a
”word-number” hybrid by writing, ”$59 billion”. It is usually a good idea to round a number
like this, and if you listen to financial news reports, you will notice that the media often rounds
numbers.
For another example, if we wanted to measure the length of a window so that we could buy the
right size replacement, we could use a very accurate measuring device and find that the length
is 75.18952 cm. The window store is probably satisfied with the rounded version, 75 cm. But,
the window is approximately 75 cm.
Exercise 3.1.1 For the following numbers, state whether it seems reasonable that the given
value is exact or an approximation. Give a reason for your answer.
1. Measurement of the distance you drive from SF to LA is 420 miles.
2. Jenisa calculated her share of the lunch that she had with her high school friends
yesterday and the figure was $12.345.
3. The family’s rent each month was $1225.
4. The same family’s monthly budget for food was $840.
5. A group of friends calculated their share of the cost of a ski trip and the figure was $342.
24
Exercise 3.1.2 For the following numbers, discuss how accurate each measurement needs to
be for it to be useful, and why it requires that amount of accuracy. Also use phrases like “at
least” or ”at most” or “to the nearest” if appropriate for the situation.
1. Measurement for rope to tie down a load.
2. Measurement for wood in making a picture frame.
3. A cup of flour to make a cake.
4. Measurement of the length of the glass used to make the windows for a new building.
5. Scores used to compute your grade.
6. The amount of line to let out in a cast while fishing.
7. The time it will take to drive to a dinner party.
8. The amount of salt used in a pot of soup.
25
3.2 Graphing Numbers on a Number Line
Exercise 3.2.1 Graph the following numbers on the number line.
Example: To graph 472, place a dot where 472 should go, then write 472 next to the dot as in the
figure:
472
b
200
300
500
400
600
1. Graph 220, 517, 350, 490:
200
300
400
500
600
20
30
40
50
5000
6000
7000
8000
4
5
6
7
5.7
5.8
5.9
2. Graph 28, 17, 35, 46:
10
3. Graph 5593, 7200, 6500, 4207:
4000
4. Graph 3.5, 5.2, 6.6, 4.1:
3
5. Graph 5.72, 5.63, 5.88, 5.55:
5.5
5.6
26
Exercise 3.2.2 Mark each number line by counting by the indicated amount.
Example: To mark the number line by counting by tenths starting at 0.7, you would mark the number
line like the following:
0.7
0.8
0.9
1. Count by hundreds, starting at 500:
2. Count by tens, starting at 30:
3. Count by ones, starting at 8:
4. Count by tens, starting at 420:
5. Count by hundreds, starting at 2100:
27
1.0
1.1
6. Count by tens, starting at 2100:
7. Count by tenths (0.1s), starting at 0.3:
0.3
0.4
8. Count by tenths, starting at 1.8:
9. Count by tenths, starting at 22.5:
10. Count by hundreds, starting at 34,800:
11. Count by thousands, starting at 998,000:
28
3.3 Rounding to a Particular Decimal Place:
To round numbers to a particular decimal place, do the following:
1. Determine the correct decimal place (tenths, ones, tens, hundreds, etc.)
2. Draw a number line, and count by the determined decimal place starting at the number
just below the number you are rounding, and ending with the number just after it.
3. Graph the number that you are rounding on the number line.
4. The rounded value is the count number that is closest to your graphed number. If the
graphed number is exactly in the middle of two count numbers, round to the higher
number.
Example: Round 48.3 to the nearest ten.
Solution: To round 48.3 to the nearest ten, count by tens starting at 30 like the following:
40
50
Then, graph 48.3, and determine which number is closest...it helps to put a tick mark at the half-way
points:
48.3
b
40
50
48.3 is to the right of the half-way point between 40 and 50, so it’s closer to 50. Therefore,
48.3 rounded to the nearest ten is 50.
Exercise 3.3.1 Round the following as indicated.
1. Round 286 to the nearest hundred:
Final answer:
29
2. Round 5.37 to the nearest one:
Final answer:
3. Round 5.37 to the nearest tenth:
Final answer:
4. Round 5.37 to the nearest ten:
Final answer:
5. Round 5.37 to the nearest hundred:
Final answer:
6. Round 2358 to the nearest hundred:
Final answer:
7. Round 2358 to the nearest thousand:
Final answer:
30
Exercise 3.3.2 Fill in the table to practice rounding. The first two are done for you as
examples:
The Number to Round off
Underline
and the Level of Rounding the Digit(s)
1) (example) 18.09
18.09
Round to the nearest 10
2) (example) 23.032
23.032
Round to the nearest 100th
3) 5671.98
Round to the nearest 100
4) 5671.983
Round to the nearest 100th
5) 1098.67
Round to the nearest 100
6) 36.0409
Round to the nearest 100th
7) 99.9986
Round to the nearest 1
8) 99.9986
Round to the nearest 10th
9) 0.00098
Round to the nearest 100th
10) 6.85
Round to the nearest 10th
The Next
Round
Digit
Up or Down
8
Up, since
8≥5
2
Down, since
2<5
Answer
20
23.03
Exercise 3.3.3 For each table of data, make a new table by rounding values as indicated:
1. The following table lists the number of pages in each of the books in the Harry Potter
series. Fill in the new table with the number of pages rounded to the nearest 10.
Given Data Table
Rounded Data Table
Book Number Number of Pages
1
309
2
341
3
435
4
734
5
870
6
652
Book Number Number of Pages
1
2
3
4
5
6
31
2. The following table lists the numbers of new products containing the artificial sweetener
Splenda for various years. Fill in the new table with the number of new products
containing Splenda rounded to the nearest 100.
Given Data Table
Year
2000
2001
2002
2003
2004
Rounded Data Table
Year
Number of New Products
Containing Splenda
183
261
365
561
1330
Number of New Products
Containing Splenda
2000
2001
2002
2003
2004
3. The following table lists the percentages of Americans who have earned a college degree
for various years. Fill in the new table with the percentages of Americans who have
earned a college degree rounded to the nearest whole percentage.
Given Data Table
Year
1960
1970
1980
1990
2000
2003
Rounded Data Table
Percentage of Americans
Earning a College Degree
7.7
10.7
16.2
21.3
25.6
27.2
Year
Percentage of Americans
Earning a College Degree
1960
1970
1980
1990
2000
2003
4. The more you stretch a spring with your hands, the more force the spring exerts on your
hands. The following table compares the amount of stretch with the forces exerted. Fill in
the new table with the amount of stretch data rounded to the nearest 100th.
Given Data Table
Force
(newtons)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Rounded Data Table
Stretch
(meters)
0.018
0.035
0.052
0.069
0.087
0.104
0.121
0.139
0.156
0.173
Force
(newtons)
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
32
Stretch
(meters)
3.4 Estimation and Measurement
Exact or Estimate
When you are finding out the value of a quantity in real life (as opposed to reading a given
value in a book), some values you know to be exact, while others you must estimate to obtain
an approximation. Some examples of exact values are as follows:
• The number of people at a small dinner party.
• The number of children in a family.
• The number of hours in a day.
• The price of gas at a gas station.
• The number of grams in a kilogram.
• The number of units passed in a semester.
By contrast, the following is a list of values that must be estimated:
• The time it takes to complete an assignment.
• The number of people who live in San Francisco.
• The weight of a sick dog at the vet.
• The distance from SF to LA.
• A person’s height.
Exercise 3.4.1
1. Explain what makes the values in the first list exact values, and the values in the second
list approximate values.
2. State five things that have a value that is exact.
3. State five things that have a value that must be estimated.
4. For the following quantities, give a value for the quantity, then state whether it is exact or
an estimation:
(a) The number of people in class on the most recent day.
(b) The height of the classroom door.
(c) The length of your index finger.
(d) The number of typed words on this page.
(e) The weight (in pounds) of the dog you have most recently seen.
33
Estimating Points on a Number Line
Exercise 3.4.2 For the following, use the number lines to estimate the value of the marked
point.
Example 1:
b
50
60
70
80
90
Solution: The value of the point is approximately 68. Since the value of the marked numbers are
counting by tens, we only have confidence in one more decimal place. Therefore, we estimate to the
nearest whole number.
Example 2:
b
13.4
13.5
13.6
13.7
13.8
Solution: The value of the point is approximately 13.72. Since the value of the marked numbers are
counting by tenths, we only have confidence in one more decimal place. Therefore, we estimate to
the nearest hundredth.
1.
b
380
390
400
410
420
524
525
526
5,900
6,000
6,100
2.
b
522
523
3.
b
5,700
5,800
34
4.
b
76.0
76.1
76.2
76.3
9000
10,000
76.4
5.
b
7000
8000
11,000
Estimating Values from Line and Bar Graphs
Exercise 3.4.3 Estimate the values from the graphs.
1. Use the following graph to estimate the value of the Dow Jones Average at certain times.
Dow Jones Average
11180
11160
11140
11120
11100
9 am 10 am 11 am 12 pm 1 pm 2 pm 3 pm
Example: Estimate the value of the Dow at 10 am.
Solution: The value of the Dow at 10 am is estimated to be 11,158.
(a) Estimate the value of the Dow at 9 am.
(b) Estimate the value of the Dow at 11 am.
(c) Estimate the value of the Dow at 12 pm.
(d) Estimate the value of the Dow at 1 pm.
(e) Estimate the value of the Dow at 2 pm.
(f) Estimate the value of the Dow at 3 pm.
35
2. Use the following graph to estimate the average U.S. wage (in dollars per hour) in June of
different years.
Average Wages in US in $/hr
20
18
16
14
12
10
8
6
4
2
0
'94 '95 '96 '97 '98 '99 '00 '01 '02 '03 '04 '05 '06
Average Hourly Wage
Average Hourly Wage in 1982 Dollars
Example: Estimate the average U.S. wage in June 1997.
Solution: The average U.S. wage in June 1997 is estimated to be $12.10 per hour.
(a) Estimate the average U.S. wage in June 1994.
(b) Estimate the average U.S. wage in June 1996.
(c) Estimate the average U.S. wage in June 1999.
(d) Estimate the average U.S. wage in June 2001.
(e) Estimate the average U.S. wage in June 2004.
(f) Estimate the average U.S. wage in June 2006.
(g) When were wages about $15 per hour?
(h) What do the “Ave. Wage in 1982 Dollars” bars tell you?
36
3. Use the following graph to estimate the high temperatures during a record breaking
eleven day heat wave in the Bay Area.
Bay Area Temps in Degrees Fahrenheit
120
Temperature in ˚F
115
110
105
100
95
90
1
2
3
4
6
5
7
8
9
10
11
Days
(a) Estimate the high temperature on day 1.
(b) Estimate the high temperature on day 3.
(c) Estimate the high temperature on day 6.
(d) Estimate the high temperature on day 7.
(e) Estimate the high temperature on day 10.
(f) On what day was the highest Bay Area high temperature recorded?
(g) On what day was the lowest Bay Area high temperature recorded?
(h) Between which days is the temperature going up?
37
An Introduction to Measurement
When we are estimating the length of something, we can get a more accurate measurement by
using a tool. One might use a ruler to measure the length of a finger, a tape measure to measure
the length of a window sill, and an odometer to measure the distance from SF to LA. In the
U.S., the units of length are the inch, the foot, the yard, and the mile. We measure with units
that are appropriate for the lengths. For example, we measure the distance between two cities
in miles, the length of a book in inches, and the length of a room in feet. In this section, you will
be using the metric system to measure length, and the most appropriate metric system unit to
use for the following lengths is the centimeter.
We could use an inch ruler, but inches are broken down into fractions of an inch rather than
tenths. We will wait to introduce inches until after we have introduced fractions.
To use a ruler to measure length, line the end of the ruler at one end of the line, and note how
far along the ruler the other end of the line is.
Example 1: Measure the length of the following line accurate to one-tenth of a centimeter:
Solution: We place the ruler near to the line as follows:
1
2
3
4
5
6
7
8
9
Looking at where the line ends, we estimate the length as 4.6 cm. The 4 stands for 4 whole
centimeters which is indicated by the 4 on the ruler. The .6 (6 tenths) stands for 6 out of ten
parts of the next centimeter. We know this because the end of the line is above the 6th mark
past the 4.
Example 2: This time, measure the length of the following line accurate to one whole
centimeter:
Solution: We place the ruler near to the line as follows:
1
2
3
4
5
6
7
8
9
Since the end of the line is closer to 6 cm than it is to 5 cm, we say the line measures
approximately 6 cm accurate to the nearest whole centimeter.
38
Exercise 3.4.4 Using a centimeter ruler, measure the lengths of the following lines accurate to
the nearest whole centimeter. Include appropriate units in your answers.
1.
2.
3.
4.
5.
Exercise 3.4.5 Using a centimeter ruler, measure the lengths of the following lines accurate to
the nearest one-tenth of a centimeter. Include appropriate units in your answers.
1.
2.
3.
4.
5.
39
4 Addition of Whole Numbers, Decimals, and Fractions
4.1 Totaling Money:
Activity 4.1 Learning Concepts of Addition with Money
Objective: To solidify concepts of addition by using money.
Materials: Bills of different denominations.
Group size: 3 to 4. If there are 4, one of them acts as banker.
Write the names of each group member:
Group member #1
Group member #2
Group member #3
Group member #4
In this activity each member of your group will begin with a specified amount of money.
Assume that the only bills (and coins) available are in denominations of $10,000, $1000, $100,
$10, $1, dimes and pennies. Each of the three members in your group take bills/coins from the
pack to total the following amounts:
• Member #1: $5,693.12
• Member #2: $9,275.35
• Member #3: $2,968.78
When you take your money, make sure that the number of bills/coins in each denomination is
fewer than ten. For each of the following problems, start each member with their original
amount of money and with their original number of each type of bill/coin.
1. (a) If you put the first two members’ stacks of money together in one pile, how many of
each bill /coin do member #1 and member #2 have together? Record the number of
bills/coins in each denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(b) Exchange bills with the bank, wherever possible, so that the total amount takes the
fewest bills necessary. e.g., twelve $100 bills should be exchanged for two $100 bills
and one $1000 bill. Record the new number of bills/coins in each denomination.
$10, 000 $1000 $100
$10
$1
$0.10
(c) How much money do they have together?
40
$0.01
2. (a) If you put the first and third members’ stacks of money together in one pile, how
many of each bill/coin do member #1 and member #3 have together? Record the
number of bills/coins in each denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(b) Exchange bills with the bank, wherever possible, so that the total amount takes the
fewest bills/coins necessary. Record the new number of bills/coins in each
denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(c) How much money do they have together?
3. (a) How many of each bill/coin do member #2 and member #3 have together? Record
the number of bills/coins in each denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(b) Exchange bills with the bank, wherever possible, so that the total amount takes the
fewest bills/coins necessary. Record the new number of bills/coins in each
denomination.
$10, 000 $1000 $100
$10
$1
$0.10
(c) How much money do they have together?
41
$0.01
4. (a) How many of each bill/coin do all three members have together? Record the number
of bills/coins in each denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(b) Exchange bills/coins with the bank, wherever possible, so that the total amount
takes the fewest bills/coins necessary. Record the new number of bills/coins in each
denomination.
$10, 000 $1000 $100
$10
$1
$0.10
$0.01
(c) How much money do they have together?
5. Explain how you found the combined monies (sums) above.
6. Compare your answers with another group in your class. Compare the number of
bills/coins in each denomination from each question before and after combining monies.
Are there any common ways of totaling that your groups followed?
7. (a) Add 27 and 15 using the traditional paper and pencil method.
(b) Add 27 and 15 using the paper money method, that is by counting total bills and
exchanging any that go over ten bills.
(c) Compare “carrying the one” from the traditional method and “exchanging
denominations that are greater than ten bills” from the paper money method.
42
Activity 4.2 Addition Activities
Objective: To practice adding in your head. No calculators please!
Materials: Paper, pencil dice.
Group size: 3 to 4.
Activity 1: Pig
Each group chooses a number between 30 and 55. Each player may roll the die as many times
as she likes. The player then adds the face value of die to the sum so far and tries to get as close
to the chosen target number as possible. Here’s the catch: if a player rolls the same number two
times in a row, they lose their turn and their points! (For a variation: the player keeps their
points so far and just loses their turn). For example, a roll might look like:
3 + 5 + 6 + 2 + 5 + 4 + 4. The player loses their points and their turn because they rolled two
fours in a row before stopping or reaching the target. It is possible to win this game with a roll
of 1 — if the other players go out first.
Activity 2: Value of Words
By assigning a value to each of the letters of the alphabet, all of a sudden, words have value
aside from their ability to help us communicate. For the problems below, use the values given
in the chart for each letter. The value of a word is the total of the values of its letters.
A=$1 G=$7
B=$2 H=$8
C=$3 I=$9
D=$4 J=$10
E=$5 K=$11
F=$6
L=$12 Q=$17 V=$22
M=$13 R=$18 W=$23
N=$14 S=$19 X=$24
O=$15 T=$20 Y=$25
P=$16 U=$21 Z=$26
1. Show that “Skyline” is worth $95.
2. How much is “Pacifica” worth?
3. Who has the most expensive name (last or first; not both)?
4. Find a 3-letter word that has:
(a) the cheapest value
(b) the most expensive value
5. What is the most expensive word that you can find?
6. What do the following words have in common? Explain.
acknowledge, beginnings, carpenter, delivery, elsewhere, friendlier, governs, hospital,
immature, judiciary
43
Exercise 4.1.1 Complete the following addition table, then continue to answer the questions:
+
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
1. Find at least four different patterns in this table. An example of a pattern could be that
starting from the upper left and going down diagonally to the right, the numbers start at
zero and go by 2’s.
44
2. Notice that there are two ways to get 2 as a sum of two numbers: 0 + 2 (we’ll assume 2 + 0
is the same) and 1 + 1.
(a) How many different ways are there to get the number 3 as a sum of two numbers?
(b) How many different ways are there to get the number 4 as a sum of two numbers?
(c) How many different ways are there to get the number 5 as a sum of two numbers?
(d) How many different ways are there to get the number 6 as a sum of two numbers?
(e) How many different ways are there to get the number 7 as a sum of two numbers?
(f) How many different ways are there to get the number 8 as a sum of two numbers?
(g) How many different ways are there to get the number 9 as a sum of two numbers?
3. What patterns do you notice in your answers to question 2? (List at least three). Is there
an easy way to use the table to find the different sums? Explain.
4. How many different ways are there to get the number 10 as a sum?
5. From your observations in the previous questions you should be able to guess the number
of different ways to write 100 as a sum:
6. Complete the following addition table:
+
4
9
6
5
1
10
3
7
0
1
3
5
8
4
9
7
10
2
0
6
45
2
8
7. Complete the following addition table:
+
3
8
5
0
2
15
3
9
7
16
3
12
8
3
6
14
6
12
18
10
8
5
Vocabulary:
• The operation is called addition.
• Each of the numbers that we add together is called an addend.
• When two numbers are added together, the answer is called the sum or total.
46
4.2 The Addition Process
During the money activity at the beginning of the chapter, you observed that when we add
money, we combine the bills with the same denominations together and every time the number
of bills in each denomination reaches 10, we replace that with one bill from the next higher
denomination, that is, the one to the left of it in the place value table. For example, ten dimes is
replaced by 1 dollar or ten $10 bills is replaced by one $100 bill.
Group Discussion Activity
Suppose you worked as a mechanic at a gas station, and were totaling the charges for a
customer. Suppose the charge for the parts came to $247 and the labor charge was $78. If your
calculator was not handy, you might use pencil and paper to write the following:
11
247
78
325
1. Why are the numbers lined up the way they are instead of like this:
247
78
2. Where do the small ones on top of the 2 and the 4 in 247 come from? What do they
signify?
3. Is there a way to approximate the answer without using pencil and paper and without
using a calculator? Explain your process.
Adding whole numbers or decimals together:
1. Line up each addend so that the digits with the same place values are in the same column.
2. Start adding from the rightmost column.
3. If the sum of the numbers in a column is 9 or less, record the sum and proceed to the next
column.
4. If the sum of the digits is 10 or more, write the rightmost digit of the sum under the
column you are adding and add the other digits to the appropriate columns to the left of
it. This amount is “carried over” to the next higher place value column.
5. Repeat until all the digits of all the columns are added.
47
Example 1: Add the following two numbers:
25, 693.12 + 9, 275.3
First, line up the numbers so that each place value is
in the same column, that is, the pennies are all in the
same column, the dimes in the next, etc. Since there
are no pennies in $9,275.3, we place a zero in the hundredths (pennies) column for that number.
te
n
t h - t ho
o
hu usa usa
n
te ndr nds ds
onns eds
e
di s
m
p e es
nn
ie
s
Solution:
25,693.12
9,275.30
Now, we start totaling by column, starting with the
pennies. There is a total of 7 pennies, 4 dimes, and 8
dollar bills. Each of these totals is 9 or less, so no need
to exchange bills. At this point the problem looks like:
25,693.12
9,275.30
8.42
When we get to the tens place, the total of 9 and 7 is
16. We exchange 10 of the tens to get one extra hundred. We place a small 1 on the top of the hundreds
column to remind us of the extra hundred, and put the
remaining 6 tens in the tens column of the answer.
25,693.12
9,275.30
68.42
We next proceed to the hundreds place, and with the
extra hundred we have a total of 9. With 14 thousands,
we have to carry 10 of them for an extra ten-thousand.
The final solution should look similar to this:
Therefore, the answer is 34,968.42.
48
1
1
1
25,693.12
9,275.30
34,968.42
Example 2: Add the following three numbers:
76, 926.8 + 685.87 + 93, 768.79
Solution:
Following the steps, we obtain a solution that looks similar to:
1 2 12 2 1
76,926.80
685.87
93,768.79
171,381.46
Therefore, the answer is 171,381.46.
Exercise 4.2.1 Add the following numbers without using a calculator:
1.
3.
+
23, 409
12, 000
2.
+
731.03
2.006
987, 962.051
16
4.
+
687.084
678, 222.2
+
7, 392.3
5. 11, 276 + 0.038 + 2.38
6. 99 + 10, 268.5 + 28.03
7. 0.00397 + 2.015 + 69.11
8. 763.19 + 2, 763.019 + 3.0019
9. 287, 309, 227.56 + 199.99
10. 0.48 + 268 + 60.46
49
4.3 Estimation of Sums
The purpose of estimating a sum before calculating it exactly is to:
• Give you a value that is relatively close to the actual sum.
• Make the calculation much easier to do. Often, the estimation process is so easy,
you can do it in your head!
Often each of these two points contradict each other. The easier the calculation, the farther the
estimate is from the actual.
Activity 4.3 Investigating Estimation Techniques
Objective: To investigate options for estimating sums.
Materials: None.
Group size: 3 to 4.
Technique 1: Rounding all numbers to the largest place value of the larger number.
• Note the largest place value in the larger number.
• Round each number to the place value noted.
• Add the rounded numbers.
Example 1: Estimate the sum:
5, 629 + 236
Solution:
5,629 is the larger number, and its largest place value is the thousand’s place.
5629 ——>
6000
+
236 ——> +
0
6000
The estimate of the sum of 5,629 and 236 using technique 1 is 6,000.
50
1. Using technique 1, estimate the sum 23, 409 + 12, 000
(a) What is the largest place value of the larger number?
(b) Round 23,409 to the largest place value of the larger number.
(c) Round 12,000 to the largest place value of the larger number.
(d) Add together the rounded numbers from part (b) and (c).
2. Using technique 1, estimate the sum 678, 222.2 + 7, 392.3
(a) What is the largest place value of the larger number?
(b) Round 678,222.2 to the largest place value of the larger number.
(c) Round 7,392.3 to the largest place value of the larger number.
(d) Add together the rounded numbers from part (b) and (c).
Technique 2: Rounding all numbers to the largest place value of the smaller number.
• Note the largest place value in the smaller number.
• Round each number to the place value noted.
• Add the rounded numbers.
Example 2: Estimate the sum:
5, 629 + 236
Solution:
236 is the smaller number, and its largest place value is the hundred’s place.
5629 ——>
5600
+
236 ——> +
200
5800
The estimate of the sum of 5,629 and 236 using technique 2 is 5,800.
51
3. Using technique 2, estimate the sum 23, 409 + 12, 000
(a) What is the largest place value of the smaller number?
(b) Round 23,409 to the largest place value of the smaller number.
(c) Round 12,000 to the largest place value of the smaller number.
(d) Add together the rounded numbers from part (b) and (c).
4. Using technique 2, estimate the sum 678, 222.2 + 7, 392.3
(a) What is the largest place value of the smaller number?
(b) Round 678,222.2 to the largest place value of the smaller number.
(c) Round 7,392.3 to the largest place value of the smaller number.
(d) Add together the rounded numbers from part (b) and (c).
5. Estimate 731.03 + 2.006 first by rounding all numbers to the largest place value in the
larger number (technique 1), then by rounding both numbers to the largest place value in
the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
6. Estimate 987, 962.051 + 16 + 687.084 first by rounding all numbers to the largest place
value in the larger number (technique 1), then by rounding both numbers to the largest
place value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
7. Estimate 11, 276 + 0.038 + 2.38 first by rounding all numbers to the largest place value in
the larger number (technique 1), then by rounding both numbers to the largest place
value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
8. Estimate 99 + 10, 268.5 + 28.03 first by rounding all numbers to the largest place value in
the larger number (technique 1), then by rounding both numbers to the largest place
value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
52
9. Estimate 0.00397 + 2.015 + 69.11 first by rounding all numbers to the largest place value in
the larger number (technique 1), then by rounding both numbers to the largest place
value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
10. Estimate 763.19 + 2, 763.019 + 3.0019 first by rounding all numbers to the largest place
value in the larger number (technique 1), then by rounding both numbers to the largest
place value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
11. Estimate 287, 309, 227.56 + 199.99 first by rounding all numbers to the largest place value
in the larger number (technique 1), then by rounding both numbers to the largest place
value in the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
12. Estimate 0.48 + 268 + 60.46 first by rounding all numbers to the largest place value in the
larger number (technique 1), then by rounding both numbers to the largest place value in
the smaller number (technique 2).
(a) Estimate using technique 1:
(b) Estimate using technique 2:
13. In general, on the sums where the two techniques give a different answer, which
technique takes less work to compute the sum after the rounding step?
14. In general, on the sums where the two techniques give a different answer, which
technique gives an estimate that is more accurate?
15. Describe the kind of sums that give the same estimate when using technique 1 as when
using technique 2.
53
We have seen in the activity that technique 1 and technique 2 can give different answers. Also,
we have seen that in general, technique 1 takes less work to do while technique 2 usually gives
a more accurate answer. In general with estimation, we want to have a method that makes the
sum easier to compute while still being relatively accurate. So, which technique is more
appropriate?
Exercise 4.3.1 Choosing the Appropriate Technique
For the following situations,
(a) Estimate the sum using technique 1. Include appropriate units in your answers.
(b) Estimate the sum using technique 2. Include appropriate units in your answers.
(c) State which technique you think is more appropriate for the situation.
(d) Explain the reasons why you chose the technique that you did in part (c).
1. Lou is grocery shopping and he only has $80 left in his checking account. He wants to use
his debit card to pay for the groceries, but doesn’t want to be embarrassed and have to
put some items back if he doesn’t have enough money. The items in his cart and their
prices are listed in the table below:
Item
1 bag of frozen shrimp
Price Tech 1 ≈ Tech 2 ≈
$19.98
1 package of beef ribs
$3.87
1 box of Wheaties
$4.49
1 12-pack of Negra Modelo
$14.99
1 bottle of Pinot Noir
$21.99
1 half-gallon of Horizon organic milk
$4.39
1 pint of whipping cream
$4.49
1 dozen eggs
$3.19
1 bottle of Advil
$16.79
Does he have enough money or does he have to put something back?
(a) Technique 1:
(b) Technique 2:
(c) Which technique is more appropriate?
(d) Why?
54
2. Marta is planning her family’s monthly budget. The following table summarizes how
much she plans to spend in different categories:
Category
Rent
Budget Tech 1 ≈ Tech 2 ≈
$854.00
Car Payment
$137.42
Power, water, and garbage
$93.75
Phones
$84.20
Food
$370.00
Cable
$75.00
Netflix
$17.99
Credit Card Payment
$175
What is Marta’s total monthly budget?
(a) Technique 1:
(b) Technique 2:
(c) Which technique is more appropriate?
(d) Why?
55
Exercise 4.3.2 Approximate Sum Practice
For each of the following exercises:
(a) Estimate the answers either by rounding all numbers to the largest place value in the largest
number (technique 1), or by rounding both numbers to the largest place value in the smallest
number (technique 2). Include appropriate units in your answers.
(b) State which technique you used to perform the estimate, and why you chose that technique
over the other one.
(c) Find the actual sum. Include appropriate units in your answers.
(d) State whether the actual sum is >, =, or < the estimate.
1. During a shopping spree, Margaret spent $259.05 at Costco and $72.99 at Ross. How
much did she spend in all?
2. The Smiths recorded their mileage over the summer and were shocked to find they went
3,759.12 miles in June and 4,250.7 miles in July. They decided to stay home during the
entire month of August, so only went 66 miles. What was their total distance traveled for
the three months?
3. A mountain is 27,369 feet high. On the top of the mountain is a building that is 150 feet
tall. On the roof of the building is a woman who is 6.13 feet tall. How high is the top of
her head above sea level?
4. Mark lives at home to try to save money. He earns $24,269.5 per year. His dad earns
$35,253.05 and his mom earns $41,299.8. What is Mark’s family’s total income each year?
5. After his cramp went away, Rick ran 5.099 miles on flat ground, 0.89 miles uphill, then
2.01 miles downhill. At the end of the downhill, his legs started cramping again! How far
did Rick run between cramps?
6. A government spent $7,831,004.05 per day for the military, and $32,890.9 per day to fund
the student loan program. How much did this government spend each day on the
military and student loans?
7. Georgio bought a used car off Craig’s list for $8,750 and then filled up the gas tank for
$48.72. How much did he spend in all?
8. 2, 224, 766 + 5, 689, 201 + 1, 920, 761
9. 2, 256, 703 + 156.56
10. 276, 107 + 24, 109.5 + 3, 655.7 + 390.7
56
Activity 4.4 Magic Squares
Objective: To practice addition and puzzle solving.
Materials: Paper, pencil.
Group size: 3 to 4.
Look at the square grid of numbers shown below. See if you can find anything unusual about
the numbers in the square.
16
5
9
4
3
2 13
10 11 8
6
7 12
15 14 1
This square is called a magic square because the numbers in each row, in each column, and in
each diagonal all add up to the same thing, called the magic number. In this case the magic
number is 34. The artist Albrecht Dürer (1471 - 1528) is credited with being the first European to
publish a magic square, by including the square shown above in his copper engraving,
”Melancholia I”. See if you can find the year the piece was completed, hidden somewhere in
the magic square.
1. The magic square below is composed of the digits 1 through 9 (exactly once each) but
some have been left out. Fill in the missing numbers in the correct places.
8
6
5
4
2
2. The numbers 1, 3, 7, 8, 9, 10, 14, and 15 have been removed from the magic square below.
Put them back in the correct locations.
14
4
11
6
10
13 2
5
3. The numbers 6, 10, 11, 12, 13, 14, 15, and 16 have been removed from the magic square
below. Put them back in the correct locations.
