UNIT V STUDY GUIDE
Percent Notation
Course Learning Outcomes for Unit V
Upon completion of this unit, students should be able to:
1. Write three kinds of notation for a percent.
2. Convert between percent notation and decimal notation, and convert between fraction notation and
percent notation.
3. Translate percent problems to percent equations and solve percent problems using percent
equations.
4. Translate percent problems to proportion equations and use proportions to solve percent problems.
5. Solve applied problems involving percent, percent of increase, or percent of decrease.
6. Solve applied problems involving sales tax and percent, commission and percent, and discount and
percent.
7. Solve applied problems involving simple interest and compound interest.
8. Solve applied problems involving interest rate on credit cards.
Reading Assignment
Chapter 8:
Percent Notation
Unit Lesson
People learn in a variety of ways. For this reason, each section in each unit of this course will include multiple
resources, such as required reading assigned in the textbook; optional media assignments that will include
the hardest concepts in the unit; homework with five help aids to the right of the problem, which include ‘Help
Me Solve This’; ‘View an Example’; ‘Textbook’, which opens the e-textbook to the page where the concept is
taught; ‘Ask My Instructor’; and the option to print the question.
The best way to learn math is by practicing. Each unit will not only provide instruction and guidance on how to
solve problems, but also numerous opportunities to practice solving them on your own through non-graded
assignments. At the end of each section will be an Exercise Set. Answers and guidance for solutions are
provided in indicated sections of the textbook, as well as in the student solutions manual that accompanies
the textbook.
When you click on Unit V in your course, you will see a ‘TO DO’ LIST to assist you in starting your course.
Click each chapter as listed in “Chapter Contents” in MyMathLab. A Chapter Opener will be available for
review. Next click on the sections within the Chapter to open the following: video presentation, multimedia eText (e-book with animations and videos to illustrate concepts), and your study plan. The above holds true for
every chapter and every section in the textbook. Each Chapter also contains a mid-chapter review, chapter
summary and review, and a chapter test with test prep videos and after Chapter 2, there will be a Cumulative
Review that will cover all concepts up to and including the current chapter.
Unit V covers Chapter 8 in the textbook.
MAT 0390, Intermediate Algebra
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CHAPTER EIGHT: PERCENT NOTATION
Section 8.1, Percent Notation
Percent Notation can be written three ways:
1. Ratio: n% = the ratio of n to 100 =
2. Fraction notation: n% = n x
1
100
=
𝑛
100
𝑛
100
3. Decimal notation: n% = n x 0.01 = 0.01n
Example: Write three kinds of notation for 70%.
70
1. Ratio =
100
2. Fraction notation = 70 𝑥
1
100
=
70
100
3. Decimal notation = 70 𝑥 0.01 = 0.7
Converting between percent notation and decimal notation is very important in everyday math. You can utilize
ratio or decimal notation to accomplish this conversion.
Example:
Write decimal notation for 78%.
78%
=
78
100
= 0.78
OR
78%
= 78 x 0.01
= 0.78
Study the boxes on pages 514 for further guidance on converting from percent notation to decimal notation
and from decimal notation to percent notation.
Let’s convert between fraction notation and percent notation.
Example: Convert
3
5
to percent notation.
Divide the numerator by the denominator.
0.6__
5 )3.0
-3 0
0
Convert the decimal notation to percent notation.
0.6 * 100 = 60% Percent notation
3
5
= 60%
Review examples 9-16, pages 516-518. Look carefully at the chart at the bottom of page 518, entitled
“Fraction, Decimal, and Percent Equivalents.” You will find these very helpful. The families of fractions will
help you too.
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FAMILIES OF FRACTIONS
The Family of Thirds:
1
=
3
0.33333 3 repeating =
1
%
3
1
Example: 66.3333% = 66 %
3
2
3
= 0.66666 6 repeating =
Example: 76. 666666 = 76
2
3
2
3
%
%
Family of Ninths (notice the pattern):
1
9
2
9
3
9
4
9
= 0.1111%
1 repeating =
= 0.2222%
2 repeating =
=
1
3
1
9
2
9
%
%
= 0.3333% 3 repeating (I believe you know this one by now.)
= 0.4444%
4 repeating =
4
9
% (Do you see the pattern?)
You finish the rest.
5=
9
6=
9
7=
9
8=
9
The Family of Odd Eighths: They are terminating decimals, but a good family to know. The even
eighths you should know. See the pattern:
1 = 0.125
8
3 = 0.375
8
5 = 0.625
8
7 = 0.875
8
MAT 0390, Intermediate Algebra
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Section 8.2, Solving Percent Problems Using Percent Equations
In section 5.8, you were pointed to this: Students repeatedly state that their biggest obstacle is to
translate words into algebraic expressions/ equations. The chart on page 366 is one of the most
important charts in the book. Study it carefully. Then practice translating phrases to algebraic
expressions on pages 367-368 in Examples 1 and 2.