9
4
3
5
1
7
8
2
57
4.4 Addition in Real Life
In a math class, we see the word “sum” used to indicate that we are supposed to add. In real
life, the operation of addition can be implied in many other ways. Some key words to look for
are total, increase, or perimeter, but sometimes none of these are used, and you have to reason,
from the context, that addition is implied.
Example 1: Marco and Mary went Christmas shopping. They found an alarm clock for $24.99, a
CD for $14.45, a pillow for $32.20, a rug for $18.69, and a game for $11.50. Estimate the pre-tax
cost for all of these items by rounding each value to the nearest whole dollar before computing,
then find the actual pre-tax cost.
Solution:
It’s not stated explicitly in the problem, but we can reason that the cost for all of the items is
the sum of each cost. We begin by rounding each to the nearest whole dollar rather than using
another estimation technique since the problem explicitly says how it wants it done:
$24.99 ≈ $25
$14.45 ≈ $14
$32.20 ≈ $32
$18.69 ≈ $19
$11.50 ≈ $12
Then, we add the rounded values, 25 + 14 + 32 + 19 + 12 = 102, to get our estimate of $102.
Finally, we add up the numbers with the cents included: 24.99 + 14.45 + 32.20 + 18.69 + 11.50 =
101.83, which is close to our estimate, so we are confident that we didn’t make a careless error!
The pre-tax cost of all of the items is: $101.83.
Exercise A: Gracie and her husband were in New York City at a discount department store.
They bought sheets for $34.96, shorts for $16.95, socks for $5.24, and a duvet for Gracie’s mom
for $48.32. Estimate the pre-tax cost for all of these items by rounding each value to the nearest
whole dollar before computing, then find the actual pre-tax cost.
58
Example 2: Sammi was excited when the value of her stock went up. At the beginning of the
month, the value was $42.90 per share, but when she checked at the end of the month, the
website indicated that the value had increased by $2.73 per share. What was the per share value
of her stock at the end of the month?
Solution:
To find a new value of something after an increase, and the amount of the increase to the
starting value: First, even though we were not told to estimate, it’s a good idea, so we
round each number to the nearest dollar before adding: 43 + 3 = 46 for an estimate of $46.
When we calculate the actual sum, we get: 42.90 + 2.73 = 45.63. The new value of her stock
is: $45.63 per share. Notice we have included the units in the answer instead of just writing 45.63.
Exercise B: Bridgit was excited when her son’s pediatrician told her that her son had grown a
lot since their last visit. At their last visit, her son was measured at 45.5 inches, and the doctor
told her that he had grown by 3.5 inches. What was her son’s new height?
Example 3: The Community Gardening Society wanted to make a short fence to surround their
garden. The plot is a rectangle that measures 11.3 meters by 4.8 meters. How many meters of
fencing do they need to buy?
11.3 m
4.8 m
4.8 m
11.3 m
Solution 1:
The amount of fencing that they need is the total distance around the rectangle, or, what is
called the perimeter of the rectangle. Since it’s a rectangle, the side measurements are repeated
on the opposite sides (see figure), so the total is 11.3 + 4.8 + 11.3 + 4.8. Since they didn’t specify
a level of rounding, we will use the method outlined on page 49. 11.3 is the largest number, so
we will round to the nearest ten meters. Our estimate is now 10 + 0 + 10 + 0 = 20 meters, and
with a little more work, the actual perimeter is 32.2 meters.
Solution 2:
You may want a greater level of accuracy for the estimate and use technique 2. 4.8 is the
smallest number, so this time we round to the nearest whole meter. Our estimate is now
11 + 5 + 11 + 5 = 32 which isn’t much more difficult to compute, but is a lot more accurate.
59
Example 4: Find the perimeter of the following figure:
7ft
15ft
21ft
12ft
Solution:
The Perimeter of more complicated figures can be computed with a little effort, and with what
we math folks call problem solving. The first thing to realize, is that the perimeter of this figure
is the same as the perimeter of the rectangle that it fits into. (Talk with your classmates or your
instructor until you understand why this is true!)
21+12=33ft
7ft
15ft
7+15=22ft
21ft
12ft
From this picture we see that the dimensions of the surrounding rectangle are 33 feet by 22 feet,
giving us our estimate of 30+20+30+20 = 100 feet, and our actual value of 33+22+33+22 = 110
feet.
Exercise 4.4.1 Real Life Addition Exercises
For the following exercises, write the sum that is needed to answer the question, then estimate
the sum by using an estimation technique of your choice, then find the actual sum. Include
appropriate units in your answers.
1. Clarence goes to the store and starts putting items in his cart. As he is approaching the
checkout line, he wants to know what his total charge will be. The items in his cart are:
sugar $1.75, eggs $2.50, bread $2.25, milk $3.19, cheese $4.49, cereal $3.69,
2 boxes of Mac and Cheez at $0.79 each, soup $2.45, juice $5.50, gum $0.44.
(a) Estimate:
(b) Actual:
60
2. Hector wants to build a frame for a picture that is 8 inches tall and 10 inches wide. He
plans to buy a single piece of wood and cut it up to make the frame. How long should the
piece of wood be?
(a) Estimate:
(b) Actual:
3. As a sales rep, Marta has to drive a lot. One day, she drove 11.8 miles from San Francisco
to Oakland, then she went 41.6 miles from Oakland to Napa. She had lunch in Napa, then
drove 39.3 miles to San Rafael, and ended her day in San Jose, which is 36.8 miles from
San Rafael. How far did she drive that day?
(a) Estimate:
(b) Actual:
4. After finally convincing his spikey-haired boss, Dilbert was able to get a raise of $1945 per
year from his old salary of $40,275 per year. What is his new salary?
(a) Estimate:
(b) Actual:
5. Find the perimeter of each figure.
(a)
(b)
4ft
4ft
12ft
7ft
10ft
11ft
12ft
5ft
14ft
20ft
(c)
(d)
6ft
9ft
16ft
25ft
8ft
22ft
13ft
6ft
5ft
61
6. One month, Henry and Henrietta’s bills were out of control. Their PG&E bill was $186.82,
their phone bill was $62.37, their rent was the normal $1245, their credit card minimum
payment was $89.79, and their Chevron charge was $183.28. What is the total of these
bills?
(a) Estimate:
(b) Actual:
7. Measure the height and width of a door to the nearest centimeter, then calculate the
door’s perimeter.
(a) How are you able to measure such a long distance in centimeters?
(b) Height to the nearest centimeter:
(c) Width to the nearest centimeter:
(d) Estimate of perimeter:
(e) Actual perimeter:
8. The number of books published about cats for various years are shown in the table below.
Year Number of Books
1999
138
2000
98
2001
92
2002
73
2003
73
2004
120
What was the total number of books published about cats between the years 1999 and
2004?
(a) Estimate:
(b) Actual:
62
Exercise 4.4.2 Use the diagram to help answer the questions. Include appropriate units in
your answers.
Joe and Kathy are neighbors who are fencing their backyards in the following shapes:
Side c
Side d
Side c
Side b
Side d
Side a
Side b
Side a
Joe’s back yard
Kathy’s back yard
1. Measure each side of the pictures of Joe’s and Kathy’s backyards in centimeters. Round
1
centimeter, and record it in the table. Assume that
each measurement to the nearest 10
every centimeter on the picture represents one meter in real life. Record the real life
distances. Be sure to write units for all of your values!
Side
Joe’s
picture
Joe’s
actual
Kathy’s
picture
Kathy’s
actual
a
b
c
d
2. How many meters of fencing does Joe need?
3. How many meters of fencing does Kathy need?
4. Who needs more fencing, Joe or Kathy?
5. How much more do they need?
63
Activity 4.5 Pascal’s Triangle
Objective: To practice addition and pattern recognition.
Materials: Pencil.
Group size: 2 to 4.
Find a pattern in the numbers given in the triangle below. Use the pattern(s) you find to help
you fill in the missing numbers. Hint: Each row’s numbers come from working with the
numbers in the row above it.
1
1
1
1
1
1
2
3
1
3
4
6
1
4
1
1
1
1
1
1
1
1
1
64
Activity 4.6 Darts
Objective: To practice addition, strategy, and organization skills.
Materials: Paper, pencil.
Group size: 2 to 4.
1
2
3
4
Imagine you are given four darts to throw at the dartboard shown above. The rules for playing
are simple. You throw your darts at the board and add up the points you get depending on
which ring your dart lands in. For now, assume you hit the board all four times.
1. What is the highest score you could get and how would you get it?
2. What is the lowest score you could get and how would you get it?
3. What are all of the possible scores?
4. Make a list of all the possible score combinations you could get playing this game.
For example, you could score 11 by getting 2 + 2 + 3 + 4 = 11 and also by scoring
4 + 4 + 2 + 1 = 11. Ignore the order in which you score – treat 2 + 2 + 3 + 4 the same
as 4 + 3 + 2 + 2 and so on.
5. From your list in (4), how many different ways are there to score 10?
(Remember, treat 2 + 2 + 3 + 4 the same as 4 + 3 + 2 + 2).
6. Which score is most common (has the most ways of getting it)?
7. Suppose that it is possible to score a 0 by missing the dartboard entirely. What are all of the
possible scores now?
65
4.5 Introduction to Fractions
We know from the way it’s said, that quarters are a quarter of a dollar. But where is this term
coming from? A little history of money in the U.S. will help. The following is an excerpt from
http://www.collectsource.com/americas.htm:
“The Spanish Dollar quickly became the most popular coin in North America. It is
even thought by some that if Washington did throw a coin across the Potomac, it
was likely to have been America’s First Silver Dollar. Every day commerce was
lubricated by this remarkable coin, and the terminology which developed by using it
became so deeply embedded in American culture that it remains with us to this day.
Pieces of Eight
Because America’s First Silver Dollar was often cut into eight pie shaped ‘bits’ in
order to make change, the intact coin became known as a ‘Piece of Eight.’ Since the
entire Piece of Eight had a value of 8 Reales, each bit was valued at one eighth of the
total. Two bits equaled a quarter, four bits a half dollar and six bits three quarters of
a dollar. Did you ever spend two bits?—Then you were living the legacy of
America’s First Silver Dollar! To put the value of America’s First Silver Dollar into
perspective, an average worker during the colonial era earned about 2 bits a week!”
The following pictures are of coins of this time.
The following activity simulates history in order to introduce fractions and adding fractions.
66
Activity 4.7 Shopping with Fractions
Objective: To introduce the idea of fractions using money.
Materials: 8 breakable silver dollars.
Group Size: 3 to 4
Set up: Choose one person in the group to be the merchant, the others to be customers.
• Customer one starts with 2 dollars.
• Customer two starts with 3 dollars.
• Customer three (if there are that many in your group) starts with 3 dollars.
Procedure: Act out the following scenario, using the coins as props. Answer each question,
using the coins and your discussion to help you.
Use the following prices to complete the activity:
Pigs: 2 for a dollar
Chickens: 4 for a dollar
Coffee: 8 pounds for a dollar
The first customer comes in to buy one chicken.
1. How much is the total charge for customer one?
(a) Write as a fraction of a dollar.
(b) Write in standard decimal notation.
Have customer one pay the merchant for the goods.
2. How can the merchant give change?
3. After making the purchase, how many full coins does customer one have left? How many
“quarters”?
Now, customer two buys one pig and one chicken.
4. How much is the total charge for customer two?
(a) Write as a fraction of a dollar.
(b) Write in standard decimal notation.
Have customer two pay the merchant for the goods.
5. How many full coins does customer number two have left? How many “quarters”?
67
Customer three buys 9 chickens and four pounds of coffee.
6. How much is the total charge for customer three?
Have customer three pay the merchant for the goods.
7. How many full coins does customer number three have left? How many “quarters”?
8. If the three customers combine their change into one pile, how many full coins do they
have altogether? How many quarters?
9. How much money do they have altogether?
10. Write an addition problem, using decimal notation that models the total that you just
computed.
11. Now, re-write the sum using fraction notation, that is write
quarters, etc.
1
4
for one quarter,
2
4
for two
12. Is there more than one way to write the total in fraction form? If so, write the total in as
many correct ways as you can think of (up to five).
13. Compute the following sums. Use your coins to help. Draw a picture of the problems
using the coins. For example, if the problem was 21 + 12 , you would see that the answer is
1. Then, you would draw a picture similar to the one below.
+
+
1/2
The sum
=
5
4
+
1/2
1
2
=
1
is illustrated here:
+
5/4
(a)
1
2
+
1
4
(b)
3
2
+
3
4
(c)
1
4
+ 1 12
+
=
1/2
=
68
1 3/4
4.6 Fractions
A fraction represents a part of a whole, like a dime is a part of a dollar or a piece of pie is part of
the whole pie. The bottom number in a fraction, the denominator, tells us the number of pieces
the whole was divided into, so the 8 in 83 of a pizza means the whole pizza was cut into 8 pieces.
It gives some sense of the size of the pieces, the larger the number, the more pieces, so the
smaller each individual piece must be. The top number, the numerator, describes the number of
pieces we have out of the total. The 3 in 38 of a pizza means we have three slices.
Example 1:
Solution: Since the shape can be broken into 6 triangles of the same size and one of them is
shaded, we say that one-sixth (written 61 ) of the shape is shaded.
Example 2:
Solution: Since the shape can be broken into 8 squares of the same size and five of them is
shaded, we say that five-eighths (written 85 ) of the shape is shaded.
Exercise 4.6.1 State what fraction of the following shapes is shaded.
1. What fraction is shaded?
2. What fraction is shaded?
69
3. What fraction is shaded?
4. What fraction is shaded?
5. What fraction is shaded? (Hint: Since more than one whole rectangle is shaded, the number
of pieces shaded (the numerator) will be greater than the number of pieces in one whole (the
denominator). A fraction in which the numerator is greater than the denominator is called an
improper fraction.)
6. What fraction is shaded?
Example: Shade
5
6
of the rectangle.
Solution: Break the rectangle into six pieces of
equal size,
then shade five of them.
70
Exercise 4.6.2 Shade the fraction of the shape indicated.
1. Shade
1
4
of the rectangle.
2. Shade
1
6
of the rectangle.
3. Shade
7
8
of the rectangle.
4. Shade
3
5
of the rectangle.
5. Shade
2
4
of the rectangle.
6. Shade
1
2
of the rectangle.
7. Describe any similarities and differences between problems 5 and 6.
8. Shade
7
4
of the rectangle.
9. Shade
5
2
71
of the rectangle.
13
of the rectangle. Another way to ask this same question is to say, “Shade 13% of
10. Shade 100
the rectangle”, since “percent” means “out of 100”.
11. Shade
29
,
100
that is, shade 29% of the rectangle.
12. What fraction of a dollar is 80 cents?
13. What fraction of a dollar is 20 cents?
14. What fraction of a dollar is $1.50?
15. What fraction of an hour is 30 minutes?
16. What fraction of an hour is 20 minutes?
17. What fraction of an hour is 7 minutes?
18. What fraction of an hour is 90 minutes?
72
19. Shade
1
2
of the given rectangles.
a.
b.
c.
d.
Exercise 4.6.3 State what percent of the following rectangles is shaded. Remember, the
percent is just the number of pieces out of 100.
1. What percent of the following is shaded?
2. What percent of the following is shaded?
73
3. What percent of the following is shaded?
1
s, 0.01s, or %s). Recall that
Exercise 4.6.4 Fill in the table with the missing hundredths ( 100
percents are the same as hundredths. That is, 1 percent equals 0.01 equals
different representations of the same amount.
No.
1.
2.
3.
4.
7.
17
100
3
100
150
100
28
100
0.17
17%
0.03
3%
1.5
150%
28%
0.7
20
100
38
100
70%
0.2
8.
0.95
9.
63%
10.
0.8
11.
0.23
12.
The three are just
Fraction Representation Decimal Representation Percent Representation
5.
6.
1
.
100
122
100
13.
0.09
14.
780%
15.
13
74
1300%
4.7 Measuring Lengths with Inches
Activity 4.8 Measuring with Fractions
Objective: To further investigate fractions while introducing measuring with inches.
Materials: Long, thin rectangle made of card stock.
Procedure: In the following activity, you will create your own unit of measure, make a ruler
that measures in that unit, then measure the length of given lines, accurate to the nearest
quarter unit.
1. Choose a part of your body that is approximately one inch long. Use this length as your
“unit” of measure, and mark your blank ruler with whole unit markings up to the end of
the rectangle. Your ruler would look something like this:
1
2
3
4
2. With slightly shorter lines, mark the half way point between each unit. At this stage your
ruler would look something like this:
1
2
3
4
3. With still shorter lines, mark the one-quarter and three-quarters points between each unit.
At this stage, your ruler would look something like this:
1
2
3
4
4. Using your new ruler, make an accurate drawing of the ruler including all of the
markings.
5. Using your new ruler, measure the lengths of the following lines, accurate to the nearest
quarter of a unit. Compare your answers with other people in the class. Discuss
how/why someone could get a different answer for any of the measurements.
(a)
(b)
(c)
(d)
(e)
75
What parts make up an inch?
Up until the previous activity, we have been measuring distances using centimeters.
Centimeters are part of the metric system of measurement, so they are based on decimals. When
you need a finer measurement, you use tenths of a centimeter, or hundredths of a centimeter.
Inches are part of the English system. Instead of breaking things into tenths or hundredths
(powers of 10), inches are broken up into halves, quarters, eighths, and sixteenths. The
denominators of these fractions are all powers of 2 since each part of a fraction is half the size of
the next bigger part.
The following is an enlarged picture of an inch. Notice the markings that show the fractions of
an inch.
3
16
1
16
1
8
5
7
16 16
9 11
16 16
3
8
5
8
15
16
13
16
7
8
1
4
3
4
1
2
76
1
Exercise 4.7.1 Mark the following lengths on the given rulers.
Example:
1 ′′
2
Solution:
2
1
1.
2.
1 ′′
4
1
2
1
2
1
2
3 ′′
4
3. 1 38
′′
77
4. 3 58
′′
1
5. 2 16
6. 3 14
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
′′
′′
3
7. 4 16
8. 2 78
1
′′
′′
78
Home Project: Fraction Bottle
A fraction represents a part of a whole, like a dime is a part of a dollar or a piece of pie is part of
the whole pie. The most common fraction, 12 , is the source of almost all the fractions you will
need in daily life. The purpose of this assignment is to find and mark some common fractions
on a plastic bottle.
Materials: You will need a plastic bottle, like the ones used for water
or soda (any size will do).
You will also need two identical containers, like drinking
glasses or more plastic bottles.
You will also need a Sharpie to write on the bottle.
1/2 full
Procedure: We want to find the half full point on the bottle (see right).
1. In order to do this begin by filling the bottle with water (to the top).
2. Pour the water out into the two identical containers being careful to put an equal amount
in each glass. (figure 1).
3. Now throw out the water in one glass and pour the contents of
the other back into the bottle.
1/2 full
4. Put a mark on the side of the bottle showing the water level.
This is the half full point.
5. Now turn the bottle sideways and mark the half full point on
the bottom of the bottle.
79
1/2 full
Now we want to mark the point where half of the remaining water reaches (half of a half).
6. Using the water remaining in the bottle, repeat the procedure
above by distributing it equally between the two containers.
7. Empty the water in one of the containers and pour the remaining
water back in the bottle.
1/2 full
1/4 full
8. Mark both the side and the bottom of the bottle showing the the
quarter-full points.
9. Repeat the process above until you have made the four marks
1
. on both the side and the bottom of the bottle.
for 12 , 14 , 18 , and 16
1/2 full
1/4 full
1/8 full
1/16 full
80
4.8 Adding Fractions
As long as the pieces are the same size, adding fractions is no different than adding any other
items that are the same. Two dogs plus five more dogs adds up to seven dogs just like two
eighths plus five more eighths adds up to seven eighths. In pictures this looks like:
Adding Dogs:
+
Adding Eighths:
2
8
+
+
5
8
7
8
Adding Dimes:
+
Or, using fraction or decimal notation since a dime is just
2
10
+
5
10
=
7
10
or 0.2 + 0.5 = 0.7
81
1
10
= 0.1 of a dollar,
We have problems when we try to add a quarter to a dime:
1
4
+
+
1
10
???
2?
?
+
But we know how to add $0.25 to $0.10! We know the total is $0.35! In fraction notation, since 1
cent is one hundredth of a dollar, this is the same as saying,
Fractions
English
1
4
One quarter and one dime
+
equals 25 cents and 10 cents
=
25
100
equals 35 cents.
=
35
100
Decimals
1
10
+
10
100
= 0.25 + 0.10
= 0.35
82
Exercise 4.8.1 Shade the boxes to show the given fraction addition, then draw and shade a
box (or boxes) to represent the sum. When you are done shading, write the sum in fraction
form. Be sure to make each whole strip the same size, as well as making each piece within the
whole the same size. You may need more than one whole for some of the answers.
1.
1
3
+
1
3
1
3
+
1
3
2.
3
5
+
2
5
3
5
+
2
5
3.
3
10
+
5
10
3
10
+
5
10
4.
1
4
+
3
4
1
4
+
3
4
83
5.
3
8
+
7
8
+
5
8
+
4
5
3
8
+
7
8
+
5
8
6.
3
4
+
5
4
3
4
+
5
4
7.
1
2
+
3
2
1
2
+
3
2
8.
1
5
+
2
5
1
5
+
2
5
+
4
5
84
9.
1
10
+
7
10
1
10
+
7
10
4.9 Equivalent Fractions
When you add money, sometimes you can exchange many smaller bills with a larger bill if you
have enough of the smaller bill. The same is true for coins, the fractions that the coins represent,
and fractions in general. For example, if you add 1 quarter to 1 quarter you will get two
quarters. You may exchange your quarters for one half dollar.
1/4
+
1/4
2/4
=
1/2
=
1
4
+
1
4
=
2
4
=
1
2
Similarly, if you add 2 quarters to 3 quarters you will get 5 quarters. You may exchange four of
your quarters for one $1 bill, giving you one and one quarter dollars.
2/4
=
+
3/4
5/4 = 1 1/4
2
4
+
3
4
=
5
4
= 1 14
85
Equivalent fractions represent equal amounts. For example, suppose you wanted to spend 21
of the $24 in your wallet. You could split the $24 into two equal $12 pieces and spend one of the
two pieces. Equivalently, you could split the $24 into four equal $6 pieces and spend two of the
four pieces. The first time, you are spending 12 of your money, while the second time you are
spending 24 . Both times, however, you spend $12! The list of fractions equivalent to 12 is literally
endless.
That is:
1
2
2
4
=
=
12
24
= ...
1
2
is said to be in simplest form because of all of these fractions that represent the same amount,
has the smallest denominator.
1
2
Note: A fraction in which the numerator is greater than the denominator is called an Improper
Fraction, and they can be written in an equivalent form called a Mixed Number. For example,
from above we saw that one and a quarter is equivalent to five quarters. 1 14 is a mixed number,
while 54 is an improper fraction.
If a fraction has a numerator that is smaller than the denominator, it is called a Proper Fraction.
3
is an example of a proper fraction. Proper fractions don’t have a mixed number equivalent.
4
We can use pictures to show that two fractions are equivalent. We just have to make sure that
the size of each rectangle is the same, and that the total size of the shaded parts are the same.
Here are two examples:
Example:
2
3
Solution 1:
2/3
2
3
making sixths
add lines to
make sixths
4/6
4
6
making ninths
add lines to
make ninths
6/9
6
9
86
Here is another way to show the same fractions equivalent to 23 :
Solution 2:
2
3
What line was
inserted to go
from thirds to...
4
6
sixths?
6
9
ninths?
What lines were
inserted to go
from thirds to...
Exercise 4.9.1 Write two other fractions, with different denominators, that are equivalent to
the given fraction. For the given fraction and each of your equivalent fractions, draw a shaded
rectangle (or more than one if the given fraction is an improper fraction) showing that they all
represent the same amount. For the first two problems, the given fraction has been drawn three
times. Leave the first one alone, then draw lines on the other two to make equivalent fractions.
For problems 3-10, draw each picture completely.
1.
1
2
2.
1
3
3.
2
5
4.
3
4
5.
3
2
6.
1
8
7.
3
8
8.
4
3
9.
7
4
87
Exercise 4.9.2 Complete the following equivalence problems. Include appropriate units in
your answers.
2. Write three lengths with different de′′
nominators equivalent to 64 .
1. Write three lengths with different de′′
nominators equivalent to 21 .
1
2
(a)
(a)
(b)
(b)
(c)
(c)
3. Write three lengths with different denominators equivalent to
1
2
1
3 ′′
.
4
2
(a)
(b)
(c)
′′
4. Write three lengths with different denominators equivalent to 3 42 .
1
2
4
3
(a)
(b)
(c)
88
5
4.10 Adding Mixed Numbers:
Adding mixed numbers is similar to adding whole numbers. The only difference is that the
fraction parts of the numbers must be thought of as their own place value.
Example 1: Add 14 53 + 28 45
Solution: First estimate as usual: 10 + 30 = 40 so that we know that the sum is approximately
40. For the actual sum, line up the tens, ones, and fifths for each number:
14
28
3
5
4
5
Next, add up the fifths. Since 57 is bigger than one-whole, we break it up into
extra one is ”carried” over into the one’s column:
1
14
28
3
5
4
5
}
3
4 = 7
+
5 =
5
5
5
5
= 1 and 25 . The
1 25
2
5
Now, with the carry, there is a total of 13 ones. As usual, we carry 10 of them as an extra ten:
11
28
3
5
4
5
43
2
5
14
The final answer is the mixed number 43 52 .
89
Example 2: Kreg was making a shopping list before baking. He was going to make 2 batches
of cookies, and some banana bread. Each batch of cookies called for 2 41 cups of flour, and the
banana bread called for 1 34 cups. How much flour did he need altogether?
Solution: The total is the sum of 2 14 + 2 14 + 1 43 . Following the process outlined, the work should
look something like this:
1
2 14
2 14
1 34
6 14
The answer is 6 41 cups.
Exercise 4.10.1 Add the mixed numbers in the sums or solve the word problems. Write the
answers as mixed numbers, with the fraction part in simplest form.
1. 2 85 + 7 58
2. 5 43 + 14 34 + 5 43 + 14 43
5
9
+ 4 16
3. 1 16
4. 2 43 + 3 12
5. For the four day weekend, you decide to do some baking. Your chocolate chip cookie
recipe calls for 1 41 cups of sugar. The banana bread needs 43 cups of sugar. The cake you
are making from scratch calls for 2 43 cups of sugar. If you make a double batch of cookies,
and three batches of banana bread, and one cake, how much sugar will you need?
90
6. Find the perimeter of the following figure:
3/8 in
5/8 in
1 1/8 in
4/8 in
1 3/8 in
2/8 in
7. Find the perimeter of the following figure:
21 3/4 in
8 1/4 in
8. Count by 12 ’s up to 5. Write each fraction in simplest form.
9. Count by 41 ’s up to 2. Write each fraction in simplest form.
10. Count by
1
’s
16
up to 1 12 . Write each fraction in simplest form.
91
Exercise 4.10.2
1. In each problem, add the fractions of an inch by jumping along the ruler below.
Example:
1 ′′
2
+
3 ′′
4
=
Solution: Start at the beginning of the ruler and “jump” a length of
′′
jump an additional 34 . The total is where you land!
1 "
+
2
3 "
=
4
1 ′′
.
2
Then continue on and
1 "
1
4
2
1
The answer is 1 41 inches.
(a)
5 ′′
8
+
7 ′′
8
=
(b)
1
′′
7 ′′
16
+
13 ′′
16
2
=
2
1
′′
(c) 2 12 + 1 34 =
1
3
2
92
4
5
2. Using a ruler to measure the sides to the nearest eighth of an inch, confirm that the
perimeter of the rectangle below is 8 43 inches. Mark the measurements on the figure.
3. Use a ruler and measure the sides of the shape below to the nearest eighth of an inch in
order to determine its perimeter. Mark the measurements that you use on the figure.
(Hint: You may not need to measure every side!)
93
Exercise 4.10.3 Add the following using any method. Recall that percents are the same as
hundredths. That is, 1 percent equals 0.01 equals
of the same amount.
1. 40% + 20%
2.
17
100
+
13
100
3. 0.39 + 0.52
4. 28% +
4
100
5. 0.23 + 14%
6.
37
100
+ 0.52
7.
7
100
+ 9% + 0.05
8.
96
100
+
38
100
9. 47% + 79%
10. 0.82 +
11. 0.3 +
61
100
5
10
+ 10%
94
1
.
100
The three are just different representations
Activity 4.9 Fraction Paths
Objective: To practice adding fractions.
Materials: Pencil.
Group size: 2 to 4.
Start at any of the openings along the outside of the square and draw a path through the
fractions that adds up to the number at the Finish.
1.
2.
1
3
1
3
3
3
1
6
3
6
2
6
4
3
1
3
2
3
4
6
1
6
3
6
3
3
1
3
4
3
1
6
2
6
1
6
3
Finish
3
Finish
3.
4.
1
2
3
4
1
2
3
8
1
4
5
8
1
4
2
4
2
2
1
2
3
4
1
8
3
2
1
4
6
4
7
8
1
4
7
8
6
Finish
4
Finish
95
5 Subtraction
5.1 Subtracting Money
Activity 5.1 Money Activity: Borrowing from the Bank
Objective: To solidify concepts of subtraction by using money.
Materials: Bills of different denominations.
Group size: 4.
In your groups of four, select a person to start as the banker (group member #1) and select who
will be member #2, member #3, and member #4. Using denominations $10,000, $1000, $100, $10,
$1, dimes and pennies, the banker will hand out the original amount of money using the fewest
number of bills/coins possible. Then, when needed, exchange bills so that the loan can be
made.
Write the names of each group member:
Group member #1
Group member #2
Group member #3
Group member #4
1. Group member #1 is the banker. Begin by counting out $347.10 and give it to member #2.
Make sure that you take the least number of each denomination that makes up this
amount. Record how many of each denomination member #2 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
(a) Group member #3 takes $20 from member #2.
i. Can member #2 give the money to member #3 without exchanging bills with the
bank?
ii. If not, why not? What will they have to exchange to make the gift possible?
iii. How much does member #2 have left after the gift? Record how many of each
denomination member #2 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
iv. Write a subtraction equation that represents this event.
96
(b) Now, group member #4 takes $50 from what member #2 has left.
i. Can member #2 give the money without exchanging bills with the bank?
ii. If not, why not? What will they have to exchange to make the loan possible?
iii. How much does member #2 have left after the gift? Record how many of each
denomination member #2 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
iv. Write a subtraction equation that represents this event.
2. Now, start over giving all the money back to the bank. The banker counts out $1258.92 to
group member #3. Make sure that you take the least number of each denomination that
makes up this amount. Record how many of each denomination member #3 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
(a) Group member #2 takes $225 from member #3.
i. Can member #3 lend the money without exchanging bills with the bank?
ii. If not, why not? What will they have to exchange to make the gift possible?
iii. How much does member #3 have left after the gift? Record how many of each
denomination member #3 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
iv. Write a subtraction equation that represents this event.
(b) Now, group member #4 takes $50 from what member #3 has left.
i. Can member #3 give the money without exchanging bills with the bank?
ii. If not, why not? What will they have to exchange to make the gift possible?
iii. How much does member #3 have left after the gift? Record how many of each
denomination member #3 has:
$10, 000 $1000 $100
$10
$1
dimes pennies
iv. Write a subtraction equation that represents this event.
97
3. Compare your results with another group in the class.
4. What strategy could you use to estimate the answers? How would this compare with the
methods used to estimate sums?
5. How would you compare this activity with the method for subtraction that you know?
5.2 The Process of Subtraction:
The previous activity demonstrated the process of taking money away. If Greg has $23 in the
form of two $10 bills and three ones, then in order to give Fred $7, Greg has to exchange one of
his tens for ten ones.
We use the same idea of exchanging when we subtract any two numbers.
Vocabulary
• The operation is called subtraction.
• The number to subtract from is called the minuend.
• The number being subtracted is called the subtrahend.
• The result of the operation is called the difference.
For example, in 8 − 5 = 3, the 8 is the minuend, the 5 is the subtrahend, and the 3 is the
difference.
Steps for subtracting whole numbers and decimals:
1. As with addition, line up the numbers, digit by digit, so that each place value is in the
same column. The number you are subtracting from (Minuend) should be on top, and the
number you are subtracting (Subtrahend), below that.
2. Start from the right most column.
3. If the digit on top is greater than or equal to the digit on the bottom, take away the digit
on the bottom from the digit on the top and write the difference below the bottom digit in
the same column.
4. If the digit on top is smaller than the digit on the bottom, exchange from the digit to the
left on top, giving you an extra ten, enough to take away. To indicate the extra ten, write a
small 1 just to the left of the top number’s digit. Then take away the bottom and write the
difference below the bottom digit.
98
Example 1:
7, 458.39 − 5, 297.58
Solution:
7 , 4 5 8 . 3 9
− 5 , 2 9 7 . 5 8
1
3
7
7 , 4/ 15 8/ . 13 9
− 5 , 2 9 7 . 5 8
0 . 8 1
−→
7
7 , 4 5 8/ . 13 9
− 5 , 2 9 7 . 5 8
8 1
−→
3
7
7 , 4/ 15 8/ . 13 9
− 5 , 2 9 7 . 5 8
2 , 1 6 0 . 8 1
So, the difference of 7,458.39 and 5,297.58 is 2,160.81. Granted, with all of the markings for the
exchanging, the process can get pretty messy.
• If your handwriting is messy already, try writing bigger.
• Everyone needs to be careful to keep the place value columns in line.
One nice advantage to subtraction, is that you can check your answer with addition. If your
difference is correct, then the difference plus the subtrahend will equal the minuend. In our
case, to check, we see that 2, 160.81 + 5, 297.58 = 7, 458.39
1
1
2 , 1 6 0 . 8 1
+ 5 , 2 9 7 . 5 8
7 , 4 5 8 . 3 9
99
Example 2:
24, 215.4 − 35.17
Solution:
Notice the extra zero in the minuend as a place-holder for the hundredths place.
3
2 4 , 2 1 5 . 4/ 10
−
3 5 . 1 7
3
1
2 4 , 2/ 11 5 .
−
3 5 .
0 .
−→
3
4/ 10
1 7
2 3
1
3
2 4 , 2/ 11 5 . 4/ 10
−
3 5 . 1 7
2 4 , 1 8 0 . 2 3
So, the difference of 24,215.4 and 35.17 is 24,180.23. Use addition to check:
1
1
2 4 , 1 8 0 . 2 3
+
3 5 . 1 7
2 4 , 2 1 5 . 4 0
Exercise 5.2.1 Find the following differences to practice subtraction. Add back to check.
1. 236, 972.112 − 35, 387.17
2. 76, 002.17 − 12, 378.18
3. 38, 937.08 − 12, 609.5
4. 237.16 − 59.98
100
5.3 Alternate subtraction method: Subtraction by Counting Up
If you are working retail and have to give change for a purchase, you can use the method of
counting up. In fact, to be polite and let the customer know that you are doing it correctly, you
would count up out loud!
Example 1: Suppose a customer pays for an $8 item with a $20. How would you give change?
Solution: As you counted back the proper bills you might say something like, “$8, 9, 10, and
10 is 20”, as you hand the customer two ones and a ten. On a number line, this counting up
procedure looks like:
+10
+1 +1
4
5
6
7
8
9
10 11 12 13 14
15 16 17 18
19 20 21
The total change is the total of the jumps, that is, $12.
When we subtract using the standard method introduced in the previous section, we get a
result that looks something like the example below.
Example 2:
4
5 4
− 2 9
−→
−
6 5 14
2 9
2 5
There are two important things to notice about the result:
(1) We can check our work by adding the bottom two numbers to see if we get the top
number (does 25 + 29 = 54 ?).
(2) This gives us an alternative way of subtracting numbers. Since the result of subtraction
(the difference) adds with the lower number (the subtrahend) to make the top number, we can
figure out the difference by counting up from the bottom to the top. For example, let’s use this
approach with the example from above:
5 4
− 2 9
+1
−→
30
+20
−→
50
+4
−→
54
By adding 1 to 29 we jump to 30, a nice round number, and a relatively easy number to jump to
50 from - just add 20. Once we’re at 50 we only have to add 4 more to get to 54. If we add up the
numbers we used to make our jumps: 1 + 20 + 4 = 25, we get the difference.
101
Try the next example on your own before reading the solution:
Example 3:
1 2 3
− 8 7
Solution:
1 2 3
− 8 7
+3
−→
90
+10
100
−→
+23
−→
123
Adding the jumps gives us 3 + 10 +
23 = 36 so we know that 123−87 = 36.
The graph below shows what these jumps look like on a number line.
+3
80
87
+23
+10
90
100
110
120
123
130
140
These examples offer one possible way to count up from one number to another. You might
have chosen different jumps than those shown here and so long as you add your jumps
correctly when you count them up, the jumps you choose are up to you.
Exercise 5.3.1 Continue finding differences to practice subtraction. Include appropriate units
in your answers.
Find these differences by counting up.
1. 123 − 88
2. 275 − 189
3. 204 − 178
4. 1782 − 945
5. 1.58 − 0.71
6. How much change would you get on a bill of $8.39 if you pay with a $20?
7. How much change would you get on a bill of $17.62 if you pay with a $20?
8. How much change would you get on a bill of $33.14 if you pay with a $50?
9. How much change would you get on a bill of $47.23 if you pay with a $100?
102
Find these differences using any method. Show your work clearly, so that someone looking
at it could tell what method you are using. Then, check your result with a calculator.
10. 4, 398.9 − 51.71
11. 192, 543.09 − 76, 119.12
12. 321, 311.61 − 99, 378.5
13. 3, 012.9 − 2, 981.16
14. 6, 429, 199 − 399.05
15. 12, 001, 260.01 − 10, 652, 981.7
Find The Patterns
Exercise 5.3.2 For the following lists of numbers,
• Fill in the missing numbers.
• Describe the pattern in words. (The first one is done for you.)
, 12, ...
1.
2, 4, 6, 8,
2.
10, 13, 16,
3.
28, 23,
,
4.
3 5 7
8, 8, 8,
, 1 38 , 1 85 ,
...
5.
32.7, 32.3,
,
, 31.1, ...
6.
23,
7.
1 1
2, 14,
8.
14.8,
9.
14 21 , 11,
Pattern: The numbers go up by two each time.
, 22, 25, ...
, 8, 3
, 35,
Pattern:
, 47,
Pattern:
Pattern:
, 59, ...
, 2 34 ,
, ...
, 18,
,
, 4,
Pattern:
Pattern:
Pattern:
, 22.8, ...
Pattern:
103
Pattern:
Find The Digits
Exercise 5.3.3 Fill in the blanks to complete the addition and subtraction problems below.
1.
2.
5
+ 3
1 7 3
+
4
2 6 7
1 8
3
6 1
4.
3.
+ 5
1 5
5.
2
3
1 2
−4 3
8 5
6
2
−
4
8.
6 5
+8
1 5 0
−
10.
3
4
4
8
9
11.
5
+ 2
7
4
9
5 3 2
8 1
1 6
0 3
− 6 3
3
5
8
12.
4
2
6
8
4
8
6
3
6
1
9.
6
5
8
7
6.
1
− 5
3
7.
2
9
+
3
104
5
+ 6
4 8
5
3
9
3
7
2
5.4 Estimation or Approximation
An Estimation Technique for Subtraction:
• Note the largest place value in the largest number.
• Round each number to the place value noted.
• Subtract the rounded numbers.
Example 1: Estimate the difference:
7, 649 − 3, 452
Solution:
7,649 rounded to the nearest thousand is 8,000.
3,452 rounded to the nearest thousand is 3,000.
The difference of 8,000 and 3,000 is 5,000.
The estimate of the difference of 7,649 and 3,452 is 5,000.
Example 2: Estimate the difference:
159.99 − 20.898
Solution 1:
159.99 rounded to the nearest hundred is 200.
20.898 rounded to the nearest hundred is 0.
The difference of 200 and 0 is 200.
The estimate of the difference of 159.99 and 20.898 is 200.
105
Notice that although this estimate is easy to make, the value is far from the actual. An
alternative is to round to the largest place value of the smaller number. This makes the problem
a little more accurate and usually only a little more difficult.
Solution 2: The smallest number in the difference is 20.898 and its largest place value is the
ten’s place. Therefore, we will round both numbers to the nearest 10:
20.898 rounded to the nearest 10 is 20.
159.99 rounded to the nearest 10 is 160.
The difference of 160 and 20 is 140.
A more accurate estimate of the difference of 159.99 and 20.898 is 140.
Exercise 5.4.1 Approximate Difference Practice
For each of the following differences:
(a) Estimate the difference by rounding each number before subtracting.
(b) Find the actual difference.
(c) State whether the actual difference is >, =, or < the estimate.
1. 45, 467.9 − 128.4
2. 890.13 − 632.9
3. 70.5 − 19.12
4. 3, 749, 221.11 − 67, 921.5
5. 2, 039 − 1934.5
6. 11, 012 − 9, 261
7. 12, 135, 295 − 2, 367, 109
8. 17, 726.14 − 17, 632
9. 10, 138.22 − 8, 924
10. 1.56 − 0.84
106
5.5 Subtracting Fractions with Like Denominators
The idea of subtracting fractions is the same as with addition. For example, if you have three
quarters and take away two quarters, you have one quarter left. Back in the early days that
would look like:
3
4
=
2
4
-
=
1
4
With rectangles representing each whole, a representation of this problem might look like:
=
Or, on a ruler if you start at
3 ′′
4
3
4
2
4
1
4
and jump back
2 ′′
,
4
you will land at
1 ′′
.
4
2
1
Notice that just like with addition, the denominators must be the same in order for the problem
to be possible.
Just as important, when we represent the subtraction problem with a picture, we must
remember to make the rectangles representing each whole the same size and shape, and then
make sure the fractional pieces within each whole are the same size and shape as well.
107
Exercise 5.5.1 For each of the following differences:
(a) Draw a picture using shaded rectangles representing the difference.
(b) Represent the difference on the given ruler.
(c) Find the result of the difference.
1.
3
4
−
1
4
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
2.
5
8
−
3
8
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
108
3.
3
2
−
1
2
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
4.
5
4
−
3
4
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
109
5.
7
2
−
1
2
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
3
2
(c) The result:
6.
7
4
−
1
2
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
110
4
5
7.
7
8
−
1
2
(a) Drawing with rectangles:
(b) Representation on a ruler:
1
2
(c) The result:
Some of the answers in the last exercise could be written as an equivalent fraction with a smaller
denominator. For the problems in which this is true, check “yes”, and write the original
difference as well as the simpler, equivalent fraction. If there is no simpler, equivalent fraction,
check “no”. The second one is done for you.
Problem Yes No Original Fraction Equivalent Fraction
1
2
√
2
8
3
4
5
6
7
111
1
4
5.6 Comparing Fractions
How can we decide which fraction is bigger?
• If the fractions have the same denominator, then the size of the pieces is the same, and we
are just comparing the number of pieces.
Example 1:
We can compare
3
4
3
4
and
2
4
by seeing that
3
4
has one more fourth than
2
4
does:
2
4
>
• If the fractions do not have the same denominators, we need to find equivalent fractions
for each that do have the same denominator:
Example 2:
How can we compare
5
8
5
8
and 21 ? By seeing that
?
1
2
ahhh...
5
8
>
4
8
112
1
2
is equivalent to 84 , we can see that
5
8
is bigger:
Exercise 5.6.1 For the following pairs of fractions,
(a) Draw a picture for each to figure out which is bigger. (b) Place a >, =, or < symbol between
the fractions to indicate their relative size.
1.
3
4
and
1
4
2.
5
8
and
7
8
3. 1 13 and
4.
3
4
and
3
2
5.
1
2
and
3
8
6.
1
4
and
3
8
7.
3
4
and
5
8
2
3
8. 1 58 and 1 43
Recall: Adding Fractions on a Ruler
Another way to visualize which fraction is bigger is to think of them as lengths. Recall that
when we were using the rulers to visualize the addition of fractions, we identified where the
first fraction sat on the ruler, and then added the second fraction by jumping ahead a length
equivalent to the other fraction. For example, to illustrate 21 + 43 , we drew a picture similar to
the following:
1 "
+
2
3 "
=
4
1/2
3/4
1/2
1/4 1/4 1/4
1
1
1 "
4
2
Comparing Fractions on a Ruler
To compare, just place each fraction on the ruler, and see which one is a longer length!
1"
3 "
2
4
1/2
3/4
1
2
113
Exercise 5.6.2
For the following pairs of fractions,
• Indicate where each fraction lies on the ruler to see which is bigger.
• Place a >, =, or < symbol between the fractions to indicate their relative size.
1. 1 81 and 1 14
2.
1
3.
5.
5
16
1
2
and
and
3
4
and
3
8
2
3
8
4.
1
2
1
2
7
16
114
3
4
and
1
2
1
2
7
8
5.7 Subtraction in Real Life
One of the first places you will encounter subtraction after this class is...in your next math class!
The most common words that are used to imply subtraction in math problems are “difference
between” and “subtracted from”. Because the order that numbers are subtracted changes the
difference (unlike addition in which order doesn’t matter), it is important to know what order
is implied.
Example 1: What is the difference between 3 18 and 2 12 ?
Solution:
We can find the difference by subtracting 3 18 − 2 12 :
3
1
8
2
1
2
From previous problems and pictures, we have seen that
up the 12 in 2 12 :
3
1
8
2
4
8
1
2
is equivalent to 84 . We therefore break
So, our difference is equivalent to 3 18 − 2 48 . Because 4 eighths is more than 1 eighth, we have to
borrow 8 eighths from the third whole. Now 3 18 is equivalent to 2 89 :
2
9
8
2
4
8
115
Without the pictures, this process looks like:
8
8
2
8
8
+
1 = 9
8
8
3 1
8
4
- 2 8
2
3 9
8
4
- 2 8
5
8
So, the difference between 3 18 and 2 21 is 58 .
Example 2: What is 179.95 subtracted from 438.50?
Solution:
When the words “subtracted from” are used, the order that the numbers appear is opposite
the order that they are subtracted. 179.95 subtracted from 438.50 is equivalent to the difference
between 438.50 and 179.95 which is 258.55. (Verify this! Math books often leave out steps in an
example if it’s something that they think you have already done. For practice and to make sure
that you understand, you should always work out the steps that are left out.)
So, 179.95 subtracted from 438.50 is 258.55.
116
In a more real life setting, there are three common situations where subtraction is used. They
are finding a difference in two quantities, finding how much more you need, and finding the
change in a quantity.
Difference: If you want to know how far apart two quantities are, literally find the
difference in height, weight, price, bank balance, or anything that has size, subtract the smaller
quantity from the larger quantity.
Example 1: At their 5 year check-up, the doctor weighed each twin. The older twin weighed
42.3 lbs, while the younger twin weighed 33.5 lbs. How much heavier is the older twin than the
younger twin?
Solution: It is common in math problems, that the idea is the same as something you’ve studied,
but the words used to describe the idea are different. You must practice the process of understanding the given words, by reading the passage many times and thinking clearly about what
it means, then thinking of what familiar math idea that it is equivalent to. Only through practice,
and asking questions if a certain problem doesn’t make sense, will you get good at this. In this
case, although the word “difference” is not used in the problem, the given question is equivalent
to the question, “What is the difference in their weights?”. The difference in their weights is:
42.3 − 33.5 = 8.8
So, the older twin is 8.8 lbs heavier than the younger twin.
How Much More: If you have a certain amount of something and want to know how much
more you need to get to a total, subtract how much you have from the total that you want. The
difference is how much more you need.
Example 2: A math teacher is saving to buy a 2010 Prius 4-door which costs $27,998.95. He has
already saved $8,362.17. How much more does he need to save?
Solution: This time the wording in the problem is identical to our category. The required amount
to save is how much he has already saved, $8,362.17, subtracted from the total that he needs for
the car $27,998.95. :
27, 998.95 − 8, 362.17 = 19, 636.78
Checking with addition, we see that if he has $8, 362.17 and adds $19, 636.78 more to that, he
will have a total of $27, 998.95.
So, he needs to save $19,636.78 more.
117
Change: The final common use for subtraction that we will discuss here, is finding the
change in a quantity. Whether something gets bigger or smaller, the change is the bigger value
minus the smaller value.
Example 3: The percentage of young adults (age 18 to 24 years) who voted in various presidential elections are shown in the table:
Year Percent
1972
50
1980
40
1984
41
1988
36
1996
32
2000
26
What was the change in the percentage of young adults voting between the years 1980 and 2000?
Solution: Here is a time where we use the identification numbers 1980 and 2000, to look up the
values we need for the calculation. The percentage changed from 40 to 26 during that time, so
the change is:
40 − 26 = 14.
So, the percentage changed by 14%.
Exercise 5.7.1 Answer the following problems. Use addition or subtraction as appropriate.
Include appropriate units in your answers.
Difference Between:
1. Find the difference between 5 38 and 4 34 .
2. Find the difference between 12 12 and 7 87 .
3. Find the difference between
7
10
and 0.23.
118
4. Find the difference between
23
100
and 0.09.
Subtracted From:
5. What is 14,375.04 subtracted from 32,191.23?
6. What is 17.8% subtracted from 55%?
How Much More:
7. On Tuesday, the high temperature in Pacifica was 58.7◦ F. By Saturday, the high
temperature was 64.9◦F. How much hotter was it on Saturday than on Tuesday?
8. In a math class, a student needs at least 1250 points to earn an A grade. Letty already has
987 points. How many more points does she need in order to earn her A?
9. Will is 4 feet 3 inches tall. His brother Billy is 3 feet 10 inches tall. How much taller is Will
than Billy?
Change:
10. In 1993 the number of deaths in the U.S. due to the AIDS virus was 45,381. In 1999, the
number of AIDS deaths in the U.S. was down to 16,273. What was the change in number
of AIDS deaths?
11. In San Mateo County, during the 2005-2006 school year, 32.3% of 5th graders achieved 6
out of 6 fitness standards. The next year, 36.6% of 5th graders achieved 6 out of 6 fitness
standards. By what percent did it change?
12. The life expectancy of a person born in the U.S. was 73.7 years in 1980, and 77.0 years in
the year 2000. What was the change in life expectancy over that time?
Mixed Problems:
13. During the Iowa caucuses in 2008, Obama picked up 38% of the democratic delegates,
Edwards got 30%, and Clinton got 29%. What percent was left over for the other
candidates?
14. Jing-Jing was putting up a shelf in her closet. She needed a 2-by-4 that was 54 85 inches
long, but the lumber yard sold her a 2-by-4 that was 8 feet long. How much does she have
to cut off to make the board the correct length?
15. In preparing dinner for his guests, Billy needed 2 12 cups of chicken broth for the soup, and
another 43 cups of chicken broth for the pasta sauce. How much chicken broth does he
need to prepare the dinner?
16. Paul was working the concessions stand at the basketball game. A customer ordered a
popcorn for $3.50, a hotdog for $4.25, two sodas for $0.90 each, and a candy bar for $0.85.
The customer gave Paul a 20 dollar bill. How much change does Paul have to give the
customer?
119
17. If you are planning to attach a board that is
5 ′′
8
thick onto a board that is
3 ′′
4
thick,
(a) What is the longest screw that you can use?
(b) What is the shortest screw you can use?
(c) What is a reasonable range of screw lengths?
18. Find the difference between 22 81 and 18 21 .
19. The balance in Hector’s checking account was $134.72 before the deposit. After the
deposit, the balance was $473.28. How much was the deposit?
3
inches. How long is the branch
20. Geoff saws 13 78 inches off of a branch that measured 20 16
after Geoff makes the cut?
21. Find the difference between 48% and 29%.
22. Garret wanted to try to bake bread. The recipe called for 4 21 cups of flour, but when he
looked in the cupboard, he saw that he only had 2 81 cups left. How much more flour does
he need?
23. Julie bought gas that cost $3.35/gal. The next day, she passed by the station and saw that
the price had decreased by $0.12/ gal! What was the new price of gas?
120
5.8 Negative Numbers
Activity 5.2 Debt
Objective: To introduce concepts of negative numbers by using money and debt.
Materials: Bills of different denominations. Green bills for “positive” money, and red bills for
debt.
Group size: 2.
Section A
Imagine you have a bank account. For the following, count out green bills to illustrate the ideas:
1. If you make an initial deposit of $23, then withdraw $18, how much do you have left in
your balance?
2. Write this scenario out as a subtraction equation.
3. Starting with the deposit of $23 again, if you withdraw all $23, how much do you have
left in your balance?
4. Write this scenario out as a subtraction equation.
5. Starting with the deposit of $23 again, if you withdraw $25, how can you indicate what
has happened?
6. Write this scenario out as a subtraction equation. Use a negative sign in front of the
number if it is representing debt.
Section B
For the rest of the activity, we will use green bills to represent deposits, and red bills to
represent withdrawals.
1. Start with a balance of $0. Then, deposit $5 and deposit $7.
(a) Count out green bills for each deposit. What is the new balance?
(b) Write this scenario out as an addition equation.
2. Start with a balance of $0. Then, withdraw $3 and withdraw $6.
(a) Count out red bills for each withdrawal. What is the new balance?
(b) Write this scenario out as an addition equation. Remember to use a negative sign in
front of the number if it is representing a withdrawal or debt.
121
3. Start with a balance of $0. Then, deposit $1 and withdraw $1.
(a) Count out a green bill for the deposit and red bill for the withdrawal. What is the
new balance?
(b) Write this scenario out as an addition equation.
(c) Write this scenario out as an subtraction equation.
4. Start with a balance of $0. Then, deposit $4 and withdraw $4.
(a) Count out green bills for the deposit and red bills for the withdrawal. What is the
new balance?
(b) Write this scenario out as an addition equation.
(c) Write this scenario out as an subtraction equation.
5. Start with a balance of $0. Then, deposit $8 and withdraw $3.
(a) Count out green bills for the deposit and red bills for the withdrawal. What is the
new balance?
(b) Write this scenario out as an addition equation.
(c) Write this scenario out as an subtraction equation.
6. Start with a balance of $0. Then, deposit $6 and withdraw $10.
(a) Count out green bills for the deposit and red bills for the withdrawal. What is the
new balance?
(b) Write this scenario out as an addition equation.
(c) Write this scenario out as an subtraction equation.
7. Start with a balance of $0. Then, deposit $34 and withdraw $29.
(a) Write this scenario out as an addition equation.
(b) Write this scenario out as an subtraction equation.
8. Start with a balance of $0. Then, withdraw $48 and deposit $89. Write this scenario out as
an addition equation.
9. Start with a balance of $0. Then, withdraw $79 and deposit $65. Write this scenario out as
an addition equation.
122
10. Start with a balance of $0. Then, withdraw $15 and withdraw $71.
(a) Write this scenario out as an addition equation.
(b) Write this scenario out as an subtraction equation.
11. Start with a balance of $0. Then, withdraw $84 and deposit $68. Write this scenario out as
an addition equation.
12. Start with a balance of $0. Then, withdraw $21 and deposit $37. Write this scenario out as
an addition equation.
13. Start with a balance of $0. Then, withdraw $12, then withdraw $45, and finally withdraw
$17.
(a) Write this scenario out as an addition equation.
(b) Write this scenario out as an subtraction equation.
14. Start with a balance of $0. Then, deposit $99 and withdraw $110.
(a) Write this scenario out as an addition equation.
(b) Write this scenario out as an subtraction equation.
15. Explain how to add negative numbers together with negative numbers.
16. Explain how to add positive numbers together with negative numbers.
17. Explain how to subtract a larger number from a smaller number.
18. Describe the mistake in the following solution of the difference 87 − 125. Then, find the
real difference:
−
87
125
−62
123
5.9 Signed Numbers on a Number Line
When times are good, there is no need for subtraction:
Example 1: Mel earned $14 one day, and then earned another $12 the next day. How much did
he earn in all?
Solution: Mel’s total earnings is the sum:
14 + 12 = 26
So, Mel earned a total of $26.
We can illustrate this on a number line:
+14
0
5
+12
10
14
15
20
25
26
30
But as we all know, it’s quite likely Mel will have to spend too. This is where subtraction comes
in!
Example 2: Mel earned a total of $26, then spent $10. How much did he have left?
Solution: Mel had the difference between what he started with and what he spent, that is:
26 − 10 = 16
So, Mel had $16 left.
We can illustrate this as well on a number line:
+26
-10
0
5
10
15
16
20
25
124
26
30
Finally, there is the situation that many of us run into. We spend more than we have!
Example 3: Mel had $16 and then spent $20. How much did he have left?
Solution: Again, Mel had the difference between what he started with and what he spent, that
is:
16 − 20 = ???
We can try to illustrate this as well on a number line, but we run out of money. We subtract past
0!:
-20
+16
???
0
5
10
15
20
16
25
30
One way to express when you are in debt, that is, when you have spent more than you have, is
to use negative numbers. If we think of starting with $16 and then spending the $20 in steps,
first spend $16 (until we are down to $0), then continue to spend the other $4 which puts us in
debt $4, we can think of the problem as:
16 − 20 = 16 − 16 − 4
= 0−4
= −4
Or, to see it on a number line:
-16
-4
+16
-10
-5
-4
0
5
10
15
16
20
We use the negative symbol (which looks like a subtraction symbol) to represent that we are $4
in debt.
125
Exercise 5.9.1 Subtracting on a Number Line:
Illustrate the following subtractions on the given number lines.
1. (a) 8 − 3:
-15
-10
-5
0
5
10
15
-10
-5
0
5
10
15
-10
-5
0
5
10
15
-10
-5
0
5
10
15
-10
-5
0
5
10
15
-10
-5
0
5
10
15
(b) 3 − 8:
-15
2. (a) 12 − 5:
-15
(b) 5 − 12:
-15
3. (a) 17 − 11:
-15
(b) 11 − 17:
-15
126
Exercise 5.9.2 Subtracting on a Calculator:
Find the following differences using your calculator.
1. (a) 28.32 − 15.95 =
(b) 15.95 − 28.32 =
2. (a) 47, 320 − 39, 115 =
(b) 39, 115 − 47, 320 =
3. (a) 2, 485.50 − 1, 926.69 =
(b) 1, 926.69 − 2, 485.50 =
4. (a) 1312.55 − 741.09 =
(b) 741.09 − 1312.55 =
5. (a) 0.236 − 0.091 =
(b) 0.091 − 0.236 =
6. (a) 123, 456.789 − 98, 765.4321 =
(b) 98, 765.4321 − 123, 456.789 =
Exercise 5.9.3 Investigating Subtracting:
1. Fill in the following table. You are given a starting amount of money and an amount you
spend. You must write a subtraction problem that represents the situation, as well as the
ending amount. Use a negative symbol to indicate when you are in debt, that is, when
you have spent more than you start with. The first two lines are done for you.
Starting Amount Amount Spent Subtraction Problem
(in dollars)
(in dollars)
7
4
7−4
6
8
6−8
10
6
8
2
5
9
3
8
14.2
12.8
12.8
14.2
Ending Amount
(in dollars)
3
−2
2. Explain how to find the answer to a subtraction problem when the amount you are taking
away is larger than the amount you start with.
127
Exercise 5.9.4 Compute the following. Include appropriate units in your answers:
1. 28 − 17
2. 7 − 11
3. 132 − 256
4. 13.8 − 15.1
5. 17.5 − 128
6.
3
2
−
5
2
7.
4
3
−
2
3
8.
7
8
− 1 83
9. After earning $148.17, Paula spent $152.08. What is her net worth?
10. Collin was driving to Death Valley in California from Bakersfield. He looked at an
elevation sign in Bakersfield and it read, “Elevation: 505 feet”. By driving to Death Valley,
the elevation dropped by a total of 745 feet. What is the elevation in Death Valley?
11. On the same trip, Collin kept track of the temperature as well. While he was in
Bakersfield, the Patella Credit Union digital clock flashed a temperature of 55◦ Fahrenheit.
When he arrived in Death Valley, he looked at the thermometer on the gas station wall
where he filled up his tank, and he realized that the temperature had dropped by 68◦
Fahrenheit! What was the temperature at that Death Valley gas station?
12. At Skyline College in San Bruno, the temperature was 58◦ Fahrenheit at 10 am when Julie
arrived for her classes. When she left at 2 pm, the fog had lifted and it was 72◦ Fahrenheit.
How much had the temperature increased while Julie was on campus?
13. Does it matter what order numbers are subtracted? For example, is 8 − 5 the same as
5 − 8? Explain. Describe what is the same and what is different in two subtraction
problems where the only thing that has changed is the order that the numbers are written.
128
5.10 Order of Operations and Grouping Symbols
When the only operation is addition, the order and grouping of the numbers being added
doesn’t change the sum. We can use this fact to help make a sum of many numbers easier.
Example 1: Suppose the members of the Math Club decided to sell popcorn in the stands at the
college basketball game as a fund-raiser. At the end of the game, the members brought their
money collected together to be totaled. Michael collected $27, Min collected $29, Marleen collected $23, Maurice collected $36, and Maria collected $31. What was the total amount collected?
Solution: We could get the total by adding the numbers in the order given:
27 + 29 + 23 + 36 + 31 =
=
=
=
56 + 23 + 36 + 31
79 + 36 + 31
115 + 31
146
so that the total is $146.
We can make this easier, however, if we change the order that the numbers are written, and
then group them in easy pairs:
27 + 29 + 23 + 36 + 31 =
=
=
=
=
27 + 23 + 29 + 31 + 36
(27 + 23) + (29 + 31) + 36
50 + 60 + 36
110 + 36
146
which is the same total. We can add the numbers in any order we want to!
We have seen, however, that with subtraction the order matters. For example,
6−4=2
while
4 − 6 = −2
129
Changing the grouping, that is, changing which operation comes first, also matters. For
example,
7−3+1=4+1=5
while
7 − (3 + 1) = 7 − 4 = 3
Example 2: We just calculated that the Math Club collected $146 selling popcorn. Suppose,
however that the popcorn maker costs $36 to rent for the night, and the supplies (popcorn, oil,
salt, and bags to hold the popped popcorn) cost $19. Which of the following expressions can be
used to calculate how much profit the Math Club made?