If you mastered this concept, then Section 8.2 will be easy for you. Page 525 gives you the key words in
percent translations. Make sure that you learn these key words. Once your equation is set up, follow the rules
for solving equations: You must show the addition/subtraction/division steps on both sides of the
equation to receive full credit. You must show your work algebraically.
This section is simply an extended review of solving equations.
Section 8.3, Solving Percent Problems Using Proportions
In this section, we will solve percent problems using proportions. Many students state that this is their favorite
way to solve percent problems. Remember that a percent is a ratio of some number to 100. For example,
46% is the ratio of 46 to 100, which can be represented by the fraction
46
100
.
The numbers 67,620,000 and 147,000,000 have the same ratio as 46 and 1000.How do you determine that
fact? Remember cross products? If the cross products are equal, the ratios are equal.
46
1000
46 ∙ 147,000,000
6,762,000,000
67,620,000
147,000,000
= 1000 ∙ 67,620,000
= 6,762,000,000
=
Example:
Translate to a proportion and solve.
23%
of 5 is what?
# of
Hundredths
base amount (a)
Set up the proportion by finding the cross products, and solve.
23
100
⤮
𝑎
5
100a = 115
(This is the equation to solve (115 = 23 x 5)
100a = 115
100
100
(Reduce by dividing each side by 100.)
𝑎 = 1.15
MAT 0390, Intermediate Algebra
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Study the key terms base and amount carefully. It is very easy to transpose those.
Example: What percent of 50 is 16?
𝑛
100
𝑛
100
=
⤮
16
50
16
50
50n = 1600
50n = 1600
50
50
n = 32%
Note that when solving percent problems using proportions, ‘n’ is a percent and needs only a % sign.
Section 8.4, Applications of Percents
In this section, we will look at real-world applications and solve these using proportion equations or the
percent equation (see Section 8.2). Remember that in section 5.8 these instructions were given: Students
repeatedly state that their biggest obstacle is to translate words into algebraic expressions/
equations. The chart on page 366 is one of the most important charts in the book.
Now is the time to implement what you learned on page 366. Read each example carefully. Take notes on
how the proportions are set up; if those ratios are incorrect, you will get a wrong answer. You need to master
this now for future success in this course and future math courses. It is strongly advised that you use the
students’ solutions manual and work the odd-numbered problems on pages 542-544. If you need further help
with any of these, contact your instructor.
Section 8.5, Percent of Increase or Decrease
In this section, we will solve applied problems involving percent of increase or percent of decrease.
Percent of Decrease
The Dow Jones Industrial Average (DJIA) plunged from 11,143 to 10,365 on September 29, 2008. This was
the largest one-day drop in its history.
Example: What was the percent of decrease?
First, let’s find out how much the decrease was:
11,143 – 10,365 = 778
Next, we create our Proportion Equation:
𝑛
778
=
100
11143
𝑛
100
⤮
778
11143
11143𝑛 = 77800
11143𝑛
77800
=
11143
11143
MAT 0390, Intermediate Algebra
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n ~ 6.981 %
n ~ 7%
Percent Equation:
Example: 778 is what percent of 11,143?
778 = 11,143p
778 = 11,143p
11143
11143
0.06981 ~ p
0.07 * 100 ~ p
P ~ 7%
Translating for Success, page 549
Section 8.6, Sales Tax, Commission, and Discount
Taxes are a big issue in all of our lives. We have sales tax, city tax, county tax, state tax, and federal tax.
Since we deal with taxes every day, we need to know how to figure the taxes in all of these areas.
Example: The sales tax on the purchase of a GPS navigator is $55.93. The GPS navigator costs $799. That
is the sales tax rate?
You can solve this in one of two ways.
First way:
𝑟=
55.93
799
𝑟 = 0.07
𝑟 = 0.07 * 100
r = 7% (sales tax rate)
Second way:
Set up a proportion equation:
55.93
55.93
799
0.07
0.07 * 100
7%
= 799r
799𝑟
=
799
=r
=r
= r (sales tax rate)
Many sales people work for a base salary + commission. Some sales people work on graduated commission.
Let’s work a problem with a graduated commission.
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Example: Miguel’s commission is increased according to how much he sells. He receives a commission of
5% for the first $2000 of sales and 8% for the amount over $2000. What is his total commission on sales of
$6200?
C = (6200 - 2000)(0.08) + 2000(0.05)
C = 4200(0.08) + 100
C = 336 + 100
C = $436
We all love to buy things at a discount. Pay close attention to definitions on page 556 of original or marked
price, the rate of discount, the discount, and the sale price. Note that discount problems are a type of percent
of decrease problem.
1
Example: A leather sofa marked $2379 is on sale at 33 % off. What is (a) the discount? (b) The sale price?
3
Look back at the family of fractions to help with the percent.