Solution 1: 146 − 36 + 19 = 110 + 19 = 129, so the profit is $129,
or
Solution 2: 146 − (36 + 19) = 146 − 55 = 91, so the profit is $91?
The first solution says we start with the $146, take away $36 and then add $19, implying that
the $19 is extra money coming in rather than an expense. In the second solution, we total all of
the expenses before taking it away from the income. Which is correct?
Exercise 5.10.1 Find the following.
1. 10 − 3 + 5
2. 16 − 23 + 2
3. 21 + 27 − 18 + 5
4. 10 − (3 + 5)
5. 16 − (23 + 2)
6. 21 + (27 − 18) + 5
7. 21 + 27 − (18 + 5)
8. Write two different expressions for the following, then compute the value of each of the
expressions to show that they both answer the question to this problem: Gina was
keeping track of her balance in her checkbook. She started with a balance of $48.95, then
made a deposit of $120. She then wrote checks for $25, $17.22, and $36.50. What was her
final balance?
130
Notes:
131
6 Multiplication
6.1 Introduction to Multiplication
Activity 6.1 Money Activity: Repeating the Same Amount
Objective: To solidify concepts of multiplication by using money.
Materials: Bills of different denominations.
Group size: 3.
In this activity you will investigate finding a total when each part has the same amount. As
before, when each member takes money, take as few bills in each denomination as possible.
There are three members in each group.
Write the names of each group member:
Group member #1
Group member #2
Group member #3
1. Each of the three members in the group starts with $24
(a) Collect all three members’ money in one pile, and record the total number of bills of
each denomination in the table.
$10,000 $1,000 $100
$10
$1
$0.10
$0.01
h
h
h
h
h
h
h
(b) Now, do the necessary bill exchange in your total to keep the number of bills in each
denomination less than ten. Record the number of bills in each denomination as well
as your total value of all of the money.
$10,000 $1,000 $100
$10
$1
$0.10
$0.01 Total $
h
h
h
h
h
h
h
h
(c) How much would there be all together if there was $24 three times?
i. Write out how you could find this total using addition.
ii. Write out how you could find this total using multiplication.
132
2. Start over and this time each member in the group starts with $65.37.
(a) Collect all three members’ money in one pile, and record the total number of bills of
each denomination in the table.
$10,000 $1,000 $100
$10
$1
$0.10
$0.01
h
h
h
h
h
h
h
(b) Now, do the necessary bill exchange in your total to keep the number of bills in each
denomination less than ten. Record the number of bills in each denomination as well
as your total value of all of the money.
$10,000 $1,000 $100
$10
$1
$0.10
$0.01 Total $
h
h
h
h
h
h
h
h
(c) How much would there be all together if there was $65.37 three times?
i. Write out how you could find this total using addition.
ii. Write out how you could find this total using multiplication.
3. Describe the two different arithmetic operations that you could have used to find the
totals in questions (1) and (2).
4. Perform the following multiplications without using your calculator. You may use the
money to help.
a) (346.14)(4)
b) 6, 380.12 × 5
c) 278.29 · 7
d) 12(0.25)
5. Practice multiplying by powers of ten by thinking of money.
(a) What is the value of one hundred $5 bills?
(b) How much is ten $20 bills worth?
(c) How much is one hundred quarters worth?
(d) One dollar is equal to how many dimes? How many pennies?
6. How many $100 bills does it take to make $15,000?
7. How many $10 bills does it take to make $630?
8. How many dimes ($0.10) does it take to make $47?
9. How many pennies ($0.01) does it take to make $700?
10. How much is twice $36.19?
11. If you triple $45.25, how much do you have?
133
12. Complete the following multiplication table:
×
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
13. Find four different patterns in the completed table above. Include descriptive words such
as row, column, diagonal, consecutive, odd, even, increase, decrease.
(a) Pattern #1:
(b) Pattern #2:
(c) Pattern #3:
(d) Pattern #4:
134
14. Fill in the top-most row and left-most column with the numbers 1 through 12, but not in
order. Then fill out the multiplication table:
×
135
Exercise 6.1.1
1. Complete the multiplication table below using your calculator.
×
10
1, 000
100
10, 000
100, 000
a) 6
b) 25.3
c) 0.2
d) 0.007
2. What pattern do you see when multiplying numbers by a power of 10 like 10, 100, 1000,
etc.?
3. Is there anything different/special that you have to think of if the number has a decimal?
4. Complete the multiplication table below in your head.
×
10
1, 000
100
a) 8
b) 37
c) 409
d) 1471
e) 5040
f)
0.3
g) 0.047
h) 1.06
i)
14.109
j)
10.7083
136
10, 000
100, 000
5. Remember that multiplication is repeated addition so 3 × 5 = 5 + 5 + 5 = 15. Explain the
meaning of 10 × 67.
6. Negatives:
(a) Explain the meaning of 3 × (−5) using the idea of repeated addition.
(b) Find 2 × (−7) without using your calculator.
(c) Find 5 × (−4) without using your calculator.
(d) Find 10 × (−13) without using your calculator.
7. Fractions:
(a) Explain the meaning of 3 × 12 using the idea of repeated addition.
(b) Find 2 × 34 without using your calculator.
(c) Find 3 × 85 without using your calculator.
(d) Find 4 × 23 without using your calculator.
8. Multiply the following in your head:
a) 50 × 7 =
b) 30 × 80 =
d) 500 × 70 =
e) 4000 × 900 =
g) 1200 × 700 =
h) 30 × (−20) =
c) 60 × 110 =
f) 2000 × 7000 =
i) 700 × (−800) =
9. Describe a method for multiplying by numbers in which only the first digit is not a zero.
137
6.2 Significant Digits
Significant digits are used to help us indicate accuracy in numbers. When we use a measuring
tape to measure a block of wood, the measurements are only as accurate as the tape allows us
to make them. Most measuring tapes have only one or two decimal places of accuracy marked
on them since we are unable to see in finer detail. People have designed other tools to use when
they need more accuracy.
When we say “round a number to one significant digit” or to two significant digits, and so on,
we determine one, two, or however many significant digits in the following manner:
Reading from left to right, the first nonzero number and all the digits following it are
significant.
Example 1: The number 340.69 has five significant digits. The first significant digit is 3. The first
two significant digits are 3 and 4.
Example 2: The number 0.034069 also has five significant digits. The zeros on the left are not
considered significant for the purposes of rounding.
Example 3: In each of the following numbers, the digits that are in bold and their significant
digits are being indicated or the place value of the digit is indicated.
• 345.69
• 4572.28
• 34.59
• 0.0275
• 23.79
1 significant digit is in bold.
2 significant digits are in bold.
4 significant digits are in bold.
3 significant digits are in bold.
The tenth’s place is in bold.
What is the appropriate significant digit to round to in different numbers?
That depends on what kind of accuracy is needed in a particular situation.
• When measuring the radius of the earth, you need less accuracy than measuring the
dimensions of a room that you want to carpet.
• When you have to perform a series of calculations in a problem, it is a good idea to round
off at the end to get a more accurate solution, as opposed to estimating a solution, when
you need to round off before calculations. When you are rounding before to estimate, you
are making your problem easier to calculate, but be aware that you are sacrificing
accuracy.
• Your final answer is never more accurate than the information you were given.
138
The idea of rounding is to choose a level of accuracy, either a number of significant digits, or a
particular decimal place. Your rounded number is the closest number to your original that has
the level of accuracy desired. The following process can be used to round:
Process of rounding if the digit you are rounding to is to the left of the decimal point:
1. Mark the digit that you are going to round to.
2. Look at the digit to the right of the digit that you marked in step 1.
3. If the digit in step 2 is a 5, 6, 7, 8 or 9, add one to the digit that you marked in step 1 and
make all the other digits after that zeros.
4. If the number in step 2 is 0, 1, 2, 3 or 4, make all digits after the digit you marked in step 1
zeros. Any digits to the right of the decimal point and the decimal point itself are not
written.
Example 1:
Round 74392 to the nearest hundred.
Solution:
The digit marked for step one is the 3 in the hundreds place: 74392 .
The digit to look at for step 2 is the 9: 74392 .
Since 9 is 5 or more, we add 1 to the 3 to get a 4 in the hundreds place. The digits to the right are
changed to zeros. 74392 rounded to the nearest hundred is: 74400.
Example 2:
Round 276, 395, 661 to three significant digits.
Solution:
The third significant digit is in the millions place, so the digit marked for step 1 is the 6 in the
millions place: 276,395,661 .
The digit to look at for step 2 is the 3: 276,395,661 .
Since the digit marked in step 2 is 3 which is less than 5, we just change 3, 9, 5, 6, 6 and 1 to zeros.
276,395,661 rounded to the nearest million is 276,000,000 which is often written 276 million.
139
Example 3:
Round 38,995,623.27 to the nearest ten-thousand.
Solution:
The digit marked for step 1 is the 9 in the ten-thousands place: 38,995,623.27 .
The digit to look at for step 2 is the 5: 38,995,623.27 .
Since the 5 in step 2 is 5 or more, we add 1 to the 9. But this is tricky! A better way of looking at
it is that we are adding 1 to the rest of the number, that is, adding 1 to 3899 to get 3900. Write the
rest of the digits as zeros, leaving out the decimal point and the digits to the right of the decimal
point. This is best seen for many by looking at the number line. If we count by 10,000’s starting
just below our original number, we get the following:
38,980,000
38,990,000
39,000,000
39,010,000
39,020,000
Our number is right in between 38,990,000 and 39,000,000. Notice, it is a little closer to the 39
million. By our procedure, the digit 5 indicates that we round up:
38,995,623.27
b
38,980,000
38,990,000
39,000,000
Exercise 6.2.1 Round the numbers as indicated.
1. Round 26,347.92 to the nearest thousand.
2. Round 26,347.92 to one significant digit.
3. Round 26,347.92 to the nearest ten.
4. Round 499,728.50 to the nearest thousand.
5. Round 499,728.50 to one significant digit.
6. Round 499,728.50 to the nearest million.
7. Round 499,728.50 to the nearest hundred.
140
39,010,000
39,020,000
Process of rounding if the digit you are rounding to is to the right of the decimal point:
1. Mark the digit that you are going to round to.
2. Look at the digit to the right of the digit that you marked in step 1.
3. If the digit in step 2 is a 5, 6, 7, 8 or 9, add one to the digit that you marked in step 1 and
drop all the other digits after that.
4. If the number in step 2 is 0, 1, 2, 3 or 4, drop all digits after the digit you marked in step 1.
Example 1:
Round 325.7496 to the nearest hundredth.
Solution:
The digit marked for step one is the 4 in the hundredth’s place: 325.7496 .
The digit to look at for step 2 is the 9: 7325.7496 .
Since 9 is 5 or more, we add 1 to the 4 to get a 5 in the hundredth’s place. The digits to the right
are dropped. 325.7496 rounded to the nearest hundredth is: 325.75.
Example 2:
Round 12.68531 to five significant digits.
Solution:
The fifth significant digit is in the thousandth’s place, so the digit marked for step 1 is the 5 in
the thousandth’s place: 12.68531 .
The digit to look at for step 2 is the 3: 12.68531 .
Since the digit marked in step 2 is 3 which is less than 5, we just drop the 3 and the 1. 12.68531
rounded to five significant digits is 12.685.
141
Example 3:
Round 439.9957 to four significant digits.
Solution:
The fourth significant digit is in the tenth’s place, so the digit marked for step 1 is the 9 in the
tenth’s place: 439.9957 .
The digit to look at for step 2 is the 9 in the hundredth’s place: 439.9957 .
Since the digit marked in step 2 is 9 which is 5 or more, we round up to 440.0. We need the extra
0 in the tenth’s place to indicate how accurately we have rounded.
Rounding to five significant digits would have required an extra zero, and the answer would
have been 440.00.
Example 4:
Round 1.38562 to three decimal places.
Solution:
The third decimal place is the thousandths place, so the digit marked for step 1 is the 5: 1.38562 .
The digit to look at for step 2 is the 6 in the ten-thousandths place: 1.38562 .
Since the digit marked in step 2 is 6 which is greater than or equal to (≥) 5, we round up to
1.386.
Exercise 6.2.2 Round the numbers as indicated.
1. Round 7.9265 to the nearest tenth.
2. Round 7.9265 to the nearest hundredth.
3. Round 0.0798 to one significant digit.
4. Round 0.2495 to one significant digit.
142
Activity 6.2 Estimating Products
Objective: To introduce the concept of estimation of products.
Materials: None.
Group size: 3-4.
1. Use technique 1 then technique 2 to estimate the sum 1785.33 + 473
2. Find the product 1785.33 × 473 using your calculator.
3. Estimate the product 1785.33 × 473 by rounding each factor to the largest place value of
the larger number before multiplying in your head.
(a) Is the estimate close to the actual value? Explain.
(b) Is the estimate easy to calculate without a calculator or pencil and paper? Explain.
(c) Is this the “best” method to use for estimating products? Explain.
4. Estimate the product 1785.33 × 473 by rounding each factor to the largest place value of
the smaller number before multiplying in your head.
(a) Is the estimate close to the actual value? Explain.
(b) Is the estimate easy to calculate without a calculator or pencil and paper? Explain.
(c) Is this the “best” method to use for estimating products? Explain.
5. Estimate the product 1785.33 × 473 by rounding each factor to one significant digit before
multiplying in your head.
(a) Is the estimate close to the actual value? Explain.
(b) Is the estimate easy to calculate without a calculator or pencil and paper? Explain.
(c) Is this the “best” method to use for estimating products? Explain.
6. Suggest a strategy for estimating a product.
7. Use your new strategy to estimate the following products:
(a) 492 × 28.7
(b) 65, 328.17 × 250
(c) 3, 255.28 × 0.45
143
Exercise 6.2.3 Estimate, without using a calculator, the products below by first rounding each
number to one significant digit. Include appropriate units with your answers.
Example:
236
x 58
200
x 60
12000
1)
4)
7)
67
× 115
451
× 3501
169.75
× 225
2)
5)
8)
367
× 639
47.95
× 23
295, 734, 134
×
36, 729
3)
548
× 971
6)
25.3
× 54
9)
×
438
3.35
10) The U.S. population is approximately 295,734,134. The average annual salary in the U.S. is
$36,729. Estimate the total amount of money earned by the U.S. population each year.
11) On a road trip to L.A., you used 17.8 gallons of gas. If gas cost $3.35/gal, how much did the
gas for the trip cost?
12) Registration for the math conference was $110 each. If 336 math teachers registered, how
much money was collected in registration fees?
13) 47 × 32 87
14) 229.7 × 5 34
15) 59 × (−38)
16) 280 × (−16)
17) 3 × 22.4%
18) Re-do problems 1-12 exactly using your calculator. Note the accuracy of your estimates.
144
6.3 Multiplication is Repeated Addition
Exercise 6.3.1 For the following problems, figure out the answer in two ways. Show your
steps. Include appropriate units with your answers.
a) Using repeated addition.
b) Using multiplication.
1. The math club was ready to buy t-shirts. Each shirt cost $12. If there are 8 members in the
club who want to buy shirts, what is the total cost for the shirts?
(a) Repeated addition:
(b) Multiplication:
2. While driving to L.A., Elissa averaged 60 miles per hour. She traveled for 3 hours at this
speed before taking a rest break. How far had she traveled before taking the break?
(a) Repeated addition:
(b) Multiplication:
3. A classroom has 5 rows of desks. Each row has 8 desks in it. How many desks are in the
classroom?
(a) Repeated addition:
(b) Multiplication:
4. Max is building a fence. Each board is 7′′ wide, and he has 64 boards. How long is the
longest fence he can build? (See picture)
7"
(a) Repeated addition:
(b) Multiplication:
145
6.4 Introduction to Area
In the following activity we will see that area is the measure of how many squares fit in an
enclosed flat space. In the activity, you will use square pieces that measure 1 inch on each side,
so the units of the measurements are called square inches. One can measure area with square
inches, square centimeters, square miles, or even square units, as long as the length of the units
are defined.
1 inch
1 centimeter
1 square inch
a “unit” of one size
1 unit
1 square unit
1 unit
1 square unit
a “unit” of another size
146
1 square centimeter
Activity 6.3 Area Activity: Just a Bunch of Squares
Objective: To introduce concepts of area with hands-on square inch tiles.
Materials: Square inch tiles.
Group size: 3-4.
Using your square inch tiles in your groups,
1. Draw two different shapes that each have an area of 5 square inches.
2. Estimate the area of the cover of a binder or notebook or portfolio.
3. Estimate the area of the top of a desk.
4. Estimate the area of the back of a calculator.
5. Estimate the area of the classroom floor. (Hint: The tiles on the floor are the same size.)
6. Estimate the area of the following rectangle.
7. By using you ruler instead of the square inch tiles, find the area of the following rectangle
to the nearest square inch. Describe your procedure.
147
6.5 Area Formula For a Rectangle
In order to find the area of a shape without using square tiles and counting, it is important to
understand how repeated addition or multiplication is involved in the area of a rectangle. If
you have a very thin rectangle that is only 1 unit wide, then finding out the area in square units
is simply counting how many squares are in that one row:
12 square units
The wider the rectangle gets, the more rows you have.
12 + 12 =
12 two times =
2 × 12 =
24 square units
12 + 12 + 12 =
12 three times =
3 × 12 =
36 square units
12 + 12 + 12 + 12 =
12 four times =
4 × 12 =
48 square units
So, to figure out the area of a rectangle, you just need to know how many squares in each row,
and how many rows. To find out how many squares in each row, measure how many units fit
across the rectangle, and to find out how many rows fit, measure how many units fit down the
rectangle.
Note: Is a square a rectangle?
In short, yes. The definition of a rectangle is a four sided shape with four angles that have the
same measure. The definition of a square is a four sided shape where all four angles have the
same measure and all four sides have the same measure.
Example 1:
Find the area in square units of the following rectangle, if each unit is
is .
148
, and each square unit
Solution: The rectangle measures 18 units across and 6 units down.
So, the area of the rectangle is 18 + 18 + 18 + 18 + 18 + 18 = 6 × 18 = 108 square units.
Exercise 6.5.1 Find the perimeter and area of the following rectangles.
• Measure the perimeters in units.
• Measure the areas in square units.
• Each unit is
1.
3.
, and each square unit is
.
2.
4.
5.
149
Find the perimeter and area of the following triangles.
• Measure the perimeters in units.
• Measure the areas in square units.
• Each unit is
, and each square unit is
.
6. (Look at problem #1)
7. (Look at problem #4)
8. (Look at problem #5)
9. Describe, using pictures and/or words and/or math symbols, how you can calculate the area
of a rectangle.
10. Describe, using pictures and/or words and/or math symbols, how you can calculate the
area of a triangle.
11. Jason finds the perimeter of the triangle:
by completing the rectangle:
and adding all the sides to get 46 units, then cutting 46 in half to get 23 units. Persuade Jason
that he can’t be right, then explain what he should do instead.
150
Exercise 6.5.2 For the following shapes, find the areas in square units given that:
• Each unit is the same length as the following line:
• Each square unit is the size of the following square:
1. Now I’ve got you, you bastard!
2. Now I’ve got you, you bastard!
3. Now I’ve got you, you bastard!
151
4. Now I’ve got you, you bastard!
5. For this one, find a reasonable over-estimate, and a reasonable under-estimate.
152
6. For this one, find a reasonable estimate for the area in square units. Then, using your
answer and assuming that each square unit represents 623 square miles in real life,
calculate the approximate area of California in square miles.
153
6.6 Pythagorean Theorem and Square Roots
Exercise 6.6.1
1. Find the area of a square with side 15.
15
2. Find the areas for all the squares with sides 1 through 14:
(1) :
(6) :
(11) :
(2) :
(7) :
(12) :
(3) :
(8) :
(13) :
(4) :
(9) :
(14) :
(5) :
(10) :
3. (a) Find the length of the side of a square
with area 121:
(b) Find the length of the side of a square
with area 81:
side =
side =
4. If the area of a square is 60 square units, the side of the square is between
what two consecutive natural numbers?
A = 60
and
5. If the area of a square is 30 square units, the side of the square is between
what two consecutive natural numbers?
A = 30
and
6. If the area of a square is 20 square units, the side of the square is between
what two consecutive natural numbers?
A = 20
and
154
7. Use square units like the one shown at the top of the diagram
to help you answer these questions.
B
(a) Find the area of the square, ABCD.
A
(b) How long is the side AB?
C
D
Squaring:
The area of a square, like the area of any rectangle, comes from the
product of its height and width. The area of the square to the right is
A = s × s. We abbreviate this A = s2 and say the area is given by s –
“squared”. Do you see where the name comes from?
A = s2
s
s
Square Root:
The reverse process also has a name.
If the area of a square is 100, the side is 10 because 102 = 10 × 10 = 100.
√
The length of the side is called the “square root” (written
) of the
area.
For √
the example above the side of a square with area 100 is given by
s = 100 = 10.
√
In general we say s = A.
Example: If the area of a square is 60, the side is s =
is between 7 and 8.
155
√
A
s=
√
s
A
60. We know from # 4 that this number
Exercise 6.6.2
1. What is the length of the side of a square with area 90?
What two consecutive natural numbers is this value between?
and
2. Show approximately where these numbers appear on the number line below. Do not use a
calculator.
√ √ √ √ √ √ √
2, 3, 5, 10, 19, 33, 40.
0
1
2
3
4
5
6
3. Use the figure to the right to answer these questions.
Q
(a) Find the exact area of the square, PQRT.
P
(b) How long is the side RT?
R
T
156
Activity 6.4 Activity: Finding Exact Area
Objective: To find the exact area of rectangles indirectly using square roots. To introduce
Pythagorean Theorem, the concept of square roots, and square root notation.
Materials: A pencil and straight edge.
Group size: 4.
Find the length of AB without using a ruler.
B
A
157
Here are two ways of find the length of a line on the grid page.
One way to find the length of a line.
1. Starting with AB,
2. Make a square with side AB:
3. Find the area of the square:
R
R
B
B
B
A
A
A
Area =
4. Then find the length of the
side by taking the square root.
AB =
Another way (a shortcut to finding the area)
5. Make a square with AB.
6. Find the areas of P and Q.
Area P:
, Area Q:
−→ How do the areas relate to
the area of R?
R
B
R
B
A
P
A
The area of square R is the sum
of the areas of P and Q.
Q
7. Now find the length of the side by taking the square root.
The length of AB is the same as AB =
√
Area P + Area Q
158
AB =
The Pythagorean Theorem
In the previous activity you saw how we use area to find the length of a line segment. The
shortcut in step 6 is the key to what is called the Pythagorean Theorem, one of the oldest and
most useful relationships in numbers. In order to describe it completely we need some
definitions.
Hypotenuse
In a right triangle (a triangle with a 90◦ angle) the two
sides that form the 90◦ angle are called the Legs.
L1
H
The side opposite the 90◦ angle is called the HyL2
potenuse.
Legs
From the previous activity we saw that the three sides
of the right triangle are related by (L1 )2 + (L2 )2 = H 2
2
A=
The Pythagorean Theorem says that:
The area of the square made from the hypotenuse is
equal to the sum of the areas of squares made from
the two legs.
H
L1
H
L2
2
A= (L )
2
Equivalently, we take the three areas:
The area of the square made from the hypotenuse:
The area of the square made from one leg (a):
The area of the square made from the other leg (b):
c
and according to Pythagoras,
a
b
+
=
Which means:
The length of hypotenuse is c =
r
+
Example: Find the length of the hypotenuse of the triangle below.
H
12
5
H 2 = 52 + 122
H 2 = 25 + 144
H 2 = 169
√
H = 169
H = 13
159
The hypotenuse is 13.
2
A= (L )
1
Exercise 6.6.3 Find the exact value of the indicated lengths:
2. Find the length of the
hypotenuse.
1. Find the length of the
hypotenuse.
3. Find the length of the
hypotenuse.
H
10
c
9
24
x
24
12
c=
H=
7
x=
4. Find the length of the
hypotenuse.
5. Find the length of the
side.
10
e
6. Find the length of the
side.
9
7
6
6
d
10
f
f=
d=
e=
Lad
der
7. A 24 foot ladder is leaned against a building. If the base of the ladder is 8 feet from
the base of the building, how high will the
ladder reach?
8
160
8. Find the perimeter of a right triangle
if the legs are 8cm and 15cm long.
9. Find the perimeter of this triangle.
(a) Give the exact computation for the perimeter:
(b) Without using a calculator, give the approximate computation
for the perimeter:
.
161
.
Exercise 6.6.4
Complete the times tables below.
× 2
4
3
11
15
1.
8
12
56
×
5
4
6
36
2.
10
×
40
18
20
20
135
50
60
3500
3.
1600
7200
100
162
6.7 The Distributive Property
Vocabulary: The numbers being multiplied together in a multiplication problem are called the
factors, and the result of the multiplication problem is called the product.
We have seen how to multiply two numbers together up to 12 × 12 = 144. We’ve also seen that
to estimate the product of two numbers that are bigger, we just round each number to one
significant digit, then use our knowledge of multiplication by powers of 10. For example, we
estimate the product 32, 429.8 × 192, 487.103 as 30, 000 × 200, 000 = 6, 000, 000, 000.
The next step involves finding a way to get an exact product of numbers with two or more
digits. We need the power of the Distributive Property. Let’s say you want to find the product
4 × 26. We can see the four rows of 26 squares each in the following picture:
The distributive property just says that we can take 4 × 26 and think of it as 4 × 20 plus 4 × 6 as
seen in the following picture:
4 × 20
+
4×6
So, 4 × 26 = 4 × 20 + 4 × 6 = 80 + 24 = 104.
Here’s another one. Let’s say you want to find the product 7 × 15. We can see the seven rows of
15 squares each in the following picture:
The distributive property just says that we can take 7 × 15 and think of it as 7 × 10 plus 7 × 5 as
seen in the following picture:
7 × 10
+
7×5
So, 7 × 15 = 7 × 10 + 7 × 5 = 70 + 35 = 105.
163
Let’s try a harder one. 16 × 24 needs two steps of distributive property. First, we see 16 × 24 as
10 × 24 plus 6 × 24 as seen in the following picture:
10 × 24
=
16 × 24
+
=
6 × 24
We already know that 10 × 24 = 240, so we just have to break up 6 × 24 into 6 × 20 plus 6 × 4.
10 × 24
+
10 × 24 = 240
=
6 × 24
+
6 × 20 = 120
So, 16 × 24 = 10 × 24 + 6 × 20 + 6 × 4 = 240 + 120 + 24 = 384.
164
+ 6 × 4 = 24
Exercise 6.7.1 For the following pictorial representations of the distributive property, fill in
the missing factors to make the products reflect the pictures.
1.
=
×
×
+
×
2.
=
×
×
+
165
×
3.
=
×
×
×
+
4.
63
8
=
×
3
×
+
166
×
5.
×
(a)
=
(b)
=
+
×
×
×
(c)
+
×
+
167
×
Exercise 6.7.2 In the previous exercise, you used pictures and distributive property to break
up a difficult product into the sum of simpler products. In this exercise, write the math that is
illustrated by the pictures in the problems from exercise 6.7.1 to find the value of the original
products. The first one is done for you:
1. 5 × 14 = 5 × 10 + 5 × 4 = 50 + 20 = 70
2. h
3. h
4. h
5. h
Exercise 6.7.3 Use distributive property to find the products.
Example 1:
Solution:
37
4
30
7
120
28
4
4 × 37 = 4 × 30 + 4 × 7
4 × 37
= 120 + 28 = 148
Example 2:
Solution:
75
70
1400
5
100 20
28
560
40
8
28 × 75 = 20 × 70 + 20 × 5 + 8 × 70 + 8 × 5
28 × 75
= 1400 + 100 + 560 + 40 = 2100
168
1.
2.
54
73
6
8
6 × 54
8 × 73
3.
4.
21
72
15
38
15 × 21
38 × 72
5.
84
36
36 × 84
6. Multiply the numbers below by drawing a rectangle and breaking it into pieces like in the
previous problems
(a) 44 × 27
(b) 18 × 57
169
(c) 29 × 72
Exercise 6.7.4 Using distributive property, re-write the following products as the sum of
simpler products that you can figure out in your head. Then, add them together, to find the
answer to the original product. You may use pictures to help if you like, but you are not
required to.
1. 75 × 8
2. 52 × 14
3. 107 × 32
4. 530 × 61
5. 93 × 87
6. Compare and contrast (discuss what is similar and what is different) the distributive
property method of multiplication with the standard “vertical method” of multiplication
that you know from your past. Use the product 38 × 74 as the example to use in your
discussion.
Find More Digits
Exercise 6.7.5 Fill in the blanks to complete the multiplication problems below.
1.
2.
3.
2
×
2
8
4
× 2
2
8
1
170
4
7
8
5
× 3
9
1 6
0
6
2
Exercise 6.7.6 Include appropriate units in your answers.
1. Estimate the following answers by first rounding the numbers to one significant digit.
(a) How far do you travel if you drive for 6 hours at 62 mph?
(b) How much will you earn if you are paid $12.45 per hour for 38 hours?
(c) How much will you pay for a year of service if you pay $76 per month for your cell
phone?
2. Repeat number 1 parts (a) and (c), but calculate the answers exactly this time.
3. How much carpet should Duane buy to cover a floor that is 12 feet 2 inches wide by 13
feet 4 inches long? Explain your reasoning.
′′
4. Maria is building a fence using boards that are 7 58 wide. If she has 55 boards, give a
reasonable over–estimate and a reasonable under–estimate of the longest fence she can
build.
5. Estimate the area and perimeter of the following right triangles with the given side
lengths.
(a) Side a: 8.9 cm.
Side b: 14.2 cm.
(b) Side a: 25 mi.
Side b: 123.5 mi.
Side c: 16.76 cm.
Side c: 126 mi.
6. How far is it (in feet) from this classroom to the Learning Center in building 5? (Hint:
First figure out how many of your strides there are, then use the length of your stride in
feet to calculate the number of feet.)
171
Exercise 6.7.7 Tricky Math Problems using the Distributive Property
The Distributive Property can be used to make a product into a sum, like we’ve seen, or a
difference. There are times when making a difference is easier.
Example: Conchi is buying new swivel chairs for her boutique business. She has to buy chairs
for all 6 stations, and each chair costs $197. How much will all of the chairs cost?
Solution: We can visualize this as the difference of two areas as follows:
197
197 × 6 = ?
6
=
197
3
197 × 6 = ?
18
6
200 × 6 = 1200
So, 197 × 6 = 200 × 6 − 3 × 6 = 1200 − 18 = 1182, so, the chairs will cost $1182.
For the following,
a) First estimate the answers by rounding the factors to one significant digit before multiplying.
b) Use distributive property to find the exact answer.
1. Find the product: 895 × 8
2. Find the product: 3998 × 27
3. Find the product: 48 × 6 (Can be done with addition OR subtraction.)
4. Find the area of a plot of land in the shape of a rectangle that measures 94 meters by 28
meters.
5. Joey wanted to take his buddies out to lunch. The lunch special where they like to go
costs $6.93 for each lunch and includes tax and tip. To feed all of his friends, he needs to
buy 6 lunches. How much will it all cost him?
172
6.8 Traditional Process of Multiplication
In the following activity we will see that the traditional method of multiplication is the same as
the distributive property, just without seeing all of the details. Recall the traditional method of
multiplying 36 × 72:
1
×
4
2 1
2 5
7
3
3
6
9
2
6
2
2
Each of the numbers written down are part of the following distributive property solution:
70
2
70 × 30 = 2100
60
70 × 6 = 420
12
30
6
36 × 72 = 420 + 12 + 2100 + 60 = 2592
The traditional process “keeps track” of the place values of the one digit numbers so that when
you multiply, if you put your answers in the right columns, the answer comes out correct.
Exercise 6.8.1 Compute the following by hand showing all of your steps using (a) the
traditional method, and (b) the distributive property.
1. 28 × 5
(a) Traditional:
(b) Distributive Property:
2. 62 × 34
(a) Traditional:
(b) Distributive Property:
3. 107 × 49
(a) Traditional:
(b) Distributive Property:
4. 55 × 256
(a) Traditional:
(b) Distributive Property:
5. 3006 × 87
(a) Traditional:
(b) Distributive Property:
173
6.9 Factoring
Activity 6.5 Factoring Activity: Making Rectangles with a Given Area
Objective: To introduce concepts of factoring by using rectangle areas.
Materials: Graph paper.
Group size: 3-4.
For a given area, there may be several rectangles with different dimensions that have that area.
Example:
For an area of 16 square units
there are three different rectangles:
1 × 16:
2 × 8:
4 × 4:
• Notice that 8 × 2 and 16 × 1 are the same as the first two we already have, just sideways.
• Also, recall that since a square is a rectangle, the 4 × 4 counts.
1. Repeat this example for rectangles with areas of 2 through 12 (there are eleven different
areas, so 11 different problems). Use graph paper to help you draw all of the different
rectangles that have each given area.
2. Review your areas and their drawings and list all of the areas for which there was only one
rectangle possible.
174
Factors
When two numbers are multiplied, the result is called their product and the two original
numbers are called factors of their product. For example, the 3 and 5 in 3 × 5 = 15 are called
factors of 15. The process of working backwards to split a number into the product of its factors
is called factoring. You may have noticed in the previous activity that by finding the
dimensions of a rectangle with a particular area, you were finding factors of that area. The
factors of 16, for example, are exactly those numbers we used in the three rectangles we drew
for the previous example. We say that 16 has five factors: 1, 2, 4, 8, and 16.
Numbers with only two factors, the number itself and 1, are called prime numbers. The first five
primes are 2, 3, 5, 7, and 11. Please take a moment and think of where you have seen these
numbers before. Notice that the number 1 is omitted from this list even though it fits the
criteria. This is done by agreement among mathematicians because it makes things less
complicated to leave 1 out.
Exercise 6.9.1
1. List the first 10 prime numbers.
2. Notice that in the example for the previous activity, each of the rectangles has a different
perimeter. Of the rectangles with area 16 square units, the one with a perimeter of 34 units
has dimensions 1 by 16. Find the dimensions of the rectangle that satisfies each
description below.
(a) Area 14 square units, perimeter 30 units.
(b) Area 20 square units, perimeter 24 units.
(c) Area 24 square units, perimeter 20 units.
3. What are the dimensions of the rectangle with the smallest perimeter that has an area of
36 square units? Include all of your reasoning in your solution.
4. Of all the rectangles with a perimeter of 20 units, what are the dimensions of the one with
the smallest area? Include all of your reasoning in your solution.
5. Repeat the previous problem if you are allowed to use decimals.
6. Carla made $320 at work. If she worked a whole number of hours and her hourly wage is
a whole number (no fractions), what possible hours and hourly wages could she have
had?
7. If Raymond drove 220 miles, list the possible times and speeds he could have had.
(Assume both are whole numbers).
175
Taxman
Work in pairs as a team to beat the Taxman. You will be given a number to work with for each
game, such as 17. First write all the whole numbers from one through your number (in this case
1-17) like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
Then, make a table like this:
Team Taxman
with columns for the Taxman and the Team. To start the game, the team picks one of the
numbers on the list, writes it in their column, and crosses it off the list. Next, they take all of the
divisors of that number that have not already been crossed off, and write them in the taxman’s
column. They then cross those numbers off the list.
For example, let’s say the team picks 10 first. The divisors of 10 are 1, 2, 5, and 10, but the team
already has 10, so the Taxman gets 1, 2, and 5. The list and table after this round look like this:
/
1, /
2 , 3, 4, /
5 , 6, 7, 8, 9, 10
/ , 11, 12, 13, 14, 15, 16, 17
Team Taxman
10
1
2
5
At each round, the team picks a number still on the list, but there is one rule:
The Rule: Every number the team picks must have at least one divisor that has not been crossed
off the list yet. That is, the Taxman must get something on each round.
For example, the team cannot now pick a 3, since its only factor is crossed off the list. (What
other numbers can’t be picked?)
176
Let’s say the next number that the team picks is 12. The divisors of 12 are 1, 2, 3, 4, 6, and 12,
but 1 and 2 are already crossed off, and the team just picked the 12. The list and table after this
round look like this:
/
1, /
2, /
3, /
4, /
5, /
6 , 7, 8, 9, 10
/ , 11, 12
/, 13, 14, 15, 16, 17
Team Taxman
10
1
2
5
12
3
4
6
When the team can pick no more numbers, the Taxman gets all the numbers that are left. The
numbers in each column are then added, to give the Team’s score, and the Taxman’s score.
The score at this point is the team has 22, and the taxman has 21.
After the next two rounds, it might look like this:
/
1, /
2, /
3, /
4, /
5, /
6 , 7, /
8 , 9, 10
/ , 11, 12
/, 13, 14
/, 15, 16
/, 17
Team Taxman
10
1
2
5
12
3
4
6
14
7
16
8
Since none of the remaining numbers have factors remaining, the taxman gets them all!
Team
10
12
14
16
Taxman
1, 2, 5
3, 4,6
7
8
9, 11, 13, 15, 17
The final score is the team has 52, and the taxman has 101. The taxman wins!
177
Activity 6.6 Playing Taxman
Objective: To practice finding factors of numbers and to design a strategy for winning a game.
Materials: Paper, pencil.
Group size: 2-4.
1. Play Taxman with the numbers 1 through 13.
2. Play Taxman with the numbers 1 through your choice (pick a number between 20 and 30).
3. What strategy can you use to beat the taxman? What number should you pick first? How
should you decide what numbers to pick before other numbers in order to win?
178
6.10 Prime Factorization
We have seen that to factor a whole number is to write it as a product of other numbers. For
example, we can factor 12 as 3 × 4. This factorization is not unique; there are other ways to factor
12, such as 2 × 6. The prime factorization of a whole number is a way to factor a given number
uniquely so that the factors are as simple as possible. For example, the prime factorization of 12
is 2 × 2 × 3. It is a factorization of 12 since we have written 12 as a product of factors, and it is
the prime factorization of 12 since 2 and 3, the factors, are each prime.
To find the prime factorization of a whole number like 18:
1.
Write the number at the top of your
workspace.
2. Think of any factorization of the number
where each of the factors are smaller than the
number.
18
like 2 × 9
(3 × 6 works just as well, but
1 × 18 is no good since it doesn’t
make the 18 any simpler.)
3. Write the factors you thought of below the
original number, then draw lines from the original number to each of the factors. Some people
think these lines look like branches of a tree, so
this process is called making a factor tree.
18
2
4. Look at the each of the numbers at the bottom
of the lines. If the number is prime, you are done
with that branch. Circle the prime factor.
9
18
2
9
18
5. If the number is not prime, think of any factorization of that number, and write it below,
connecting with lines.
2
9
3
3
18
2
6. Continue until all the factors are prime.
9
3
7. The prime factorization is the product of all
the circled primes.
179
18 = 2 × 3 × 3
3
Exercise 6.10.1
For each of the whole numbers between 2 and 25, write the word prime if the number is prime,
and find the prime factorization of the number if the number is composite. The first few are
done for you:
• 2 prime
• 3 prime
• 4= 2×2
• 5 prime
• 6 through 25
6.11 Equivalent Fractions Revisited
One way to look at finding a simpler, equivalent form of a fraction is to remove lines from the
picture, making fewer, larger pieces. For example, by removing the middle horizontal line in
the 24 picture, it becomes 21 :
2
4
1
2
If we are simplifying this way, it matters how we draw the fraction. We just drew fourths by
making the four pieces in a 2 × 2 grid. If we had chosen a 1 × 4 grid instead, we would have
had to remove two lines to make our equivalent fraction:
2
4
1
2
180
Exercise 6.11.1
• Draw a rectangle representing the given fraction.
• Draw another rectangle with the same amount shaded as the original picture, but with a
line or some lines removed.
• Write the equivalent fraction.
• Hint: use graph paper to help.
Example:
6
9
Solution:
6
9
=
2
3
1)
2
6
2)
2
8
3)
5
10
4)
6
10
5)
6
8
6)
8
12
7)
2
12
8)
3
12
9)
6
4
10)
10
8
181
6.12 Factorization and Equivalent Fractions
We saw in chapter 2 that fractions can be written in several equivalent ways. For example, we
8
. We have seen equivalence
have seen that 48 has many equivalent representations like 42 and 16
represented on a ruler (the fourth eighth inch mark is the same place as the second quarter inch
mark etc.) and with picture representations like:
1 whole
4
8
2
4
8
16
In all of these, once the size of the whole is established, we can see that all of these fractions are
equivalent to 12 .
Now that we understand factoring and factorization, we can investigate a pattern that occurs
with equivalent fractions. Starting with 84 , we have the whole broken into 8 pieces, and 4 of
them are shaded. If we remove two of the lines in the drawing, so that the whole is cut into
only half as many pieces (denominator = 4), then the number of pieces shaded is also cut in half
(numerator = 2). By looking at factored forms of the numerator and denominator of 48 , we can
see a common factor of 2 which means we can easily cut each in half.
4 = 2×2 = 2×2 = 2
8
2×4
2×4
4
182
Similarly, if we start with 48 and double the number of pieces in the whole which also doubles the
number of pieces shaded, we see both the numerator and denominator of 84 is doubled:
4 = 4×2 = 8
8
8×2
16
To get to simplest form, we start with 48 and replace 4 and 8 with their prime factorizations.
Then look for all common factors in the numerator and denominator, and remove them. We put
1 as a factor of 4, so that something is left in the numerator:
4 = 2×2 = 1×2×2 = 1
8
2×2×2
2×2×2
2
Exercise 6.12.1 For the given fractions, do the following:
• Draw a picture representing the given fraction.
• Find the prime factorization of the numerator and denominator.
• Write an equation that is, “given fraction = fraction with numerator and denominator
replaced with their prime factorizations”.
• Circle all factors that are common in the numerator and denominator. You should have
exactly the same number of factors circled in the top as you do in the bottom.
• Remove the common factors that you circled (exactly one from the numerator for each
one from the denominator).
• Write the newly found simplest form of the original fraction.
• Draw a picture representing the simplest form.
Example: Given fraction is
6
8
Solution:
6
8
The picture representation is:
The prime factorization of 6 is 2 × 3, and the prime factorization of 8 is 2 × 2 × 2
183
The equation looks like:
6 = 2×3
8
2×2×2
Circle the common 2 and then remove them to get:
6 = 2×3 = 3
8
2×2×2
4
The picture for
3
4
lined up with the picture for
6
8
so that the equivalence can be seen:
6
8
3
4
Fractions used for exercise 6.12.1:
1.
2
8
2.
3
6
3.
4
12
4.
6
16
5.
3
12
6.
184
8
12
6.13 Multiples
The products created by multiplying a particular number by other numbers are called multiples
of that number. The entries of the times table provide lists of multiples for the numbers at the
beginning of that row or column. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on.
Another way of saying this is that any number with 4 as a factor, is a multiple of 4.
A common multiple of two numbers is a number where both numbers are factors of (go into
evenly). For example, 12 is a common multiple of 2 and 3 since both numbers are factors of 12.
Example: Finding two common multiples of 3 and 5:
Solution: For the numbers 3 and 5 we look at the first several multiples of each number.
Multiples of 3
Multiples of 5
3 6
5 10
12 15 18 21
15 20 25 30 35
9
24 27 30 33
40 45 50 55
36
60
We could choose 15 and 30 as two different common multiples.
Checking a times table, the numbers 3 and 5 are both factors of 15, since 3 × 5 = 15
and 3 and 5 are both factors of 30, since 3 × 10 = 30 and 5 × 6 = 30.
Exercise 6.13.1 Find two different numbers the given values share as a common multiple.
1. 2 and 5
2. 4 and 6
3. 4 and 8
4. 6 and 7
5. 9 and 12
Example: Finding the Least Common Multiple of 6 and 10:
Solution: Again, we look at multiples.
Multiples of 6
Multiples of 10
6 12 18 24
10 20 30 40
30 36 42 48 54 60 66 72
50 60 70 80 90 100 110 120
For 6 and 10 the smallest multiple is 30. While 60 (= 6 × 10) is a multiple, it isn’t the smallest.
If we check a times table for the multiples of 6 and 10, the first number they share in
common is 30.
Exercise 6.13.2 Find the smallest multiple that each of the given values share. (Note: This
smallest shared multiple is called the Least Common Multiple and is abbreviated LCM )
1. 2 and 3
2. 4 and 6
3. 4 and 8
6. 8 and 20
7. 12 and 18
8. 12 and 15
185
4. 6 and 9
5. 9 and 12
9. 12 and 20
10. 9 and 24
Activity 6.7 LCM and Prime Factorization
Objective: To investigate the relationship between LCM and Prime Factorization.
Materials: Paper and pencil.
Group size: 3-4.
I. For each pair of numbers:
a. Write the prime factorization of each number. Circle any prime factors that appear in each.
b. Write the LCM of the two numbers.
c. Write the prime factorization of the LCM.
Example: 6 and 10
Solution:
a. 6 = 2×3
10 = 2×5
b. 6, 12, 18, 24, 30, 36, ...
10, 20, 30, 40, ...
The LCM of 6 and 10 is 30.
c. 30 = 2 × 3 × 5.
1. 2 and 3
2. 4 and 6
3. 4 and 8
6. 8 and 20
7. 12 and 18
8. 12 and 15
4. 6 and 9
5. 9 and 12
9. 12 and 20
10. 9 and 24
II. There is a relationship between the prime factorizations of two numbers, and the prime
factorization of their LCM. Looking at part I , describe any relationship that you can see. How
can you use the prime factorizations of numbers to find their LCM?
III. Use the strategy that you came up with in part II to find the LCM of the following pairs of
numbers.
1. 10 and 15
2. 4 and 14
3. 6 and 8
186
4. 8 and 12
5. 15 and 20
6.14 Finding a Common Denominator to Add or Subtract Fractions.
We have seen that when adding or subtracting fractions, a common denominator is critical. For
example, we have seen that 18 + 83 is no problem since the denominators match...
1
8
but that
1
8
=
+
1
8
+
3
4
3
8
=
4
8
=
1×2×2
2×2×2
=
1
2
needs some work before we can add because the denominators are different.
+
3
4
1
8
=
+
6
8
7
8
Notice that to go from 34 to 68 , we just add the middle line which doubles the number of pieces
in the whole from 4 to 8, and doubles the number shaded from 3 to 6. One way to write this is:
3
3×2
= 4×2
= 86 . It’s as if we introduce 2 as an extra factor in the numerator and 2 as an extra factor
4
in the denominator.
If the pictures are simple enough, then we know how to make the denominators the same. If
not, we need a systematic way to make equivalent fractions with matching denominators. It
turns out that there are two standard ways to find common denominators: 1) Use the product
of the denominators or 2) Use the LCM.
There are pros and cons to each. Over the next couple of exercises, you will look at both of
them.
187
We will use the sum of 14 + 61 to illustrate the two methods for finding common denominators.
For each method, we will draw a picture so that you can see how it is working.
Method 1: Use the product of the two denominators.
Step 1: Draw the two fractions, the first one where the rectangle is broken up into 4 rows, and
the second where the rectangle is broken up into 6 columns.
1
4
1
6
+
Step 2: Find the product of the two denominators, in this case, 4 × 6 = 24.
Step 3: Break each of the four rows in the first drawing into six equal pieces, and break each of
the six columns in the second drawing into four equal pieces.
1x6
4x6
1x4
6x4
+
Step 4: The previous step has changed each fraction into equivalent fractions that both have a
denominator of 24. The problem now becomes:
1
4
+
1
6
=
1×6
4×6
+
1×4
6×4
=
6
24
+
4
24
=
10
24
Step 5: Simplify the result. Since 10 and 24 have a common factor of 2, we can write 10 as 5 × 2,
and 24 as 12 × 2. Now,
10
24
=
5×2
12×2
=
5
12
Our final answer is
5
.
12
188
Method 2: Use the LCM of the two denominators.
Step 1: Draw the two fractions, each as a long, thin rectangle. Place them on top of each other
instead of side by side. With this drawing, it is very easy to see that the size of each fourth is
bigger than the size of each sixth. We need a common denominator!
1
4 +
1
6
Step 2: Find the LCM of the two denominators, in this case, looking at the multiples of 4 and 6,
12 is the smallest common multiple.
Multiples of 4
Multiples of 6
4 8 12
6 12 18
20 24 28
24 30 36 42
16
32
48
Step 3: Extend the division lines from each rectangle into the other rectangle to make more
lines, and therefore, more pieces.
?
? +
?
?
Step 4: Fill in more division lines so that all of the pieces are the same size.
Notice that you have broken each of the fourths into 3 pieces, and each of the sixths into two
pieces.
3
12 +
2
12
Notice also that 12 = 4 × 3 so that you can multiply the 1 in the numerator and the 4 in the
denominator in the fraction 14 by 3 to get an equivalent fraction with denominator equal to 12,
and that 12 = 6 × 2 so that you can multiply the 1 and 6 of 61 by 2 to get an equivalent fraction
with denominator equal to 12. This gives you the following:
1
4
+
1
6
=
1×3
4×3
+
1×2
6×2
=
3
12
+
2
12
=
5
12
Step 5: This time, there is no need to simplify the result, since
Again,
our final answer is
5
.
12
189
5
12
is already in its simplest form.
6.15 The Golden Rule of Fractions
A given fraction can be written as an equivalent fraction in two ways: remove a factor that
appears in both the numerator and denominator, or introduce an extra factor in the numerator
and denominator. In either case, whenever the operation is multiplication where we are taking
out or putting in factors, we follow the Golden Rule of Fractions: “Do to the top the same thing
that you do to the bottom.”
Let’s see how this works with
1
8
+ 43 :
• Step 1: Find the LCM of the given denominators.
We have already seen that the LCM of 4 and 8 is 8.
• Step 2: Find a separate extra factor for each denominator to multiply times the original
denominator to obtain the LCM.
8 is already the LCM, so we don’t need to change it at all. Leave
1
8
as 81 .
We need to find an extra factor to multiply by 4 to get 8. Since 8 is a multiple of 4 (it had to
be, since it’s the LCM), we know this is possible. In this case, we know that 4 × 2 = 8. The
extra factor that we need for the fraction 34 is 2.
• Step 3: Use the Golden Rule. Introduce the extra factor to both the denominator AND the
numerator.
3
4
=
3×2
4×2
• Step 4: Multiply the factorizations that you came up with.
3×2
4×2
=
6
8
Now, we can add the fractions:
1
8
+
3
4
=
1
8
+
6
8
= 78 .
• Step 5: The final step is to simplify the answer if possible.
Since the numerator, 7 and the denominator, 8 have no factors in common (7 is prime!),
is already in simplest form.
190
7
8
Example 1:
3
4
−
1
6
Solution:
• Step 1: The LCM of 4 and 6 is 12.
• Step 2: 4 × 3 = 12 and 6 × 2 = 12, so 34 needs an extra factor of 3 in the numerator and
denominator, and 61 needs an extra factor of 2 in the numerator and denominator. (Note:
For this step we ignore the numerators.)
• Step 3:
3
4
=
3×3
4×3
and
• Step 4:
3
4
=
3×3
4×3
=
9
12
1
6
=
1×2
.
6×2
and
1
6
=
1×2
6×2
=
Now, we can subtract the fractions:
2
.
12
3
4
−
1
6
=
9
12
−
2
12
=
7
.
12
• Step 5: Again, since 7 is prime, there are no common factors.
form, and is the final answer.
Example 2:
5
6
−
7
12
is already in its simplest
5
8
Solution:
• Step 1: The LCM of 6 and 8 is 24.
• Step 2: 6 × 4 = 24 and 8 × 3 = 24, so 56 needs an extra factor of 4 in the numerator and
denominator, and 85 needs an extra factor of 3 in the numerator and denominator. (Note:
For this step we ignore the numerators.)
• Step 3:
5
6
=
5×4
6×4
and
• Step 4:
5
6
=
5×4
6×4
=
20
24
5
8
=
5×3
.
8×3
and
5
8
=
5×3
8×3
=
Now, we can subtract the fractions:
15
.
24
5
6
−
5
8
=
20
24
−
15
24
=
5
.
24
• Step 5: Again, since 5 is prime, there are no common factors.
form, and is the final answer.
191
5
24
is already in simplest
Example 3:
5
6
−
1
3
Solution:
• Step 1: The LCM of 6 and 3 is 6.
• Step 2: Since 56 already has the required denominator, we don’t need to use the Golden
Rule to change it. 3 × 2 = 6, so 31 needs an extra factor of 2 in the numerator and
denominator. (Note: For this step we ignore the numerators.)
• Step 3:
1
3
=
1×2
.
3×2
• Step 4:
1
3
=
1×2
3×2
= 26 .
Now, we can subtract the fractions:
5
6
−
1
3
=
5
6
−
2
6
= 36 .
• Step 5: 3 and 6 have a common factor of 3, so we re-write the fraction and use the Golden
1×3
= 12 . 12 is in its
Rule to remove common factors and make the answer simpler. 63 = 2×3
simplest form, and is the final answer.
Exercise 6.15.1 Add or subtract as indicated. Include drawings of rectangles with your
solutions for the first four problems. Include appropriate units in your answers.
1.
1
4
+
1
6
2.
1
2
−
1
3
3.
3
4
−
5
8
4.
5
9
+
1
6
5.
4
9
+
7
12
6.
3
10
−
2
15
7.
5
8
+
5
6
′′
8. On the golf course, the grass on the fairway is supposed to be 83 high. If the grass on the
′′
fairway at the bottom of Sharp Park is 34 high, how much needs to be cut off?
9. While tiling their kitchen floor, Joyce and Pedro chose a pattern that alternated tiles that were
′′
′′
1 14 wide with bigger tiles that were 2 85 wide. How wide were the two tiles put together if they
′′
also had to put 81 of grout between them?
10. Ahmed wanted to frame the window in his home office with decorative border strips. If his
′′
′′
window frame was 52 58 by 32 34 , what is the total length of the strips that he needs to go all the
way around the window?
11.
3
50
+ 0.17
12. 92% −
7
25
13. 48% −
3
10
14. 0.63 +
13
20
15. 50% −
1
8
1
of
16. 25% of the people preferred dark chocolate. 12 of the people preferred milk chocolate. 10
the people preferred white chocolate. The rest didn’t have a preference. What fraction of the
people didn’t have a preference?
192
6.16 Ratio and Proportion
Suppose we know that a classroom has 35 students in it: 14 men and 21 women. Using the idea
of fractions that we have been studying, we can say that the fraction of the class that is men is
14
, which using Golden Rule, simplifies as follows:
35
14
35
=
2×7
5×7
= 52 .
So, we can say that
2
5
of the students in the class are men.
There is another way to use fractions to talk about the relative number of men in the class. A
Ratio is a way to show the relative size of one quantity in comparison with another.
In our classroom example, we can say that the ratio of men to women in a classroom is 14 to 21.
to represent this ratio, but we have to be careful to say that it is the
We can use the fraction 14
21
ratio of men to women.
This fraction notation can be very confusing, since up to now we have used the denominators
of fractions to represent how many pieces an entire thing is broken up into. Now we are saying
that the numerator and denominator can stand for anything, as long as we are consistent within
the same problem.
Suppose the classroom next door has 10 men and 15 women. We say that the ratio of men to
women in these two classrooms are the same, even though the numbers of men and women are
10
different since 14
and 15
are equivalent. The relative number of men compared to the number of
21
women is the same. In each class, for every 2 men there are 3 women.
We can see this using the Golden Rule:
14
21
=
2×7
3×7
=
2
3
and
10
15
=
2×5
3×5
= 32 .
When ratios are equivalent like this, we say we have a proportion. When we know things are
proportional we can use the Golden Rule to find equivalent fractions to solve problems.
Example 1: The ratio of men to women in a classroom is 2 to 3. How many men are there if
there are 18 women?
Solution: Since the ratio of men to women is 2 to 3, we know that the ratio of men to women
written as a fraction is equivalent to the fraction 32 . That is,
men
women
=
2
3
=
?
.
18
The Golden Rule says we can make these fractions equivalent by multiplying the top and
bottom by the same number. Since 3 times 6 is 18, we know that the number we will use is 6.
This gives us:
men
women
=
2
3
=
2×6
3×6
=
12
.
18
This tells us that the number of men is 12!
193
Example 2: How many yards long is a red carpet that measures 33 feet?
Solution: Since there are 3 feet for every 1 yard, any ratio of feet to yards is 3 to 1, which written
as a fraction is equivalent to the fraction 13 . That is,
feet
yards
=
3
1
=
33
.
?
Since 3 times 11 is 33, we know that the number we will use for the Golden Rule is 11. This
gives us:
feet
yards
=
3
1
=
3×11
1×11
=
33
.
11
This tells us that the length of the carpet is 11 yards.
Exercise 6.16.1 Use proportions to find the missing quantities. Include appropriate units in
your answers.
1. If three out of every four people in a class are under 21 years old, then how many people
in a class of 32 are under 21?
2. If it costs $2 for five potatoes, then how much will 20 potatoes cost?
3. If there are two quarters in
1
2
dollar, how many quarters in
5
2
dollar?
4. If there are 16 cups in one gallon, how many cups are there in 3 gallons?
5. How many cups are there in one-half gallon?
6. If there are 12 inches in one foot, how many inches are there in 5 feet?
7. How many inches are there in one-quarter foot?
8. If it takes 43 cup of flour to make 10 pancakes, how much flour will it take to make 40
pancakes?
9. If 60% of the class is staying home for Spring Break, how many people are in the class if 24
are staying home for Spring Break? (Hint: Simplify the ratio obtained from the percentage
before continuing with the Golden Rule.)
194
6.17 Multiplication by Negative Numbers
We have seen that negative numbers can be used for quantities like debt, elevation below sea
level, or temperatures below zero. On a number line, negative numbers are to the left of zero.
For example, thinking of money, if you had $20 and spent $35 we could express this idea with
the following subtraction equation:
20 − 35 = −15
The negative sign in front of the 15 indicates that since we spent more than we had, we are $15
in debt. We could express the same thing using addition instead of subtraction if we think of
combining a positive 20 (money we had) with a negative 35 (money contributing to our debt)
we get the same result:
20 + (−35) = −15
The parenthesis around the negative 35 are not grouping symbols, nor do they indicate
multiplication. They are there simply so that the plus sign doesn’t get too close to the negative
sign. It’s too hard to see the negative sign if it’s written 20 + −35 = −15!
Using this idea, and the fact that multiplication is repeated addition, we can make sense out of
multiplication of a whole number by a negative number:
Example 1: Even though he was broke, Rick bought three CDs that cost $15 each. What was his
net worth after he bought the CDs?
Solution: He went in debt by $15 three times, so his total is:
3 × (−15) = (−15) + (−15) + (−15)
= −45
So, Rick is in debt a total of $45!
Example 2: One winter day in Minnetonka, Minnesota, it was 0◦ F at 5 pm. Over the next 4
hours, the temperature decreased by 2◦ F per hour. What was the temperature at 9 pm?
Solution: The temperature went down by 2◦ F four times, so the final temperature is:
4 × (−2) = (−2) + (−2) + (−2) + (−2)
= −8
So, the temperature at 9 pm in this cold little town was −8◦ F.
195
Exercise 6.17.1 Compute the following using repeated addition:
1. 5 × (−12)
2. 6 × (0.45)
3. 2 × (−57.75)
4. 2 ×
3
4
5. 4 × 2 21
Exercise 6.17.2 For the following, write a short description of a scenario that could be
modeled by the given expression. Use money, temperature, elevation, measurement, or some
other real-life scenario that makes sense to you:
Example: 7 × (−30)
Possible Solution: The final elevation if you started at sea level and the elevation dropped by
30 feet per hour for 7 hours.
1. 5 × (−12)
2. 6 × (0.45)
3. 2 × (−57.75)
4. 2 ×
3
4
5. 4 × 2 21
196
Exercise 6.17.3 For the following scenarios, translate the scenario into a mathematical
expression, then compute the result. Include appropriate units in your answers.
Example: The final temperature if the temperature started at 0◦ F then went down 14◦ F per hour
for two hours.
Solution: 2 × (−14) = (−14) + (−14) = −28, so the final temperature is −28◦ F.
1. The final bank balance if you start with no money and write 6 checks for $13 each.
2. The final temperature if it started at 0◦ and fell 3◦ per hour for 4 hours.
3. The distance travelled if you drove for 6 hours at an average speed of 65 miles per hour.
4. The final altitude if you started at sea level and dropped 75 feet per hour for 3 hours.
5. The final bank balance if you start with no money and make 5 deposits of $42 each.
′′
6. Jon built a gate using 6 fence boards that were each 6 83 wide. How wide was the gate that
Jon built?
6.18 Order of Operations with Multiplication and Addition
Notice that in each of these scenarios, since each quantity starts at zero, only multiplication is
necessary to figure out the final quantity. But in real life, this is not always the case! Suppose
instead that you start with $2 and then deposit $4 into the account 3 times. To calculate this, we
add the total deposit to our initial amount. Since we deposited $4 into the account 3 times, our
total deposit is 3(4). The expression used to represent the whole scenario is:
2 + 3(4)
Thinking about the scenario, we know that this is $2 plus the total deposit of $12 for a final
balance of $14. But, if we just work on the expression, performing the operations from left to
right (like with addition and subtraction) we get:
2 + 3(4) = 5(4)
= 20
which we know is not correct! What is going on here?
197
Order of Operations Agreement for Multiplication, Addition, and
Subtraction
In an expression which contains multiplication and addition and/or subtraction,
perform all of the multiplications first, from left to right. Then, after all of the
multiplication is finished, perform all of the subtractions and additions from left to
right. If a situation requires an addition or subtraction to be performed before the
multiplications, grouping symbols must be used.
Exercise 6.18.1 For the following expressions:
a) Compute the value of the expression using the order of operations agreement.
b) Write a scenario using money or some other real life situation that could be modeled by the
given expression.
Example: 7 + 2(−11)
Solution:
a) 7 + 2(−11) = 7 + (−22) = −15
b) Starting at a balance of $0, you make a deposit of $7 and write two checks for $11 each. The
final balance is $15 of debt!