(a) Discount =
D=
Rate of discount x
1
33 % x
3
D=
1
3
Original price
$2379
x
$2379
2379
3
$793
D=
D=
Original price –
$2379 –
(b) Sale price =
S=
S=
Discount
$793
$1586
Here is a second way to solve Step B of this problem:
1
S = $2379 – (2379 ∙ )
3
S = $2379 – $793
S = $1586
The second method solves the problem in a more efficient method; the first method involves more work.
Section 8.7, Simple Interest and Compound Interest, Credit Cards
If you have to take out a loan, always try to get simple interest. The amount of money that you borrow is the
principal (P). The interest rate = r. Time = t, which represents 1 year. The simple interest formula is

I = Prt (Interest = Principal x rate x time)
Example: What is the simple interest on $2500 invested at an interest rate of 6% for 1 year?
P = $2500,r = 6% = 0.06, t = 1 (t will always be a base of 1 year)
I = Prt
I = 2500 * 0.06 * 1
I = $150
The simple interest for 1 year is $150.
MAT 0390, Intermediate Algebra
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Example: What is the simple interest on a principal of $2500 invested at an interest rate of 6% for 3 months?
Remember that t = 1 year; 3 months =
P = $2500 r = 6% = 0.06
I = Prt
1
I = 2500 * 0.06 *
I=
t=
3
12
=
1
4
1
4
4
2500∗0.06
4
I = $37.50
When interest is paid on interest, we call it compound interest. This type of interest is usually paid on
investments. The Compound Interest Formula is

𝒓 𝒏 ∙𝒕
𝐀 = 𝐏 * (𝟏 + 𝒏)
5
Example: The Ibsens invest $4000 in an account that pays 5 %, compounded quarterly. Find the amount in
8
1
the account after 2 years.
2
P = $4000 r = 5
5
8
% = 0.05625 n = times a year; quarterly = 4 times a year
𝑟 𝑛 ∙𝑡
A = P ∙ (1 + 𝑛)
A = 4000 ∙ (1 +
0.05625 4∙(5)
) 2
4
A = 4000 ∙ (1 +
0.05625 10)
)
4
A = 4000 (1.0140625)10
A ~ $4599.46
1
The amount in the account after 2 years is $4599.46.
2
According to creditcards.com, the average credit card debt per household with credit card debt in the United
States was $16,007 in 2009. The money you obtain through the use of a credit card is not “free” money.
There is a price (interest) to be paid for the privilege. A balance carried on a credit card is a type of loan. A
small change in an interest rate can make a large difference in the cost of a loan. When you make a payment
on a credit card, do you know how much of that payment is interest and how much is applied to reducing the
principal?
Knowledge of credit cards is very important to you. Study carefully Examples 6 on pages 566-567. Also, it
may be helpful for you to work problems 1-49, odds, on pages 568-570. The answers can be found in the
back of the book.
Summary and Review, pages 571-576
Test, pages 577-578
Cumulative Review: Chapters 1-8, 579-580
MAT 0390, Intermediate Algebra
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Reference
Bittinger, M. L., Ellenbogen, D. J., & Johnson, B. L. (2012). Prealgebra (6th ed.). Boston, MA:
Addison-Wesley.
Learning Activities (Non-Graded)
Practice What You Have Learned
After reading Chapter 8, improve your mastery of the content by working the odd-numbered problems on
the following pages:








Section 8.1, pages 519-524
Section 8.2, pages 529-530
Section 8.3, pages 535-536
Section 8.4, pages 542-544
Translating For Success, page 549
Section 8.5, pages 550-552
Section 8.6, pages 558-561
Section 8.7, pages 568-570
Once you have completed the problems, you can check your answers in the back of the textbook to see
how well you did.
Review What You Have Learned
Before attempting the Homework and the Unit Assessments, study the chapter summaries, review the
concepts taught in the chapters, and work the odd-numbered problems in the review exercises.




Chapter 8 Mid-Chapter Review, pp. 537-538
Chapter 8 Summary and Review, pages 571-576
Chapter 8 Test (Practice) pages 577-578; Note: See the top of page 577 to find the step-by-step
solutions for each problem in the test.
Cumulative Review: Chapter 1-8, pages 579-580
Study Plan for Unit V
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your progress and do practice exercises.
Your study plan is updated each time you take an Assessment. The study plan is optional and is generated
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Other Resources and Activities
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interactive figures on a variety of topics.
When you click on ‘Tools for Success’ at the very top of the page, it will also provide links to a variety of
helpful aids ranging from Translating for Success and Visualizing for Success Interactive Animations to Basic
Math Review Card and the Introductory Algebra Review Card, which give a brief summary of many key math
concepts.
MAT 0390, Intermediate Algebra
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In the ‘Chapter Contents’ you will also be able to view the Answer section from your textbook, view the
Glossary from your textbook, and view the Index from your textbook.
Non-Graded Learning Activities are provided to aid students in their course of study. You do not have to
submit them. If you have questions, contact your instructor for further guidance and information.
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