1. 8 + 2(12)
(a)
(b)
2. 9 + 3(−8)
(a)
(b)
3. 3(7) + (−9)
(a)
(b)
4. 5(−8) + 17
(a)
(b)
198
5. 3(−17) + 5(12)
(a)
(b)
Exercise 6.18.2 For the following scenarios:
a) Translate the scenario into a mathematical expression.
b) Evaluate the expression to find the result of the scenario.
c) What does your numerical answer mean in the context of the scenario?
1. Starting with a balance of $0, you withdrew $15 three times and deposited $19. What is
your final balance?
(a)
(b)
(c)
2. Starting with a balance of $0, you deposited $18 four times and withdrew $12 five times.
What is your final balance?
(a)
(b)
(c)
3. Starting with a balance of $8, you withdrew $11 three times and deposited $13 twice.
What is your final balance?
(a)
(b)
(c)
199
4. Assume east is the positive direction and west is the negative direction. Starting at your
house, you traveled east at 4 mph for 6 hours, then traveled west at 5 mph for 2 hours.
What is your final position relative to the house?
(a)
(b)
(c)
5. Assume east is the positive direction and west is the negative direction. Starting at your
house, you traveled west at 7 mph for 5 hours, then traveled east at 10 mph for 3 hours.
What is your final position relative to the house?
(a)
(b)
(c)
6. Assume east is the positive direction and west is the negative direction. Starting 3 miles
east of your house, you traveled west at 22 mph for 3 hours, then traveled east at 15 mph
for 7 hours. What is your final position relative to the house?
(a)
(b)
(c)
7. Assume up is positive, and down is negative. Starting on the ground, the balloon rose
higher at a rate of 18 feet per minute for 17 minutes, then fell at a rate of 5 feet per minute
for a half hour. What is your final height?
(a)
(b)
(c)
200
Notes:
201
7 Division of Whole Numbers and Decimals
Activity 7.1 Money Activity: Divide it Up
Objective: To model concepts of division by using money.
Materials: Bills of different denominations.
Group size: 3.
Write the names of each group member:
Group member #1
Group member #2
Group member #3
The bank has a pile of bills and coins in the following denominations: $10,000, $1000, $100, $10,
$1, dimes and pennies. There are three members in each group. When you pick your money,
make sure that the number of bills in each denomination is less than ten at any given time.
1. Start with $582. Divide this amount evenly among the three people in the group.
(a) First split the $100 bills evenly among the 3 people. How many $100 bills does each
person get? How many are left over?
(b) If there were any $100 bills left over, exchange each one for ten $10 bills. Combine
these with the $10 bills that you already had. How many $10 bills are there in all?
(c) Now, split the $10 bills evenly among the 3 people. How many $10 bills does each
person get? How many are left over?
(d) If there were any $10 bills left over, exchange each one for ten $1 bills. Combine these
with the $1 bills that you already had. How many $1 bills are there in all?
(e) Now, split the $1 bills evenly among the 3 people. How many $1 bills does each
person get? How many are left over?
(f) Record the remainder, that is, how much money is left that can’t be split evenly
without using coins.
2. Start over, this time with $6791.
(a) First split the $1000 bills evenly among the 3 people. How many $1000 bills does
each person get? How many are left over?
(b) If there were any $1000 bills left over, exchange each one for ten $100 bills. Combine
these with the $100 bills that you already had. How many $100 bills are there in all?
202
(c) Now, split the $100 bills evenly among the 3 people. How many $100 bills does each
person get? How many are left over?
(d) If there were any $100 bills left over, exchange each one for ten $10 bills. Combine
these with the $10 bills that you already had. How many $10 bills are there in all?
(e) Now, split the $10 bills evenly among the 3 people. How many $10 bills does each
person get? How many are left over?
(f) If there were any $10 bills left over, exchange each one for ten $1 bills. Combine these
with the $1 bills that you already had. How many $1 bills are there in all?
(g) Now, split the $1 bills evenly among the 3 people. How many $1 bills does each
person get? How many are left over?
(h) Record the remainder, that is, how much money is left that can’t be split evenly
without using coins.
3. Start over, this time with $1264.
(a) First split the $1000 bills evenly among the 3 people. How many $1000 bills does
each person get? How many are left over?
(b) If there were any $1000 bills left over, exchange each one for ten $100 bills. Combine
these with the $100 bills that you already had. How many $100 bills are there in all?
(c) Now, split the $100 bills evenly among the 3 people. How many $100 bills does each
person get? How many are left over?
(d) If there were any $100 bills left over, exchange each one for ten $10 bills. Combine
these with the $10 bills that you already had. How many $10 bills are there in all?
(e) Now, split the $10 bills evenly among the 3 people. How many $10 bills does each
person get? How many are left over?
(f) If there were any $10 bills left over, exchange each one for ten $1 bills. Combine these
with the $1 bills that you already had. How many $1 bills are there in all?
(g) Now, split the $1 bills evenly among the 3 people. How many $1 bills does each
person get? How many are left over?
(h) This time, if there were any $1 bills left over, exchange each one for ten dimes, that is,
ten $0.10. Since you didn’t start with any dimes, this is all you have.
(i) Now, split the $0.10 (dimes) evenly among the 3 people. How many $0.10 does each
person get? How many are left over?
203
(j) If there were any $0.10 left over, exchange each one for ten pennies, that is, ten $0.01.
Since you didn’t start with any pennies, this is all you have.
(k) Now, split the $0.01 (pennies) evenly among the 3 people. How many $0.01 does
each person get?
(l) To the nearest 0.01, what is 1264 divided by 3?
4. Each member of the group takes $1000.
(a) With your own stack of $1000, each member splits it evenly into 5 separate piles.
i. Record the amount in each pile.
ii. Record the remainder, if any.
(b) With your own stack of $1000, each member splits it evenly (to the nearest $1) into 6
separate piles.
i. Record the amount in each pile.
ii. Record the remainder, if any.
(c) With your own stack of $1000, each member splits it evenly (to the nearest $1) into 3
separate piles.
i. Record the amount in each pile.
ii. Record the remainder, if any.
(d) Compare your answers with your group members.
(e) How can you check your answers using multiplication and addition?
(f) By splitting $1000 into 5, 6, or 3 equal piles, when did you have the most money in
each pile?
204
5. Member #1 starts with $14,000.
(a) From the $14,000, give member #2 $1,500. Record how much money member #1 has
left.
(b) Member #1 continues to give $1500 at a time to member #2 until there is not enough
money to continue. Record in the following table how much money member #1 has
left after each time:
Number of times
$ given to #2
$ left over
1
2
3
4
5
6
7
8
9
10
ouch! ouch! ouch! ouch! ouch! ouch! ouch! ouch! ouch!
(c) How many times does 1,500 go into 14,000?
6. Member #2 starts with $2,400.
(a) From the $2,400, give member #3 $450. Record how much money member #2 has left.
(b) Member #2 continues to give $450 at a time to member #3 until there is not enough
money to continue. Record in the following table how much money member #2 has
left after each time:
Number of times
$ given to #3
$ left over
1
2
3
4
5
6
7
8
9
ouch! ouch! ouch! ouch! ouch! ouch! ouch! ouch! ouch!
(c) How many times does 450 go into 2,400?
7. How are the operations of subtraction and division related?
205
10
7.1 The Process of Division
Just as subtraction reverses the process of addition, multiplication (which is repeated addition)
is reversed by division (which is repeated subtraction).
To divide 6 by 2, we write 6 ÷ 2 and by repeated subtraction we get,
6−2 =4
4−2 =2
2−2 =0
3 two’s
Which means we subtract 2 three times from 6 before nothing is left.
For larger numbers, like 448 ÷ 7 we can subtract large amounts by multiplying first:
60×7=420
}|
{
z
448 ÷ 7 = 448 − 7 − 7 − · · · − 7 = 28
60 sevens
4×7=28
z
}|
{
= 28 − 7 − 7 − 7 − 7 = 0
4 sevens
64 sevens
The operation of division is symbolized in a variety of ways. The symbol, ÷, reminds us of a
fraction where the dots represent the numbers in the fraction. In fact, 12 ÷ 4, 12/4 and 12
are
4
equivalent ways of representing twelve
divided by
four. Another common representation for
division is the long division symbol,
, where 4 12 also represents twelve divided by four.
Vocabulary:
• The operation is called division.
• The number being divided (the 12 in 12 ÷ 4) is called the dividend.
• The number used to divide (the 4 in 12 ÷ 4) is called the divisor.
• Both the entire expression (the 12 ÷ 4) and the result of the operation (3), are called the
quotient.
The algorithm or method we typically see for long division is adapted from the idea of repeated
subtraction above.
206
Example 1: To divide 7 448 (that is, 7 into 448 or 448 divided by 7) we begin by subtracting the
largest number of sevens we can calculate easily. Using place value, since 7 × 100 is too large
(700 > 448), we choose the largest tens place number whose product with 7 is less than 440.
Since 7 × 60 = 420 is the closest, we write a 6 above the tens place of the dividend and then
subtract (7 × 60 =) 420 from the dividend.
6
7 448
−→
6
7 448
420
28
−→
We then start the process over again. Since we chose the largest tens place value to multiply by
the first time, we move to the largest ones place now. Since 7 × 4 = 28 we choose 4 and write it
in the ones place above the dividend. We then subtract 28 from 28 and since nothing remains
we conclude that 448 ÷ 7 = 64.
64
7 448
420
28
−→
64
7 448
420
28
28
0
Notice that we can use multiplication to check that if we have the number 7, 64 times, we will
get 448. Notice also, that if we use the distributive property to perform the multiplication, we
get 7 × 64 = 7 × 60 + 7 × 4 = 420 + 28. These two terms in the distributive property sum are
exactly the numbers that appeared in our division problem!
207
It is often the case that a quotient leaves a remainder or an amount left over that is not evenly
divisible by the divisor. If four roommates want to split up 21 dishes, we have 21 ÷ 4 = 5 with
one plate left over. Since one plate isn’t of much use in pieces, there’s no way to share the
remaining plate and it is left out as a remainder. Some things can be divided into fractional
pieces, however, as we saw in the chapter opening money activity on page 201. Consider the
example below.
Example 2: Four roommates want to split up the $623 security deposit they receive when they
move out of their apartment. In order to divide 4 623, we begin as before:
1
4 623
400
223
−→
15
4 623
400
223
200
23
−→
155
4 623
400
223
210
23
20
3
Now there is $3 remaining to split between the four people. If we exchange the dollars for
dimes we have 30 dimes. This is accomplished in writing by placing a decimal point after the 4
in the top line and recognizing that we are now using tenths of a dollar so $3 is equivalent to 30
tenths of a dollar, or 30 dimes:
−→
155.
4 623
400
223
200
23
20
30
−→
155.7
4 623
400
223
200
23
20
30
28
2
155.7
4 623
400
223
200
23
20
30
28
20
−→
−→
155.75
4 623
400
223
200
23
20
30
28
20
20
0
From the thirty dimes, each person receives 7, or seven dimes, leaving 2 dimes remaining. If we
exchange the dimes for pennies, we now have 20 cents or twenty hundredths of a dollar. We
accomplish this simply by placing a zero after the 2 and working from the hundredths place in
the number in the top line. As a result, we see each roommate receives $155.75.
208
Exercise 7.1.1
1. For these problems, use repeated subtraction.
(a)
12 72
(b) (Hint: You may subtract ten 8s at a time!)
8 576
2. For each exercise below, first perform the division and then write a word problem to go with
it.
(a)
4 108
(b)
8 156
3. Estimate the following quotients by first rounding the numbers to the nearest 10.
(a)
13 874
(b)
7 203
4. Estimate the answers to these questions by first rounding to the nearest 10.
(a) If 7 people want to share $2468 evenly, how much should each person get?
(b) If Ramon drives his car 438 miles in 8 hours, what is his average speed?
(c) If Jenny can drive her car 370 miles on 12 gallons of gas, what is her fuel efficiency in
mpg?
(d) How many feet are there in 378 inches?
5. A rectangular garden has an area of 48 square feet. If the length of the garden is 8 feet, what
is the width?
6. Using the idea of repeated subtraction, explain what is wrong with 1 ÷ 0.
209
There are three main symbols to use to indicate the operation of division: ÷,
bar), and
(the fraction
Exercise 7.1.2 Fill in the table so that each line means the same division problem. The first
one is done for you.
Problem number
÷
1)
10 ÷ 2
2)
14 ÷ 7
3)
(the fraction bar)
2 10
10
2
the result
5
24
6
9 54
4)
5)
7
210
7.2 Divisibility
A number is divisible by another number if when you divide the first by the second, there is no
remainder. In other words, numbers are divisible by their factors.
Examples:
• 6 is divisible by the numbers 1, 2, 3 and 6.
• 15 is divisible by the numbers 1, 3, 5 and 15.
• 24 is divisible by the numbers 1, 2, 3, 4, 6, 8, 12 and 24.
The idea of divisibility (a division concept) is closely linked with the idea of factors (a
multiplication concept).
Exercise 7.2.1 For certain problems, it is nice to know before dividing what numbers it is
divisible by. In the following exercises you will explore divisibility and come up with some
divisibility rules.
1. Numbers Divisible by 2:
(a) List eight 3-digit numbers that are divisible by 2.
(b) How can we tell whether a given number is divisible by 2?
(c) Use your divisibility rule to figure out which of the following numbers are divisible
by 2:
1,392
57,120
228,301
77,754
300,005
713,293
2. Numbers Divisible by 5:
(a) List eight 3-digit numbers that are divisible by 5.
(b) How can we tell whether a given number is divisible by 5?
(c) Use your divisibility rule to figure out which of the following numbers are divisible
by 5:
1,392
57,120
228,301
77,754
211
300,005
713,293
3. Numbers Divisible by 3:
(a) List the first twelve numbers bigger than 40 that are divisible by 3 in the first column
of the table below. The first two are done for you.
(b) In the second column of the table, write the sum of the digits of the number written
in the first column. The first two are done for you.
(c) In the third column, write “Yes” if the sum of the digits is divisible by 3, write “No”
if the sum of the digits is not divisible by 3.
Multiple of 3 Sum of its Digits
42
4+2=6
45
4+5=9
Is the sum divisible by 3? Yes or No?
Yes
Yes
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
ouch!
(d) Find a 2-digit number that is NOT divisible by 3. Is the sum of its digits divisible by
3?
(e) Find a 3-digit number that is NOT divisible by 3. Is the sum of its digits divisible by
3?
(f) How can we tell whether a given number is divisible by 3?
(g) Use your divisibility rule to figure out which of the following numbers are divisible
by 3:
1,392
57,120
228,301
77,754
300,005
713,293
(h) Simplify the following fractions that can be simplified:
i)
3
275
ii)
3
2115
iii)
3
827
212
iv)
3
528
v)
6
537
Exercise 7.2.2 Include appropriate units in your answers.
1. Make up three word problems to go with 6 56 .
In each problem the remainder should be handled differently. Write the problems on a separate
piece of paper.
2. Which of the numbers below are divisible by three?
(a) 51
(b) 91
(c) 147
(d) 3812
(e) 4017
(f) 6112
(g) 10,206
3. Make up four different three digit numbers that are all divisible by three:
4. Show your steps in answering these questions.
(a) If you drive at 60 mph for 3 hours, how far will you travel?
(b) If Joe drives 300 miles in six hours, what was his average speed?
(c) If you have $91 to split between 7 people, how much will each person get?
(d) If Nick earns $12 per hour, how much money will he make if he works 27 hours?
(e) Dolores earns $874 one week. If she worked for 38 hours that week, what was her
hourly pay?
(f) There are 12 inches in a foot.
(i) How many inches are there in 8 feet?
(ii) How many feet are there in 228 inches?
(g) There are 1000 meters in a kilometer.
(i) How many meters are there in 4 kilometers?
(ii) How many kilometers are there in 2500 meters?
(h) Maria wants to build a fence that is 90 feet long. If fence boards are 4 inches wide, how
many fence boards will she need?
(i) Sammy earned $304.75 working one week at an hourly rate of $11.50 per hour. How
many hours did Sammy work that week?
5. A rectangle has an area of 57 square inches. What are the possible whole number dimensions
of the rectangle?
213
Exercise 7.2.3 List the first twenty multiples of each number in the table below.
2
3
4
5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
214
6
7
8
9
10
7.3 Estimation Again
We have seen that with addition and subtraction, we can estimate (find a number that is
reasonably close to the answer in a quick and easy way) by rounding each of the numbers to
the highest place value that exists in any of the numbers before adding or subtracting. With
multiplication, we can estimate by rounding to one significant digit before performing the
multiplication.
Exercise 7.3.1 Estimate the answers to the following addition, subtraction, and multiplication
problems by rounding the numbers appropriately before performing the operations. Include
appropriate units in your answers:
1. Find the area of a parking lot that measures 113.7 feet by 75.1 feet.
2. Find the perimeter of the same parking lot.
3. Julia’s car blinks “low fuel” when she has used 12.38 gallons of her full tank. If she gets
gas when she first sees the light come on, and gas is $3.35/gal, how much will it cost to fill
her tank?
4. The techs in the car factory conducted an “almost” crash test for their new car. During the
test, the driver started braking when the car was 83.85 feet from the wall and came to a
complete stop when the car was 14.3 feet from the wall. How long was the skid?
5. Paulina needs 60 units to graduate. During her first semester, she took 12.5 units, but
didn’t pass a 3-unit class, so didn’t get credit for it. The next semester, she took 9 units,
and passed all of her classes. The following year, she passed all of her classes and took
13.5 units in the fall, and 17 units in the spring. How many more units does she need to
have enough to graduate?
6. (Do this one exactly, not with an estimate...) How many square inches are in one square
foot?
7. How many square inches are in 32.8 square feet?
215
Division estimation is more tricky. If the dividend (the number being divided up) is not a
multiple of the divisor (the number going into the dividend), then there will be a remainder.
This doesn’t always get better when we round to one significant digit. Look at the following
example:
Example 1: Estimate the quotient for 3257 ÷ 7
Solution: When we round to one significant digit, this becomes 3000 ÷ 7 which isn’t much
easier since 7 doesn’t go into 3 evenly (3 isn’t big enough), and 7 doesn’t go into 30 either, since
the multiples of 7 are: 7 = 1 × 7, 14 = 2 × 7, 21 = 3 × 7, 28 = 4 × 7, 35 = 5 × 7, 42 = 6 × 7, etc.
Since rounding to one significant digit doesn’t make things easy enough to be worth it, another
strategy is to:
1. Round the divisor to one significant digit.
In our example, the divisor is 7 which is already only one digit, so our problem still looks
like 3257 ÷ 7.
2. Round the dividend to the nearest multiple of the rounded divisor that has at most two
significant digits.
The dividend is 3257. If we try to round to only one significant digit (the thousands place),
3257 rounds to 3000 which is not easy to see as a multiple of 7. Staying in the thousands
place, the nearest multiple of 7 is 7000 which is too far away to be reasonable.
If we round to two significant digits instead, the multiples of 7 somewhat near 3257 are
1400, 2100, 2800, 3500, 4200, etc. The closest one to 3257 is 3500, so we will use that.
Our problem now looks like 3500 ÷ 7.
3. Perform the division.
The division is now easier to perform since 35 ÷ 7 = 5, so 350 ÷ 7 = 50 and then for our
estimate 3500 ÷ 7 = 500. We can say the quotient for 3257 ÷ 7 is approximately equal to
500.
216
Example 2: Estimate the quotient for 54, 728.2 ÷ 75.
Solution: First round 75 to 80. Next, we look at 54,728.2 thinking of two-digit multiples of 8 to
use for our 2 significant digits. Two-digit multiples of 8 near 54 or 55 are 40, 48, 56, 64, and 72.
The closest is 56, so we will round 54,728.2 to 56,000.
Now, to perform the division, we know that 8 × 7 = 56, so 80 × 700 = 56, 000. Therefore,
56, 000 ÷ 80 = 700 and 54, 728.2 ÷ 75 is approximately equal to 700.
Exercise 7.3.2 Estimate the following quotients using the strategy outlined on the previous
page, then, using your calculator, find the actual quotients rounded to the nearest whole
number:
1. 4, 352 ÷ 6
2. 53, 722.9 ÷ 9
3. 27, 523 ÷ 37
4. 337, 495.75 ÷ 823.35
5. 463, 995 ÷ 61.99
6. Make up a problem whose dividend has at least 3 significant digits and whose quotient
has an estimate of 50.
7. Alec drives his car 338 miles over a period of 7 hours. What is his average speed?
8. Erica earns $731 working for 43 hours. What is her average hourly pay?
217
Activity 7.2 Buying a Car
Objective: To practice estimation, problem solving, and organizational skills.
Materials: Paper, pencil, calculator.
Group size: 3-4.
Part 1: Harold’s Estimates
Harold is trying to collect information to help him choose a new car to buy. The car is going to
be used primarily for commuting, so he is looking for a small, low cost automobile that gets
good gas mileage. He starts by looking only at costs. The two costs he will investigate are car
payment, and costs for gas.
He does some estimating and comes up with the following information:
• He drives 46.8 miles every work day, and works 5 days per week.
• He averages 20 miles of driving each weekend.
• He gets two weeks of vacation every year, but plans to take his vacation in Hawaii every
year, and will take a cab to the airport. Since there are 52 weeks in a year, this means
Harold will be driving 50 weeks each year.
• Gas costs $3.35/gal at the station where Harold gets gas.
• Harold qualifies for a loan that charges no interest as long as he pays off the loan in 5
years.
Using this information, Harold estimates the following costs for a Honda Civic that averages 40
mpg on the highway, and costs $18,400. For each step, show how he got his answer:
1. He estimates he will commute 250 miles each week and therefore drives a total of 270
miles each week including the weekends.
2. He estimates he will therefore drive 15,000 miles each year.
3. Since the Civic gets 40 mpg, Harold estimates that he will have to buy 400 gallons of gas
each year.
4. Assuming gas stays at $3.35/gal, he estimates that he will have to pay $1200 for gas.
5. He estimates he will have to pay $4000 per year for car payments.
6. He estimates the total cost for gas and car payment each year will be $5200.
218
Part 2: Using your own information.
In this part, you will be making your own estimates about how much you drive (or travel in
someone else’s car or on the bus if you don’t drive yourself) and what the price of gas is near
where you live in order to estimate costs if you were to buy a new car.
First, estimate the following information about your own life:
1. How far do you drive (or travel) each week for school, work, etc.?
2. Do you have some weeks during the year when you drive (or travel) a lot more or less
than this?
3. How much does gas cost near you?
Next, use the information to complete the table below with estimates of the costs for gas and
car payments. Assume that, like Harold, you pay no interest on the car loan, and pay the loan
off in 5 years. Make sure that when you estimate, you round the numbers in your calculation
before you perform the operation, and that by rounding you have made the calculation:
• Easy enough to do the computation step in your head.
• Have an answer relatively close to the actual answer.
Car
Type
List Price
(in dollars)
Highway Annual Cost of
mpg
Car Payment
(estimate)
Help!
40
Honda Civic
18,400
Saturn Ion
14,725
32
Help!
Chevy Cobalt
16,150
32
Help!
Hyundai Tiburon
18,745
28
Help!
Jaguar Coupe
82,330
23
Help!
219
Annual Cost of
Gas
(estimate)
Total
Annual Cost
(estimate)
7.4 Multiplying Fractions
Exercise 7.4.1
1. (a) If you had 8 apples and gave
give to your friend?
1
2
of the 8 apples to your friend, how many would you
(b) If you divided 8 apples evenly into two groups, how many apples would you have in
each group?
2. (a) How much is
1
5
of $10?
(b) If you divided $10 evenly into 5 groups, how much money would you have in each
group?
3. (a) 12 people showed up to the church fundraiser.
showed up to the fundraiser?
1
3
of them were men. How many men
(b) At the church fundraiser, they broke the 12 people up into 3 groups. How many were
in each group?
4. (a) How far is
1
2
of 4 inches?
3
2
1
4
5
(b) If you divided 4 inches evenly into two pieces, how many inches would you have in
each piece?
5. (a) How far is
1
4
of 2 inches?
2
1
(b) If you divided 2 inches evenly into four pieces, how many inches would you have in
each piece?
6. What operation (addition, subtraction, multiplication, or division) did you use to
calculate part (b) in problems 1-5?
7. If eight friends each brought
all together?
1
2
of a pizza to the party, how many pizzas would they have
(a) Justify the solution using repeated addition and pictures.
(b) Since repeated addition of the same value is equivalent to multiplication, how could
you use multiplication to get the same answer as in part (a)?
220
8. If you had
1
5
ten times, how much would you have?
(a) Write the solution using repeated addition.
(b) Write the solution using multiplication.
9. If you had
1
3
twelve times, how much would you have?
(a) Write the solution using repeated addition.
(b) Write the solution using multiplication.
10. If you had
1
4
two times, how much would you have?
(a) Write the solution using repeated addition.
(b) Write the solution using multiplication.
221
Example: 5 ×
1
6
Solution:
(a) Using repeated addition:
1
6
+
1
6
+ 16 +
1
6
+
1
6
=
5
6
(b) The picture:
=
(c) Using multiplication: 5 ×
1
6
=
5×1
6
=
5
6
Exercise 7.4.2 For the following products:
(a) Write the solution using repeated addition. (Write fraction answers in its simplest form.)
(b) Draw a picture representing the repeated addition.
(c) Write the solution using multiplication. (Write fraction answers in its simplest form.)
1. 2 ×
3
4
2. 6 ×
2
3
3. 4 ×
3
8
4. 3 ×
5
6
. Explain in words (or pictures if you like)
5. We have seen that 7 × 34 = 43 + 43 + ... + 34 = 21
4
how to multiply in order to get this same result. Explain why this works.
222
Example: 3 ×
5
6
Solution:
(a) Using repeated addition:
(b) Using multiplication:
3×5
6
5
6
+
=
5
6
+
3×5
3×2
5
6
=
5+5+5
6
=
3×5
3×2
= 52 .
= 52 .
Exercise 7.4.3 For the following products:
(a) Write the solution using repeated addition. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
1. 2 ×
3
8
2. 6 ×
1
4
3. 5 ×
3
10
4. 4 ×
5
6
5. 8 ×
3
4
6. 2 ×
1
6
Exercise 7.4.4 Let’s look at another way of looking at a fraction times a whole number:
1. What number is
1
2
of 10?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
2. What number is
1
4
of 12?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
3. What number is
1
5
of 10?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
223
4. What number is
1
4
of 28?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
5. What number is
1
6
of 18?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
6. What number is
1
2
× 6?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
7. What number is
1
3
× 6?
(a) Write the solution using division. (Write fraction answers in its simplest form.)
(b) Write the solution using multiplication. (Write fraction answers in its simplest form.)
8. What English word means the same as “×” in the examples above?
Activity 7.3 A Rule for Multiplying a Whole Number times a Fraction
Looking at the previous few pages, you can see a pattern for how multiplication of a fraction
times a whole number or multiplication of a whole number times a fraction works. In your
groups, write a procedure for how to multiply a whole number A times a fraction bc , as in A × cb .
When you are done, make up 3 different problems like this that are different from the problems
on the previous pages.
224
More Multiplying Fractions
= 4×5
= 4. How should we
We’ve seen that 20 × 15 means 51 added 20 times and this gives us 20
5
5
1
interpret 5 × 20?
One way of looking at this is to recognize that multiplication is commutative, meaning that 3 × 4
and 4 × 3 give the same result. Therefore, 51 × 20 should be the same as 20 × 15 = 20
.
5
This doesn’t really explain what 15 × 20 means, however. Remember that 15 is represented as one
shaded box out of five:
1
5
× $20 or one fifth of twenty dollars, means that we have twenty dollars distributed equally
into five slots and we collect one of them. To see this, think of twenty dollar bills placed into
five boxes one at a time:
The first five are placed in the boxes,
$$$$$
$$$$$
$$$$$
$$$$$
$
$
$
$
$
$$$$$
$$$$$
$$$$$
$$$$$
$$
$$
$$
$$
$$
Then the third group of five are placed in
the boxes,
$$$$$
$$$$$
$$$$$
$$$$$
$$$
$$$ $$$
$$$
$$$
Finally, the last five are placed in the boxes.
$$$$$
$$$$$
$$$$$
$$$$$
Then the next five are placed in the boxes,
When we count up the dollars in the shaded box, there are 4, so
In the same way, if we multiplied
shaded boxes:
2
5
1
5
$$$$ $$$$ $$$$ $$$$ $$$$
× 20 = 4.
× 20 we would have the same rectangle but with two
$$$$ $$$$ $$$$ $$$$ $$$$
So the result is 8. Notice that this is consistent with our expectations that
225
2
5
× 20 = 20 ×
2
5
=
40
.
5
This last example was straightforward because 20 is divisible by 5. Let’s try one where it
doesn’t come out so nice:
Suppose that this time you want 14 × 14, or one-fourth of $14. Since 4 doesn’t go into 14 evenly,
when you start to distribute, we have to figure out what to do with the remainder.
The first four are placed in the boxes,
Then the next four are placed in the boxes,
Then the third group of four are placed in
the boxes,
Finally, there are only $2 left in the final
group. In order to distribute them evenly,
we must break each dollar into half-dollars,
then distribute each of the four half-dollars
to the four boxes.
$$$$
$$$$
$$$$
$$
$
$
$
$
$$$$
$$$$
$$$$
$$
$$
$$
$$
$$
$$$$
$$$$
$$$$
$$
$$$
$$$ $$$
$$$$$
$$$$$
$$$$$
Since there are 3 21 dollars in each box, we know that
1
4
$0.50 $0.50
$0.50 $0.50
× 14 = 3 21
Exercise 7.4.5
1. (a) Make a similar drawing distributing $’s into boxes to find
(b) Check that your answer is correct by finding 20 ×
3
.
4
2. (a) Make a similar drawing with sticks and boxes to find
(b) Check that your answer is correct by finding 12 ×
226
2
.
3
2
3
3
4
× 20.
× 12 .
$$$
$$$ $$$ $$$ $$$
$0.50 $0.50 $0.50 $0.50
Another way to see how the denominator divides things into groups is through length. We can
think of the product 34 × 2 as taking three quarters of two inches. So we divide two inches into
quarters and count up three of them:
1
2
From this we can see that
1
4
of 2′′ is a half inch, so
3
4
× 2 = 1 12 .
Exercise 7.4.6 Shade a rectangle above the ruler to find the products:
1. Find
2
3
× 1 21 by shading the rectangle above the ruler.
1
2. Estimate
1
2
× 2 34 by shading the rectangle above the ruler.
1
3. Estimate
5
4
2
2
3
4
5
× 3 by shading the rectangle above the ruler.
1
2
3
227
4
5
Exercise 7.4.7 Show all your work as you complete the following. Include appropriate units
with your answers:
1. A family’s budget is $3,600 per month. 21 goes towards rent, 19 towards food, 41 towards
1
bills, 60
towards entertainment, and the rest to savings. How much money does the
family pay in each category? What fraction goes towards savings?
2. A rectangle measures 4 inches long by
5
8
inches wide.
(a) Estimate the perimeter by rounding each measurement to the nearest whole inch
before computing.
(b) Find the actual perimeter.
(c) Estimate the area by rounding each measurement to the nearest whole inch before
computing.
(d) Find the actual area.
3. After finishing her holiday shopping, 25 of Sandee’s holiday savings account was left. If
she had $1245 saved originally, how much was left after the shopping?
4. Working at the golf course, Martin found 135 golf balls in the lake. 23 of them were too
soggy to use and had to be thrown away. How many golf balls were too soggy?
5. On a fishing trip, the family caught 24 fish. 58 of them were too small, and had to be
thrown back. How many fish did they keep?
6. Joan drank a 12 oz. bottle of Burpo and Carol drank a 16 oz. can of Carbon Light. ”I drank
1
more than you did,” said Carol. “I drank 41 less than you did,” said Joan. Who is right
3
and why ?
/
/
7. A student wrote the following on his arithmetic test: 16
= 41 and 626
= 25 . He claims he has
6
/4
/5
discovered that sixes always cancel. His teacher thinks this was just an accident. Who is
right? If the student is, explain to his teacher why his method always works. If his teacher
is correct, find an example for the student where his method fails.
7
8. A box of Sugar Glopsc cereal contains 10
of a pound of cereal and a box of Astro Puffs
contains 11 ounces of cereal. (1 pound = 16 ounces). They have the same food value
(none) and the same taste (sweet) and cost the same. Which is the better buy?
9. A jet plane cruises at 450 mph for 3
2
3
hours. How far does it travel?
10. After the party, only half of the birthday cake was left. The next morning, I ate
was left. What fraction of the original cake did I eat?
228
2
5
of what
11. You are planning to do some baking for an office party. You figure out that you have
enough cookies for the party and enough to bring to your mother-in-law’s house on
Sunday if you make 6 batches. The amount of each ingredient needed for each batch is
given below:
Amount
hello!
2 12 squares
Ingredient
unsweetened chocolate
1
2
cup
butter
2 cups
flour
1
2
baking soda
teaspoon
1 teaspoon
baking powder
1
4
salt
teaspoon
1 14 cups
white sugar
1 teaspoon
vanilla extract
2
3
sour cream
cups
2 cups
semi-sweet chocolate chips
Fill in the following table with the amounts needed of each ingredient to make 6 batches:
Amount
hello!
hello!
hello!
hello!
hello!
hello!
hello!
hello!
hello!
Ingredient
unsweetened chocolate
butter
flour
baking soda
baking powder
salt
white sugar
vanilla extract
sour cream
semi-sweet chocolate chips
12. You are planting bushes in your back yard. Each bush needs 34 pounds of fertilizer and 4
feet 4 inches of flexi-board to frame the hole. How much fertilizer and how much
flexi-board do you need to buy to have enough materials for 12 bushes?
229
7.5 Fraction × Fraction
We have seen that a fractional part of a whole number can be obtained through multiplication,
. The final step is to look for common
that is, 21 of 6 is the same as 12 × 6, and A × cb = bc × A = A×b
c
factors in the numerator and denominator, so that you can write your result in its simplest form.
To visualize what the fractional part of a fraction means, it is very important to be clear about
the size of “the whole” in your problem.
Example 1: What is
Solution:
1
2
×
1
2
1
2
× 21 ?
is equivalent to
1
2
of 12 .
Suppose your whole is
,
so that one-half shaded would look like
.
To shade only half of one-half, we can cut the rectangle in half like
,
then only shade half of the previously shaded part like
.
From this picture we can see that
1
2
of
1
2
is 14 , or using a math equation,
230
1
2
×
1
2
= 14 .
Example 2: What is
Solution:
2
3
×
1
2
2
3
× 21 ?
is equivalent to
2
3
of 12 .
Again, we will start with one whole as
,
so that one-half shaded would look like
.
This time, we need to cut everything in thirds like
,
so that we can shade
2
3
of the previously shaded part:
.
From this picture we can see that 32 of 21 is 26 , but since 6 = 2×3, we know that 32 × 12 = 26 = 1×2
= 31 .
3×2
Notice that if we move one of the shaded sixths in the picture to the first column, we can see
how 26 = 31 .
=
=
231
Exercise 7.5.1 Find the following products by completing the following steps:
(a) Rewrite the product with the word “of” replacing ×.
(b) Draw a picture representing all of the second fraction in the product.
(c) On the same drawing, but in a different color or darkness, shade the first fraction of the
second fraction.
(d) Write the answer to the product in its simplest form.
3
4
×
1
3
2.
2
3
×
3
8
3.
4
5
×
5
8
4.
1
2
×
5
2
5. 1 ×
1
3
6.
1
2
×
1
3
7.
3
4
×
2
3
8.
5
8
×
1
3
9. 2 ×
1
3
10. 3 ×
1.
1
3
Exercise 7.5.2 Find the following products as in the previous exercise. Compare your
answers with those with the same exercise number from the previous exercise. For example, for
number 1, compare 43 × 13 with 31 × 34 .
1.
1
3
×
3
4
2.
3
8
×
2
3
3.
5
8
×
4
5
4.
5
2
×
1
2
5.
1
3
×1
6.
1
3
×
1
2
7.
2
3
×
3
4
8.
1
3
×
5
8
9.
1
3
×2
10.
1
3
×3
232
Multiplying Mixed Fractions
When we multiply mixed fractions, it’s good to start as we have previously, by using repeated
addition.
Example 1 : (Repeated Addition)
Multiply 5 × 2 38 by repeated addition.
5×2
3
3
3
3
3
3
= 2 +2 +2 +2 +2
8
8
8
8
8
8
3 3 3 3 3
= |2 + 2 +{z
2 + 2 + 2} + + + + +
8 8 8}
|8 8 {z
10
15
8
= 10 +
= 11
7
8
15
8 7
7
= 10 + + = 10 + 1 +
8
8 8
8
We can also represent multiplication as we did previously, using an area model. Since 5 × 2 38
would give us the area of a rectangle with sides 5 and 2 38 , we get the following:
Example 2: (Area)
5
5
5X 2
2 83
5X
−→
Which again gives us 10 +
15
8
= 11 87 .
233
3
8
2
3
8
A third possibility is to multiply 5 × 2 38 by rewriting 2 38 as an improper fraction.
Example 3: (Improper Fraction)
Since 2 83 =
get:
8
8
+
8
8
+
3
8
=
19
,
8
we can use the process of multiplication we developed previously to
19
8
5 × 2 83 = 5 ×
5×2
=
5×19
8
=
95
8
19
3
= 5×
8
8
=
5 × 19
8
=
95
8
= 11
7
8
Using the same reasoning, in order to multiply 2 85 × 1 34 we can look at it as though we’re
finding the area of a rectangle with sides, 2 58 and 1 43 . Then much as we did with distribution in
the previous sections, we can break up the rectangle into smaller pieces.
2 85
5
8
2
1
2
x
1
2
x
3
4
5
x
8
2
5
8
1 1
2
5
8
1
3
4
6
4
15
32
3
4
3
4
−→
3
5
x
8
4
−→
Then the area or the product is: 2 58 × 1 43 = 2 × 1 + 2 × 34 + 85 × 1 + 85 ×
= 2 + 64 + 85 +
=2+
=2+
48
32
83
32
+
20
32
15
32
+
=2+
.
= 4 19
32
234
15
32
32
+ 32
32
32
+
19
32
3
4
Exercise 7.5.3 Multiply.
2. 2 34 × 10
1. 5 × 3 85
3. 4 21 × 2 78
Exercise 7.5.4
1. Miriam is cutting blocks of wood for the wall she is building. If she needs 20 blocks that are
′′
14 38 each, how much wood does she need?
2. Ralph ate 31 of an apple pie. Later Alice ate
Alice eat? (Hint: A picture really helps!)
3
4
of the remainder. What part of the total pie did
3. Find 2 52 × 3 43 using a picture.
2
3
of his monthly income for rent and 11
of what is left on food. If he has $540
4. Henry spends 14
left, what is his monthly income? (Hint: A picture really helps!)
5. How many inches are there in each fraction of a foot:
(b) 14 ft
(a) 12 ft
(d)
1
6
ft
(e)
6. Find
1
2
of
3 ′′
.
4
7. Find
1
2
of
5 ′′
.
8
8. Find
1
2
of
7 ′′
.
16
9. Find
1
2
of 4 14 .
10. Find
2
3
ft
(c)
1
3
ft
(f)
3
4
ft
′′
1
2
′′
of 5 38 .
′′
11. Where should you put the nail to hang a picture in the center of a wall that is 73 43 long?
235
Example: Karen spends 58 of her monthly income on rent and
she’s left with $180,what is her monthly income?
1
4
of what remains on her dog. If
Solution: Begin by drawing a rectangle to represent
Karen’s monthly income:
Now divide the income rectangle into eighths:
and shade 5 of the eighths to represent what Karen
spends on rent:
Then cut the remaining rectangle into fourths (ignore
the vertical lines):
and shade 1 of the fourth to represent what Karen
spends on her dog:
Notice that the 9 remaining rectangles represent the $180
that Karen has left:
−→ $180 ÷ 9 = $20
Then it follows that each of the nine remaining rectangles
represents $20:
= $20
$20 $20 $20 $20 $20 $20 $20 $20
Extending the quarter lines lets us see how many $20
rectangles make up Karen’s income rectangle:
$20 $20 $20 $20 $20 $20 $20 $20
$20 $20 $20 $20 $20 $20 $20 $20
$20 $20 $20 $20 $20 $20 $20 $20
There are 4 × 8 = 32 rectangles, each worth $20, so Karen’s monthly income is 32 × 20 = $640.
236
Dividing Fractions
In this section we will look at dividing fractions. We already have a sense for this from money.
We know that when we divide a dollar into quarters, we get four of them. Mathematically this
means 1 ÷ 14 = 4. Think about the operations that might explain this result.
Exercise 7.5.5 In order to understand division with fractions better, it will help to have a
visual aid. Complete the questions below.
1. Use the tangram puzzle shown below to find the fraction of the whole square represented
by each shape.
A:
B:
B
C:
A
C
D
D:
E:
E
F
F:
2. Use the figure in (1) to answer these questions.
(a)
(b)
(c)
(d)
What fraction of the square is A?
What fraction of the square is F?
How many times does F go into A?
Write the question in part (c) as a division problem using the fractions you wrote for
A and F.
3. Use the figure in (1) to answer these questions.
(a)
(b)
(c)
(d)
What fraction of the square is C?
What fraction of the square is E?
How many times does C go into E?
Write the question in part (c) as a division problem using the fractions you wrote for
C and E.
4. Use the figure in (1) to answer these questions.
(a)
(b)
(c)
(d)
What fraction of the square is A?
What fraction of the square is C?
How many times does C go into A?
Write the question in part (c) as a division problem using the fractions you wrote for
A and C.
237
From your answers to questions 1–4, you should have a sense for the way fractions work with
division. You saw that the smaller fractions of the square go into the larger ones a whole
number of times. Now let’s try to improve our understanding with more examples.
Remember that division is just repeated subtraction, as we saw with dividing whole numbers.
Example: (Repeated Subtraction)
3÷
1
2
Another way to say this is, “How many times can
3−
1
2
2 21 −
2−
1
2
1 21 −
be subtracted from 3?”
= 2 21 =⇒ subtracted 1 time
1
2
= 2 =⇒ subtracted 2 times
= 1 21 =⇒ subtracted 3 times
1
2
= 1 =⇒ subtracted 4 times
1−
1
2
=
1
2
1
2
= 0 =⇒ subtracted a total of 6 times
−
1
2
1
2
=⇒ subtracted 5 times
Therefore, 3 ÷
1
2
=6
′′
′′
1
. Using the ruler
We can also see this nicely using a ruler. Suppose we want to divide 21 ÷ 16
′′
′′
′′
1
1
1
below we can count the number of sixteenths that fit into 2 . So 2 ÷ 16 = 8.
1
Exercise 7.5.6
1. Use the ruler above to help you answer
3
4
2. Use the ruler above to help you answer
9
16
÷
3
.
16
÷
238
3
8
Another way we use to understand division with fractions is to compare them through
common denominators. When the denominators are the same, the whole is broken up into
pieces that are the same size. Therefore, we can ignore the denominators and divide the
numerators.
Example 1: (Common Denominators)
Find
3
4
3
4
Since
1
8
÷
by first writing the fractions with the same denominators.
6
8
=
we can rewrite this problem as
6
8
÷ 18 .
This means how many times does 81 go into 86 ?
We can see that the answer is 6 but it follows directly from dividing the numerators: 6 ÷ 1 = 6.
We can apply this logic to more challenging fractions.
Example: (Common Denominators)
Find
4
5
÷
3
8
by first writing the fractions with the same denominators.
The LCM of 5 and 8 is 40 so we rewrite the fractions,
4
5
−→
32
40
and
3
8
−→
15
40
32
40
÷
So now we have
15
.
40
Since the pieces (the denominators) are the same size, we can ignore them and divide the
number of pieces (the numerators) 32 ÷ 15 which is a little more than two times.
which should give us some insight into the
We can also write the answer as the fraction 32
15
arithmetic of dividing fractions. We’ll return to this question in a minute but first try the
following exercises using the common denominator method.
Exercise 7.5.7 Divide the following fractions:
1.
1
2
÷
1
8
2.
5
8
÷
5
16
4.
3
4
÷
5
8
5.
5
8
÷
5
4
3.
239
7
8
÷
2
3
. Notice
Now let’s return to the result from the previous example. We concluded that 54 ÷ 83 = 32
15
that the numerator of the answer comes from 4 × 8 while the denominator comes from 5 × 3. It
certainly appears as though we multiplied 54 by 83 .
Let’s see if we can explain why this last conclusion would make sense mathematically.
Remember the Golden Rule, that fractions are equivalent as long as we multiply the top
number by the same value we multiply the bottom number by. Also remember that any
number divided by 1 is the same as the original number, e.g. 71 = 7.
“Ours is not to Wonder Why, Just Invert and Multiply!”
Now we are ready to see why
4
5
3
8
4
5
÷
3
8
should equal
4
5
× 38 . We know that
4
5
(since
=
writing . If we apply the Golden Rule and multiply by
4
5
3
8
×
8
3
×
8
3
−→
−→
4
5
×
1
8
3
=
8
3
4
5
×
8
3
=
3
8
×
8
3
÷
24
24
3
8
is equivalent to
= 1), we get
32
15
And this shows why dividing fractions is the same as inverting the second fraction and
multiplying.
Exercise 7.5.8 Answer the following. Include appropriate units with your answers:
1. Pretend you are a teacher explaining to a student new to fractions why we “invert and
multiply,” to calculate 52 ÷ 13 .
2. A girl spends 13 of her savings and loses
much did she start with?
2
3
of the money remaining. She then has $12. How
3
miles long by
3. What is the area, in square miles, of a farm 1 10
2
3
mile wide ?
4. A length of cloth 6 78 yards long is divided into 5 equal pieces. How long is each piece ?
5. A box of modeling clay weighs 45 21 pounds. How many 1 43 lb. packages will fill one box?
6. Maurice is making shelves for his apartment. He wants the shelves to be at least 2 21 feet long.
How many shelves can he make from a piece of wood that is 8 feet long?
7. How long should Maurice make each shelf in #6 if he doesn’t want any wood left over?
8. If the minimum width for a parking space is 8 21 feet, what is the maximum number of
parking spaces you can fit parallel to a 100 foot long street (assume there are no driveways or
tow away zones)?
240
9. Each line shows a trip from point A to point B. In each case identify the fraction of the trip
travelled to at the given letter.
(a)
p
q
A
B
(b)
m
n
A
B
(c)
u
v
w
A
B
10. If B marks the end of the trip, show where B should be located on the line if the fraction
identifies the fraction of the trip from A to B completed at that point.
(a)
A
1
4
(b)
A
3
5
11. Write the following ounce measurements as fractions of 1 pound. (Remember there are 16
ounces in 1 pound).
(a) 8 oz.
(b) 10 oz.
(c) 12 oz.
(d) 6 oz.
12. Write the following times as fractions of an hour.
(a) 20 minutes
(b) 15 minutes
(c) 50 minutes
241
(d) 17 minutes
7.6 Decimal Fractions
Decimal fractions are a special case of fractions that use powers of ten, e.g. 10, 100, 1000, etc., as
denominators. These are exactly the numbers used in place value (see page 7) and as a result
we extend the place value definition to give an alternative representation of the fraction. Just as
the number to the left of the ones place is designated for tens (because ten ones add up to ten),
the number to the right of the ones place is designated for tenths (because ten tenths add up to
3
or equivalently as 0.3.
one). We will write three tenths as 10
Multiplication with decimals
A fraction is nothing more than a convenient way of dividing something up (the denominator)
and counting up the number of pieces (the numerator). When we take 43 of $60, we divide sixty
into four groups of $15 each and then add up three of the groups to get $45.
Exercise 7.6.1
1. Write the following decimal fractions in fraction form with appropriate denominators.
(a) 0.5
(b) 0.05
(c) 0.009
(d) 0.34
(e) 0.85
(f) 1.7
2. Write the following fractions in decimal form.
(a)
1
10
(b)
3
100
(c)
89
100
(d)
127
1000
(e)
237
100
Example 1:
Find 0.4 × 60.
4
4
4
Because 0.4 is the same as writing 10
, multiplying 0.4 × 60 is the same as 10
× 60 or taking 10
of
60. As with other fractions, we can divide first, to get 60 ÷ 10 = 6 and then add up four of the
sixes to get 24.
Equivalently we can write this mathematically as
4
10
× 60 =
4×60
10
=
240
10
and then divide to get 24.
Like the hundreds place, two spaces to the left of the ones, the decimal representation for
7
hundredths is two spaces to the right of the ones place. So 0.07 is equivalent to 100
. The decimal
2
7
20
7
27
0.27 is equivalent to 2 tenths and 7 hundredths so this is 10 + 100 or 100 + 100 = 100 .
242
Multiplying decimals is technically no different than multiplying fractions. We will develop the
familiar shortcuts but we will start with an example.
Example 2:
Find 0.32 × 20.
We’ll start by converting 0.32 to
Then
32
.
100
32 × 20
32
× 20 =
100
100
640
=
100
64
=
10
4
= 6
10
4
has 10 in its denominator, it is equivalent to the decimal fraction 0.4 and it is
Because 10
4
preferable to write the result, 6 10
, as 6.4.
Exercise 7.6.2 Include appropriate units with your answers.
1. Multiply these decimal fractions by first converting to fraction form. Write your answers in
decimal form.
(a) 0.3 × 20
(b) 0.08 × 15
(c) 12 × 0.2
(d) 0.24 × 42
2. Do you recognize a shorter way to get your results rather than converting to fractions and
then back to decimals? Explain.
3. Find the area of a rectangle that measures 5.3 cm by 12 cm.
4. If you put $0.35 in a jar every day for 52 days in a row, how much money would you have in
the jar all together?
5. Jolene bought notebooks for her 3rd grade class. If she has 28 students, and each notebook
costs $3.64 with tax, how much was the total cost? If the PTA donates $50, how much does
Jolene have to pay out of her own money?
6. A U.S. city block measures 201.168 meters on each side. Estimate the area of a U.S. city block.
7. There are 100 centimeters in every meter. How many centimeters are in the length of a city
block?
8. During the 2008 Democratic Primary election in California, Hillary Clinton received 56.2% of
the 347 delegates. How many delegates did she receive?
243
Example:
Find 0.6 × 0.45.
Again we start by converting 0.6 to
Then
6
10
and 0.45 to
45
100
.
6
45
6 × 45
×
=
10 100
10 × 100
270
=
1000
27
=
100
27
Because 100
has 100 in its denominator, it is equivalent to the decimal fraction 0.27 and it is
preferable to write it this way.
Exercise 7.6.3 Include appropriate units with your answers.
1. Multiply these decimal fractions by first converting to fraction form. Write your answers in
decimal form.
(a) 0.3 × 0.2
(b) 0.08 × 1.5
(c) 0.12 × 0.2
(d) 0.24 × 0.42
2. Compare your answers with those in the previous exercise set. Again, do you recognize a
shorter way to get your results rather than converting to fractions and then back to decimals?
Explain.
3. Multiply 0.4 × 0.35 using any shortcut you described above.
4. A family’s monthly budget is $4,250. If they spend 38.6% of their budget on rent, what is
their monthly rent?
5. In lab, a bio-student measures the sample given to her by her instructor. It measures 42.8 ml.
If there are 18 students in the class, and the instructor wants to have 20% of the total extra, how
much should the instructor have on hand for the lab?
6. As a contractor, Christina developed a rule when bidding the total time it takes to do a job.
First, she estimates a reasonable amount of time to do the job if everything goes according to
plan, and there are no problems. Then, she doubles the total. Finally, she increases that result
by 10%. If her initial estimate for the reasonable amount of time to do the job is 132 hours, how
much time will she use for the bid?
244
Using decimals with division
The process of division usually ends when the remainder is smaller than the divisor and we
leave it either as a remainder or as a fraction. Using decimals gives us an alternative way of
representing a fractional remainder as well as an alternative way of getting it.
Just as we did with dividing money between members of a group, whenever there isn’t enough
of a larger bill to go around, we change it out for ten of some smaller quantity. If there are four
people and only three dollars, we don’t stop dividing, we just exchange the dollars for dimes. If
there aren’t enough dimes to split up, we exchange the dimes for pennies.
The process of division with decimal fractions is exactly like splitting up money except we
don’t have to stop at pennies (hundredths) when we divide up an amount. If there were a unit
of money equal to a tenth of a penny, we would be able to keep dividing, and so it is with
decimals.
Example 1:
Divide 3 ÷ 8.
This is the same as saying we want to divide $3 among 8 people. We see that there aren’t enough
dollars to go around so our first step is to exchange our three dollars for 30 dimes.
Now 30 ÷ 8 = 3 with 6 dimes left over, so everyone gets 3 dimes.
Since we can’t divide up the 6 dimes, we exchange them for 60 pennies and again divide,
60 ÷ 8 = 7 with four pennies remaining.
Now normally we would be done since there aren’t any coins equivalent to a tenth of a penny,
but let’s imagine there are. We trade in our 4 pennies for 40 of these tenths of a penny, and now
40 ÷ 8 = 5 so each person gets 5 tenths of a penny, or 5 thousandths of a dollar.
Totalling the coins each person gets, we have 3 dimes, 7 pennies, and 5 tenths of a penny or
equivalently,
3 tenths of a dollar, 7 hundredths of a dollar, and 5 thousandths of a dollar. In decimal notation
this looks like 0.3 + 0.07 + 0.005 which adds up to 0.375.
245
Example 2:
Repeat the example above using the division algorithm.
0
8 3
0.
8 30
0.3
8 30
24
6
0.3
8 30
24
60
0.37
8 30
24
60
56
4
0.375
8 30
24
60
56
40
40
0
Since there aren’t any 8’s in 3 (since 3 isn’t big enough), we write a 0 above
the ones place.
We exchange three ones for 30 tenths (dimes) and note this above by marking a decimal point (to show we’re now in tenths).
8 divides 30 three times with a remainder of 6 tenths (meaning 8×0.3 = 2.4).
Again, we exchange 6 tenths for 60 hundredths (notice how the places line
up).
8 divides 60 seven times with a remainder of 4 hundredths (meaning 8 ×
0.07 = 0.56).
Finally, we exchange 4 hundredths for 40 thousandths and since 8 × 0.005 =
0.04 we are done. So 3 ÷ 8 = 0.375.
246
Exercise 7.6.4
1. Divide the following numbers by hand using decimals.
(a) 1 ÷ 2
(b) 1 ÷ 4
(c) 3 ÷ 5
(d)
1
8
(e)
5
8
2. Sometimes we get numbers with decimals that never end but repeat forever. This isn’t a
problem as long as we recognize the pattern and write the result with a line over the repeating
part. Divide the fractions below and express the answer with a line over the repeating part.
(a) 1 ÷ 3
(b) 2 ÷ 3
(c) 1 ÷ 9
3. The decimal expressions for numbers divided by 7 have an interesting pattern. See if you can
find it by doing the following problems.
(a)
1
7
(b)
2
7
(c)
3
7
(d)
4
7
4. See if the pattern you found in #3 can help you predict the decimals for
5
7
and 76 .
Division with Decimals
As with multiplication, we can use the same methods we developed for dividing fractions to
divide decimal fractions.
Example:
Find 0.6 ÷ 0.02.
Written as fractions we get
6
2
6
100
÷
=
×
10 100
10
2
600
=
20
= 30
Exercise 7.6.5 Find these quotients by converting the decimals to fraction form first before
dividing.
(1) 0.8 ÷ 0.004
(2) 0.12 ÷ 0.03
(3) 1.5 ÷ 0.05
247
(4) 0.42 ÷ 0.007
The more familiar version of the division procedure for decimals is to “move” the decimals in
both the divisor and the dividend the same number of times until the decimal no longer
remains in the divisor.
This looks like
0.04 0.72
−→
4 72
But there’s rarely much explanation to support it.
. Obviously it would be nice
To see what is happening, let’s write the quotient as a fraction: 0.72
0.04
if we didn’t have the decimals involved so we look at the divisor, 0.04, and since it is in
hundredths, we use the Golden Rule and multiply both the top and the bottom of the fraction
by 100:
×100
72
0.72 −→
= 18
−→ =
0.04 ×100
4
This is equivalent to the method of moving the decimal but provides some explanation for
what’s going on.
Exercise 7.6.6 Complete these division problems by hand using any method.
(1)
0.42
0.03
(2)
6.6
0.3
(3) 0.06 0.39
(4) 1.2 0.036
Exercise 7.6.7 Include appropriate units with your answers.
1. Find the answers to these two problems. Compare the methods you used to get your
answers.
(a)
(b)
1 3 7
+
9 2
0 . 1 3 7
+ 0 . 0 9 2
2. Explain how to find the answer to this problem. Explain why your method gives the correct
answer.
0 . 1 3
× 0 . 5
3. Holly and Kevin went to Canada on a vacation. Holly changed money before leaving. For
every US$0.82 , she got one Canadian dollar. Kevin changed his money when he got there. For
each US$1.00 he got $1.20 in Canadian money. Who got the better deal?
4. Convert 1.735 to a fraction and then explain a general rule for converting decimals to
fractions.
5. James needed to divide a 12 foot 2 × 4 into 5 pieces of equal length. He used his calculator to
find that each piece needed to be 2.4 feet long so he began cutting pieces that were 2 feet, 4
inches. When James was done, how long was his last remaining piece?
248
6. During group work, one student says that 1.2 hours is one hour and 20 minutes. Another
student doesn’t agree. Who is right? Explain so that both students will understand what is
correct.
7. Check the appropriate box for the result of each operation. You should do these in your head.
Less than 15
More than 15
(a) 5.65 + 9.47
(b) 21.06 − 15.49
Less than 6
More than 6
(c) 4.08 × 9.12
Less than 36
More than 36
(d) 4.08 × 9.12
Less than 37
More than 37
(e) 17.8 ÷ 3.2
Less than 6
More than 6
8. Complete these equivalent fractions.
(a)
4
5
=
100
(b)
1
5
=
100
(c)
3
4
=
100
(d)
1
2
=
100
(e)
2
3
=
100
7.7 Dividing with Negative Numbers
As we did with multiplication, we can makes sense of negative numbers in division by
thinking of money and debt.
Example: Donald Duck’s nephews, Huey, Dewey, and Louie, were in business for themselves.
Things were not going so well, so the company was in debt. The total debt of the company was
$35,400. If each nephew shared the debt equally, how much debt did they each have?
Solution: Since the total we will split into three equal shares is debt, we will use a negative
number. The situation can be modeled with the following expression:
−35, 400 ÷ 3
This problem is identical to 35, 400 ÷ 3, except that one is splitting debt, while the other is
splitting money! Since 35, 400 ÷ 3 = 11, 800 (check that!), we know that −35, 400 ÷ 3 = −11, 800.
Each nephew has $11,800 in debt.
In general, with multiplication, if you have a negative number repeated many times, you will
get a negative number.
With division, if you divide up a negative number into several equal sized pieces, each piece
will be negative.
249
Exercise 7.7.1 Find the following products or quotients.
1. −28 ÷ 4
2. 72 ÷ 9
3. 4 × (−3.86)
4. −8.4 ÷ 12
5. 7 × 0.022
6. −18, 772 ÷ 8
7. 9, 216 ÷ 36
Exercise 7.7.2 Find the following products or quotients. Write fractional answers in fraction
notation in simplest form.
1. 23 ÷ 61
2. − 54 ÷ 85
3. 21 × − 45
4. − 38 ÷ 29
Exercise 7.7.3 Answer the following.
1. Late in July 2007, the U.S. stock market went crazy due to the large number of mortgage
foreclosures in the country. On Monday, July 23, the Dow opened at 13,851.73. On Friday,
July 27, the Dow closed at 13,265.47.
(a) What was the total change in the Dow for that week? Write the answer as a positive
number if the Dow went up, and as a negative number if the Dow went down.
(b) On average, how much did the Dow change each day during the 5 days? Write the
answer as a positive number if the Dow went up, and as a negative number if the
Dow went down. Round your answer to the nearest hundredth.
2. During a trip back from Lake Tahoe, the couple started keeping track of their altitude.
They saw a sign on the road that said 7372 feet. An hour and a half later in Rocklin, their
altitude was 282 feet.
(a) What was the total change in their altitude? Write the answer as a positive number if
their altitude went up, and as a negative number if their altitude went down.
(b) On average, how much did their altitude change each hour? Write the answer as a
positive number if their altitude went up, and as a negative number if their altitude
went down. Round your answer to the nearest hundredth.
250
Notes:
251
8 Ratio and Proportion
8.1 Ratio
A ratio is a comparison between two quantities using division. The things being compared may
be of the same type. For example, if Sara is playing basketball and makes 12 baskets out of the
12
19 shots she took, her ratio of baskets made to baskets attempted is 19
which is sometimes
written 12 : 19 (This is read out loud as, “Twelve to nineteen”).
Sometimes ratios compare two different types of things. For example, if Jerry buys 8 pounds of
= 21 or 2. Notice that these ratios give us
apples for $16, then the ratio of dollars to pounds is 16
8
something more than just a way of writing the numbers in a fraction form. In the second case,
16 dollars
gives us the rate at which Jerry pays for apples, namely sixteen dollars for every 8
8 pounds
pounds or equivalently, $2 per pound.
Exercise 8.1.1 Find the following ratios and simplify them whenever possible:
1. If you travel 400 miles in 8 hours, what is the ratio of miles to hours?
What does this ratio tell you?
2. If you travel 352 miles on 16 gallons of gas, what is the ratio of miles to gallons?
What does this ratio tell you?
3. If you work for 28 hours and make $392, what is the ratio of dollars to hours?
What does this ratio tell you?
4. If you make $50,000 in 12 months, what is the ratio of dollars to months?
What does this ratio tell you?
5. A 2 foot stick measures 24 inches. What is the ratio of inches to feet?
What does this ratio tell you?
6. A 20 inch stick measures 50.8 centimeters. What is the ratio of centimeters to inches?
What does this ratio tell you?
252
Exercise 8.1.2 Use information from your class to find the ratios. Write the ratios as fractions
in its simplest form.
1. The ratio of females to males in your math class.
2. The ratio of males to females in your math class.
3. The ratio of females to all people in your math class.
4. The ratio of males to all people in your math class.
5. The ratio of smokers to nonsmokers in your math class.
6. The ratio of nonsmokers to smokers in your math class.
7. The ratio of smokers to all people in your math class.
8.2 Equivalent Fractions
We will often find it useful to be able to generate equivalent fractions with a particular
denominator. For example, if we want to write 85 as a fraction out of 56 rather than 8, we will
need a procedure. Let’s see if we can describe how to do this.
Exercise 8.2.1
1. Find the missing numbers in the fractions below.
(a)
3
=
5
20
(b)
4
=
9
54
(c)
1
=
12
60
(d)
7
=
13
91
2. Describe your procedure for finding the missing numbers in the fractions above.
3. Use your procedure in (2) to help you find the missing number in the fraction:
253
4
=
5
42
8.3 Proportion
When two ratios are equal it is called a proportion.
882
is equivalent (after
Example: Suppose that the ratio of boys to girls at a local high school, 1029
simplifying) to the ratio of boys to girls in an algebra class 76 . Then we say that the ratio of boys
882
to girls at the school is in proportion to the number of boys and girls in the class −→ 1029
= 67 .
Similarly suppose the ratio of boys to girls in Cleveland is in proportion to the number of boys
and girls in that class. If there are 126,000 girls in Cleveland, how many boys are there?
# boys in Cleveland
# boys in class
=
# girls in class
# girls in Cleveland
6
?
−→ =
7
126000
Since 126, 000/7 = 18000, you multiply the numerator by 18, 000:
We would write
×18000
108000
6 −→
−→ =
7 ×18000
126000
So there are 108,000 boys in Cleveland.
Exercise 8.3.1 Use a proportion to solve each problem. Include appropriate units in your
answers.
1. A car travels 224 km on 4 gallons of gas. How far can it be expected to travel on a tank of 12
gallons?
2. If you make $600 (after taxes) working for 70 hours, how much will you make working for
500 hours?
3. A 425 pound motorcycle weighs 68 pounds on the moon. How much will a 120 pound
woman weigh on the moon?
How much would you weigh on the moon?
4. 1 inch is equivalent in length to 2.54 centimeters. How tall (in inches) is someone who is
180cm?
5. A recipe for six dozen cookies calls for 2 12 cups of flour. How many cups of flour are needed
for 10 dozen cookies? Round the answer to the nearest 14 cup.
′′
6. The floor plan of a house is drawn to the scale of 14 = 1′ . The master bedroom measures 3 21
by 5′′ on the blueprints. What is the actual size of the room?
′′
7. On average, an adult flea is 3mm long but it can long jump 330mm (about 13 inches) and
high jump 204mm (about 8 inches). If you could jump in proportion with a flea, how far could
you long jump? High jump?
8. If a company’s stock falls by 3% or $1.20, how much was the stock worth originally?
254
9. a) If one gallon of paint covers 400 square feet, how many gallons of paint do you need to
cover 5000 square feet?
b) If one gallon of paint covers 400 square feet, how many square feet will 27 gallons cover?
10. If your car gets 25 miles to the gallon and gas costs $3.35/gal, how much will a 420 mile trip
cost?
8.4 Percent
Percent is a special case of proportion. The word comes from the Latin for by the hundred and it
is used to represent fractions as their equivalent value out of 100. For example, since 15 is
20
equivalent to 100
, we say that 51 is the same as 20 percent. To remind us that a percent is a
fraction of 100, the percent symbol is composed of the division sign and the two zeros we find
in 100: %.
Example 1:
What percent of the rectangle below is shaded?
Solution 1: Since there are 20 boxes and 9 of them are
9
shaded, we can say 20
of the rectangle is shaded. To
write this as a percentage, we need to find an equivalent fraction out of 100:
×5
9
9 ×100/20
9 −→
45
=
=
−→
−→
=
−→
20
100
20 ×100/20
100
20 ×5
100
So 45% of the rectangle is shaded.
Solution 2: One great thing about wanting to know how many hundredths you have, is that the
9
into a decimal by dividing, then
hundredths place is one of our place values! If we can turn 20
we can see the answer:
9
20
= 9 ÷ 20 = 0.45 which is 45 hundredths, or 45%.
255
Example 2:
What percent of the rectangle below is shaded?
Solution: Since there are 40 boxes and 15 of them are
shaded, we can say 15
of the rectangle is shaded. To
40
write this as a percentage, we need to find an equivalent fraction out of 100:
×2.5
15
15 ×100/40
15 −→
37.5
=
=
−→
−→
=
−→
40
100
40 ×100/40
100
40 ×2.5
100
So 37.5% of the rectangle is shaded.
Exercise 8.4.1
What percent of each rectangle is shaded?
(1)
(2)
(3)
256
Example:
Shade 30% of the rectangle shown below.
So we shade
squares:
In order to shade 30% of the rectangle, we
need to know what fraction of the total
squares to shade. Since there are 20 squares
in all we need to write 30% as a fraction out
of 20:
30
30
6
= 20 −→ 100
= 20
100
six
Exercise 8.4.2
1.Shade the indicated percentage of each rectangle.
(a) 15%
(b) 40%
(c) 67%
2. Show three different ways of shading 50% of a rectangle
(a)
(b)
(c)
257
8.5 Proportion and Percent in Problem Solving
Since a percent is really just a fraction, one way of taking a percentage of something is the same
60
as taking a fraction of it. For example, 60% of 20 is the same as writing 100
× 20 = 1200
= 12.
100
In addition, we can solve problems involving percentages by setting up and solving
proportions.
Example 1:
There are 30 students in class, and you are told that 40% of them are male. How many males are
there?
Solution:
We can set up a proportion by equating the males to students ratio. Knowing that 40% of the
class is male, means that the males to students ratio is 40 to 100. We want to know how many
males there are if there are 30 students. The proportion looks like the following:
40 males
100 students
=
unknown number of males
30 students
To solve the proportion, we use the golden rule. To find out what to multiply times 100 to get
40
30, we divide 30 ÷ 100 = 0.3 and multiply the numerator and denominator of 100
by 0.3 to obtain
the following:
40 males×0.3
100 students×0.3
=
12 males
. There are 12 males in the class.
30 students
Example 2:
During one year, the Math Club had 14 female members. That was 70% of the students in the
club. What was the total number of students in the Math Club that year?
Solution:
Again, we can set up a proportion. Knowing that 70% of the club is female, means that the
females to students ratio is 70 to 100. This time, the unknown is the number of students when
there are 14 females. The proportion looks like the following:
70 females
100 students
=
14 females
unknown number of students
To solve the proportion, we use the golden rule. To find out what to multiply times 70 to get 14,
70
by 0.2 to obtain the
we divide 14 ÷ 70 = 0.2 and multiply the numerator and denominator of 100
following:
70 females×0.2
100 students×0.2
=
14 females
. There are 20 students in the class.
20 students
258
Exercise 8.5.1 Solve the following ratio, proportion, and percent problems. Include
appropriate units in your answers.
1. Find the following percentages.
(a) 25% of 60
(b) 30% of 40
(c) 12% of 40
2. Some easy (if you know the trick!) percentages. Find the following percentages in your
head.
(a) 10% of 70
(b) 10% of 120
(c) 10% of 35
(d) 10% of 19
3. What patterns do you notice about your answers in the previous exercise?
4. 21 out of 30 students commute more than 10 miles each day. What percent commute more
than 10 miles? (Hint: The unknown is how many out of 100.)
5. Tax around California is about 9.25%. Estimate tax on each item by using 10% as your
estimate and then add this to estimate the total cost.
(a) $50 shirt
(b) $75 jacket
(c) $234 camera
(d) $1.75 coffee
6. Determine what amount the following numbers are 10% of. (e.g. $35 is 10% of $350).
(a) $8 (Careful! the answer is NOT $0.80!)
(b) $29
(c) $43.50
(d) $0.60
7. Some more easy (if you know the trick!) percentages. Find the following percentages in
your head.
(a) 20% of 70
(b) 20% of 120
(c) 20% of 35
(d) 20% of 19
8. Some more easy (if you know the trick!) percentages. Find the following percentages in
your head.
(a) 1% of 70
(b) 1% of 120
(c) 1% of 35
(d) 1% of 19
9. A sweater that normally costs $40 is marked down by 20%. How much will it cost now
(before tax)?
10. Jimmy had $120 to spend on a VCR. What price of VCR can he afford if the tax, shipping,
and handling comes to 18% of the price of the VCR?
11. Albert is making Chinese food for his friends who are coming over for dinner Saturday
night. The recipe is asking for 5 12 cups of mushrooms, 3 43 cups of bell peppers, and 2 41
cups of chicken breast. This recipe will feed 8 people. Albert only wants to feed 5 people.
How many cups of each ingredient does he need?
12. If 45% of the workers in Jack’s company “brown bag” their lunch, how many people are
in the company if 57 people “brown bag” their lunch?
259
Exercise 8.5.2 Continue to solve more ratio, proportion, and percent problems. Include
appropriate units in your answers.
1. A new car is offered with a 12% rebate. If the car normally costs $18,000, how much will it
cost now (before tax)?
2. Of the 1600 students who voted, 45% of them voted yesterday. How many students voted
yesterday?
3. 9 out of 16 people voted for the “Clean Air” bill in Clark’s town. If 15,000 people voted
for the “Clean Air” bill, how many people are there in Clark’s town?
4. If a computer costs $1000 but is marked down 30%, how much will it cost with 9.25%tax?
5. Suppose you have $35 in your wallet and you go out to eat. If you plan to pay 20% of the
bill in tip and taxes, how much should your meal add up to without tip and taxes in order
for you to cover the bill?
6. There are 65 students in Biology 101. If 43 students went on the field trip yesterday, what
percent of the students didn’t go on the field trip yesterday?
7. Anderson and Jake went fishing. Anderson caught 22 fish. That amount was 89% of the
number of fish that Jake caught. How many fish did Jake catch?
8. There are 26 students in your chemistry class. Eight students were absent on Friday. What
percent of the students were absent? What percent of the students were present?
9. Sammy bought a jacket that was on sale for 85% of the original price. If Sammy paid $240,
what was the original price of the jacket?
10. Mehdi bought 15 feet of pipe from the hardware store. How many yards of pipe did he
buy if each yard equals three feet?
11. Nikkie bought a pair of shoes with a purchase price of $43. The sales tax in her county is
9.25%. What was her total cost?
12. If you put 9 red marbles in a bag with six green marbles,
(a) what percent of the marbles are red?
(b) what percent of the marbles are green?
13. If Michelle makes 63% of her shots in basketball, how many shots did she make if she
attempted 40 shots?
260
8.6 Unit Conversion
Unit conversions can be computed using proportions since in essence a unit conversion factor
is a ratio. For example, we know that there are 12 inches in a foot. This means that the inches to
feet ratio is 12 to 1!
Example 1:
There are 12 inches in each foot. What is the measure in feet of a wall that measures 162 inches?
Solution:
We can set up a proportion by equating the inches to feet ratios. The proportion looks like the
following:
12 inches
1 foot
=
162 inches
unknown number of feet
To solve the proportion, we use the golden rule. To find out what to multiply times 12 to get
162, we divide 162 ÷ 12 = 13.5 and multiply the numerator and denominator of 12
by 13.5 to
1
obtain the following:
12 inches×13.5
1 foot×13.5
=
162 inches
. The wall measures 13.5 feet.
13.5 feet
Exercise 8.6.1 Compute the following unit conversions using proportions. Include
appropriate units in your answers.
1. After working 3500 hours without any time off, Jerry decided to take a vacation.
(a) If, on average, Jerry worked 8 hours a day, how many days has he worked?
(b) If Jerry worked 5 days each week, how many weeks has he worked?
(c) How many minutes has he worked?
(d) How many seconds has he worked?
2. The radius of the earth is 6307 kilometers.
(a) If 1 mile equals 1.609 kilometers, what is the radius of the earth in miles?
(b) If 1 kilometer equals 0.62 miles, what is the radius of the earth in miles?
(c) If 1 kilometer equals 1000 meters, what is the radius of the earth in meters?
(d) If 1 mile equals 5280 feet, what is the radius of the earth in feet?
(e) If 1 meter equals 100 centimeters, what is the radius of the earth in centimeters?
(f) If 1 foot equals 12 inches, what is the radius of the earth in inches?
261
3. José is carpeting a room that measures 12 feet 9 inches by 15 feet.
(a) How many square feet of carpet does he need?
(b) How many square inches of carpet does he need?
(c) How many square yards of carpet does he need?
(d) If 1 inch equals 2.54 centimeters, how many square centimeters of carpet does he
need?
(e) How many square meters of carpet does he need?
4. Alice is building a patio in her backyard. There are a variety of tiles that she can choose
from. If the patio measure 8 yards by 12 yards. How many tiles does she need if the size
of the tiles are:
(a) 1 square yard?
(b) 1 square foot?
(c) 1 square inch?
(d) 1 square meter?
(e) 1 square centimeter?
5. You are traveling around the world and want to spend $1200 in every country that you
visit. Use the Internet or a newspaper to find the current exchange rates for different
currencies, then find out how much money you would have to spend in the currencies of
the following countries:
(a) Mexico.
(b) The Philippines.
(c) United Kingdom.
(d) France.
(e) China.
(f) Japan.
262
GPA
In the following activity, you will learn how to calculate your course grade using a Simple
Average, and then continue to learn how to compute a Weighted Average.
An Average is a number which summarizes several other numbers. It gives information about a
group of numbers using a single value.
To calculate the Simple Average of a list of numbers,
• Find the sum of the list of numbers.
• Count how many numbers are in the list.
• Divide the sum by the count.
Exercise 8.6.2
1. Suppose a student got a 70 on the first exam, an 80 on the second exam, and a 90 on the
third exam.
(a) What is the sum of all the scores?
(b) What is the count of all the scores?
(c) What is the Simple Average of all the scores?
2. Now suppose a student got a 60 on the first exam, an 80 on the second exam, and a 100 on
the third exam.
(a) What is the sum of all the scores?
(b) What is the count of all the scores?
(c) What is the Simple Average of all the scores?
3. Finally, suppose a student got an 80 on the first exam, an 80 on the second exam, and an
80 on the third exam.
(a) What is the sum of all the scores?
(b) What is the count of all the scores?
(c) What is the simple average of all the scores?
4. Explain how a student could get different scores, but have the same Simple Average.
5. Find the simple average of the following exam scores: 93, 85, and 76.
6. Find the simple average of the following quiz scores: 15, 10, and 19.
7. Suppose each quiz was out of a possible 20 points. Find the percent correct for each of the
quiz scores: 15, 10, and 19.
8. Find the simple average of the percents in the previous problem.
263
9. Find the simple average of the following project scores: 45 and 48.
10. Suppose a student has scored 93, 85, and 76 on the first three exams. What is the least
number of points they need on the fourth exam to have a “B” (at least 80) average?
11. Explain in complete sentences how you were able to calculate the last problem.
12. Before taking the fourth exam, you are curious to see if an “A” average is possible. How
would you investigate the possibility, and what is your conclusion?
A Weighted Average counts certain scores more than others. Many instructors use a weighted
average to calculate your grade because they wish to weigh your exam performance more
heavily than, say, your group quizzes.
Example 1: Suppose your instructor weighs exams twice as much as group quizzes. Then,
suppose you have taken 1 exam and scored 50%, and one group quiz and scored 90%. Find the
simple average of the two percents, then find the weighted average.
Solution: The simple average is (simply!)
50+90
2
= 70. The simple average is 70%.
For the weighted average, we multiply each score by its weight before adding, then divide by
the total of the weights. Since exams are weighted twice as much as group quizzes, we use 2 for
the exam weight, and 1 for the group quiz weight. The weighted average is then:
2·50+1·90
2+1
=
190
3
The weighted average is approximately 63.33%.
264
≈ 63.33.
Example 2: Suppose your instructor weighs exams as 60%, quizzes as 20%, and homework and
projects as 10% each. Suppose your simple averages in each category are as follows:
• Exams: 67%
• Quizzes: 72%
• Homework: 80%
• Projects: 70%
Is this student passing with a weighted average of over 70%?
Solution: The weighted average is calculated as follows:
60·67+20·72+10·80+10·70
60+20+10+10
=
4020+1440+800+700
100
=
6960
100
= 69.6
The student is barely not passing. They have a weighted average of 69.6% which is barely less
than 70%.
265
Exercise 8.6.3 Answer the following questions based on the given information:
• The instructor uses the following weights: 50% for exams, 25% for quizzes, 15% for
projects, and 10% for group activities.
• The student has the following exam percentages: 83, 91, 75
• The student has the following quiz percentages: 78, 65, 92, 81
• The student has the following project percentages: 95, 89
• The student has the following group activity percentages: 79, 84, 68, 87
1. What is the simple average of the exams?
2. What is the simple average of the quizzes?
3. What is the simple average of the projects?
4. What is the simple average of the group activities?
5. What is their overall weighted average?
6. If the scores in the other categories stayed the same, what exam average would this
student need to have a weighted average of 90%?
7. If there are two more exams but no more assignments left in the other categories, is it
possible for this student to have a weighted average of 90%?
At Skyline College, your overall GPA is a weighted average where the scores are numbers
corresponding to the grade in the course, and the weights are the number of units in the course.
The numbers for the grades are: A = 4, B = 3, C = 2, D = 1, and F = 0.
Exercise 8.6.4 Answer the following GPA questions.
1. During their first semester, a student takes American History (3 units) and gets a B,
Philosophy (3 units) and gets a C, and Fundamentals of Math (5 units) and gets an A.
What is their overall GPA for this semester?
2. The next semester, they take English (3 units) and get a B, PE (0.5 units) and get an A, the
first half of beginning algebra (3 units) and get a B, and an MS Word class (2 units) and get
a C. What is their GPA for this semester?
3. What is their overall combined GPA for the two semesters?
4. They hear that a scholarship is available, but the GPA requirement is a 3.5. How many
units of straight A work would they need in order to boost their GPA to a 3.5?
266
Notes:
267
9 Order of Operations
Discussion Questions
1. At Santa Cruz Beach Boardwalk, I bought an All-Day Unlimited Rides Wristband for
$26.95, and 3 cotton candies for $1.25 each.
(a) What two different operations (addition, subtraction, multiplication, division) could
be used to calculate the total cost for my day at The Boardwalk?
(b) Write in words how to calculate the total cost, including what order you use the
numbers given in the problem and operations from part (a).
2. Each month, the household spends $450 for food and $1950 for rent.
(a) What two different operations (addition, subtraction, multiplication, division) could
be used to calculate the total yearly budget for food and rent?
(b) Write in words how to calculate the total yearly budget for food and rent, including
what order you use the numbers given in the problem and operations from part (a).
9.1 Order of Operations Agreement
Whenever there is more than one operation used to solve a problem, it is important to know
which order to calculate. If you calculate in the wrong order, you will get the wrong answer!
Since the order makes a difference, mathematicians came to an agreement (the Order of
Operations Agreement) about how to indicate what order the operations must be performed.
For multiplication and addition the agreement is:
• Multiplication must be performed before addition, regardless of the order it is written.
• If the addition should be performed first, use grouping symbols (parenthesis).
Example 1: Evaluate 3 + 5 × 2.
Solution: The multiplication is performed first, so 3 + 5 × 2 = 3 + 10 = 13.
Example 2: How can we make 3 plus 5 times 2 equal to 16?
Solution: To indicate that the addition should be performed first, use grouping symbols:
(3 + 5) × 2 = 8 × 2 = 16. (Note: When parenthesis are used, and a number is multiplied by the
result in the parenthesis, the × symbol is optional. The previous expression could have been
written: (3 + 5)2 or more commonly, 2(3 + 5).)
268
Exercise 9.1.1 Find the value of the following expressions:
1. 7 + 4 × 3
2. 3(5 + 11)
3.
1
2
× 43 + 1 32
4. 13 + 1.25 × 0.6
Addition and Subtraction from Left to Right
Suppose you are keeping track of your checking account balance. The starting balance is
$150.75. You then write a check for $120. Since your balance is getting low, you make a deposit
of $40. The correct way to write the mathematical expression for this situation is also the most
straightforward way, that is:
150.75 − 120 + 40.
To evaluate it, we do the addition and subtraction from left to right and get:
150.75 − 120 + 40 = 30.75 + 40 = 70.75.
Compare this to a similar situation in which you start the month with a balance of $2500, then
write checks for $32.95, $150, and $535. In this case, you must use grouping symbols to indicate
that you need to total the checks before subtracting from your balance. Inside the parenthesis,
the addition can be done in any order you like:
2500 − (32.95 + 150 + 535) = 2500 − (32.95 + 685) = 2500 − (717.95) = 1782.05.
Exercise 9.1.2 Find the value of the following expressions:
1. 28 − 4 + 13
2. 32 + 17 − 20
3.
1
2
+ (1 23 − 43 )
4. 28 − 7 × 3.5
Multiplication and Division from Left to Right
Multiplication and division work the same as addition and subtraction. That is, if both occur in
the same problem, the operations are done from left to right unless grouping symbols indicate
otherwise.
Exercise 9.1.3 Find the value of the following expressions:
1. 72 ÷ 9 × 5
2. 14 × 3 ÷ 7
3. 42 ÷ (3 × 2)
4.
1
2
× ( 32 ÷ 56 )
5. 0.28 × 1.4 ÷ 7
269
9.2 Exponents
Recall when we were finding prime factorizations, that sometimes a prime factor appeared
more than once:
24 = 8 × 3 = 2 × 4 × 3 = 2 × 2 × 2 × 3.
Since the factor 2 is repeated, it would be nice if there was a shorter way to write it. Luckily, just
like multiplication can be used to indicate repeated addition of the same value, exponents are
used to indicate repeated multiplication of the same value. In our example, instead of writing
8 = 2 × 2 × 2, we write 8 = 23 with the little 3 above and to the right of the 2 written to indicate
that the 2 is multiplied by itself 3 times. It is important to write the exponent a little smaller
than the 2 (which is called the base) so that it is clear to all reading it that it is an exponent.
Using this notation, the prime factorization of 24 can be written as:
24 = 23 × 3
Exercise 9.2.1 Write the following products using exponential notation:
1. 2 × 2 × 3 × 5 × 5
2. 2 × 3 × 3 × 7
3. 2 × 2 × 2 × 2 × 5
4. 3 × 3 × 5 × 5 × 5
5. 3 × 3 × 7 × 7 × 7
Exercise 9.2.2 Write the prime factorization of the following numbers. If a factor is repeated,
use exponential notation.
1. 36
2. 400
3. 250
4. 2400
5. 625
Exercise 9.2.3 Find the value of the following exponential expressions.
1. 32
2. 25
3.
4.
2 4
3
1 3
5
5. (0.012)2
270
Suppose you started a small business with $5,000. Then, during the first year, the value of the
business doubled. In the second year, the value of the business doubled again. In the third year,
after lots of hard work, miraculously, the value doubled again! To calculate how much the
business is worth after the three years, we have to multiply 5,000 by 2 for each time the value
doubled.
Value of the business after three years = 5, 000 × 2 × 2 × 2 = 40, 000
or using exponential notation because of the repeated 2’s:
5, 000 × 23 = 40, 000, so that the business is worth $40,000 at the end of the 3 years.
Notice that even though the multiplication in 5000 × 2 is written before the exponent, to get the
correct answer, the exponent has to be evaluated before the multiplication. In fact, the full
order of operations agreement is as follows:
9.3 The Full Order of Operations Agreement
• Evaluate any expression inside grouping symbols or parenthesis first.
• Perform all exponents next.
• Perform all multiplications and divisions from left to right next.
• Finally, perform all additions and subtractions from left to right last.
Beware of the popular mnemonic, “Please Excuse My Dear Aunt Sally”, or PEMDAS for short.
The problem with this, is that although it’s easy to remember, it doesn’t include the “from left
to right” part, and many students think that the phrase “My Dear” means that multiplication
always comes before division, and that “Aunt Sally” means that addition always comes before
subtraction. This is not the case! Multiplication and division are weighted equally, and must be
performed from left to right. Similarly, addition and subtraction are weighted equally, and
must be performed from left to right.
Example: 5 − 3 + 8 × 2
Solution: The multiplication comes first, then the subtraction, then the addition.
Exercise 9.3.1 For the following expressions, state the order that the operations must be
performed. No need to find the value!
1. 2 + 32
2. 4 + 0.5 ÷ 0.25 × 8
3. 3(5 − 1)2
4.
1
2
× 34 +
3 3
2
5. 24 ÷ (3 × 21 ) + 4.25 × 6
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Example: 5 − 3 + 8 × 2
Solution: The multiplication comes first, so 5 − 3 + 8 × 2 = 5 − 3 + 16.
Now, the subtraction followed by the addition, so 5 − 3 + 16 = 2 + 16 = 18
Exercise 9.3.2 Find the value of each expression.
1. 2 + 32
2. 4 + 0.5 ÷ 0.25 × 8
3. 3(5 − 1)2
4.
1
2
× 34 +
3 3
2
5. 24 ÷ (3 × 21 ) + 4.25 × 6
Exercise 9.3.3 For each problem below, write a mathematical expression that answers the
question, then use order of operations to find the answer.
1. You went to the Giant’s game, and paid $24 for your seat and $20 for parking. Since they
were winning, you felt generous and began to buy peanuts for the people around you at
$3.25 per bag. Including your ticket and parking, how much had you spent altogether
after buying:
(a) 2 bags of peanuts
(b) 5 bags of peanuts
(c) 7 bags of peanuts
(d) 10 bags of peanuts
2. Your cell phone company charges $35 per month plus overage charges. The deal is, you
get 400 anytime minutes covered by your monthly fee, but are charged $0.42 per minute
for each minute over 400. What is the monthly charge if your total minutes for the month
is:
(a) 125 minutes
(b) 475 minutes
(c) 522 minutes
(d) 418 minutes
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3. As coach of the girl’s little league softball, you are responsible for buying the equipment
and uniforms for the team. (Don’t worry; the parents will pay you back!) Each girl will
need a cap that costs $9 and a jersey that costs $19.45. The team also need to buy a pack of
softballs which costs $32.80 and 3 bats at $8.35 each. Excluding tax, how much will you
need to spend if the team has:
(a) 9 girls
(b) 10 girls
(c) 11 girls
(d) 12 girls
4. For the situation in the previous problem, if the girls’ families split the total cost evenly
(they can’t pay the coach, but they don’t want the coach to pay either!), how much will
each family have to pay (to the nearest penny) if the team has:
(a) 9 girls
(b) 10 girls
(c) 11 girls
(d) 12 girls
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10 Extra Exercises
Exercise 10.0.4 Fill in the table to practice rounding.
The Number to Round off
and the Level of Rounding
1) 17.0483
Round to the nearest 10
2) 3467.892
Round to the nearest 100
3) 3467.892
Round to the nearest 100th
4) 99.9978
Round to the nearest 10th
5) 0.000876
Round to the nearest 100th
6) 19.76
Round to the nearest 10
7) 19.76
Round to the nearest 100
8) 208,789.08
Round to the nearest 10,000
9) 90,099.99
Round to the nearest 10th
10) 89.0045
Round to the nearest 10th
11) 2.05678
Round to the nearest 1000th
12) 2.06589
Round to the nearest 10
13) 9,870,349,390
Round to the nearest million
14) 123.538905
Round to the nearest 100
15) 123.538905
Round to the nearest 10
16) 123.538905
Round to the nearest one
17) 123.538905
Round to the nearest 1000th
18) 976.3904
Round to the nearest 100
19) 976.3904
Round to the nearest 100th
20) 976.3904
Round to the nearest 10,000
Underline
the Digit(s)
The Next
Round
Digit
Up or Down
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Answer
Fraction Drawing Practice: A Study
For the following sums, use the given rectangle as the shape and size of all of your wholes.
Shade one fraction in the sum with one color, the other fraction in another color. Then, shade
the sum using both colors. See the example on the board.
The rectangle used for all the wholes:
1. Example to be shown on the board:
1
2
+
1
8
+
=
We can see from the picture that the 21 is still part of the total, but that it is broken into
pieces and looks like 48 . So, the sum becomes: 12 + 18 = 48 + 81 = 58 .
2.
1
2
+
1
4
3.
1
4
+
1
8
4.
1
2
+
3
4
5.
1
2
+
3
8
6.
1
4
+
3
8
275
Common Denominators
Add or subtract as indicated. Show all of your steps in finding the common denominator and
making the equivalent fractions using Golden Rule. Write your answer in simplest form.
1.
1
4
−
1
6
2.
3
8
+
1
6
3.
1
3
−
1
9
4.
3
10
5.
5
9
−
6.
4
5
− 15%
+
3
5
1
12
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Multiplying and Dividing by Powers of 10
1. Fill in the following multiplication table to practice multiplying by powers of 10:
10
×
4
1000
100, 000
100
1, 000, 000
72
6.23
0.039
2. Estimate the following products:
(a) 28 × 42
(b) 328.9 × 7.29
(c) 4200 × 63.8
(d) 7 45 × 2 83
3. Use your calculator to fill in the following division table. The first few are done for you:
÷
430
10
100
430 ÷ 10 = 43 430 ÷ 100 = 4.3
1, 000
10, 000
100, 000
7269
6.23
0.39
4. Based on any patterns you see from filling out the division table above, write a procedure
for dividing a number by a power of 10 without using a calculator.
5. Use your procedure to divide the following without using a calculator:
(a) 2800 ÷ 10
(b) 52, 000 ÷ 1000
(c) 478 ÷ 10
(d) 68, 000 ÷ 100
(e) 420, 000 ÷ 10
277
6. Divide the following in your head. Check your answer with a calculator.
(a) 420 ÷ 7
(b) 80, 000 ÷ 40
(c) 350, 000 ÷ 500
(d) 3, 600, 000 ÷ 9000
(e) 1800 ÷ 20
278
Fraction Formulas
1. Whole number Times a Fraction:
(a) Find 3 × 25 . Write your answer as an improper fraction.
(b) Find 5 × 43 . Write your answer as an improper fraction.
(c) Find a formula for the product A × cb .
(d) Write in words the steps for multiplying a whole number times a fraction using the
formula.
2. Simplifying Fractions:
(a) Find the prime factorization of 6.
(b) Find the prime factorization of 10.
(c) Simplify
6
.
10
(d) Find the prime factorization of 4.
(e) Find the prime factorization of 6.
(f) Simplify 46 .
(g) Simplify
a×c
.
b×c
(h) Write in words the steps for simplifying a fraction using the formula.
279
3. Multiplying Fractions
(a) Write the product
1
2
×
3
4
using “of” instead of ×.
(b) Draw a picture of the fraction 43 .
(c) Either on the same drawing or on a new drawing, make a drawing in which you can
clearly see that 12 of 34 is shaded.
(d) What is
1
2
× 34 ?
(e) Write the product
4
5
×
3
2
using “of” instead of ×.
(f) Draw a picture of the fraction 32 .
(g) Either on the same drawing or on a new drawing, make a drawing in which you can
clearly see that 45 of 32 is shaded.
(h) What is
4
5
× 32 ?
(i) What is
a
b
× dc ?
(j) Write in words the steps for multiplying fractions using the formula.
280