MAT001 – Chapter 5 - Percent
Denominators of 100
The word percent means “per hundred.” A percent
is another way to describe a part of a whole.
Section 5.1
Understanding Percent
7
100
7%
7 out of 100 of the rectangles are shaded.
7 percent of the whole is shaded.
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Denominators of 100
2 of 60
CQ5-01. Write
Example: Write 8.3 as a percent.
100
8.3
100
1.
2.
3.
4.
8.3%
Example: Write 435 as a percent.
100
435
100
10
0.19%
19%
1900%
0.0019%
0%
435%
0%
1.
3 of 60
CQ5-02. Write
1.
2.
3.
4.
19
as a percent.
100
5 .9
100
2
3
4
5
6
7
8
9
10
11
12
13
14
0%
3.
15
16
4.
4 of 60
17
Writing Percents as Decimals
as a percent.
Percents can be written as fractions and decimals.
27
27%
100
27
0.27
100
10
0.059%
1
0%
2.
0.59%
59%
5.9%
27%
0.27
Changing a Percent to a Decimal
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
1. Drop the % symbol.
2. Move the decimal point two places to the left.
4.
17
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6 of 60
1 of 10
MAT001 – Chapter 5 - Percent
Writing Percents as Decimals
Writing Decimals as Percents
Example: Write 19% as a decimal.
19% = .19. = 0.19
Changing a Decimal to a Percent
1. Move the decimal point two places to the right.
2. Add the % symbol at the end of the number.
Example: Write 2.67% as a decimal.
2.67% =. 2.67 = 0.0267
Example: Write 0.25 as a percent.
0.25 = 0.25. = 25%
An extra zero is
added to the left
of the 2.
Example: Write 0.23% as a decimal.
0.23% =. 0.23 = 0.0023
Two extra zeroes
are added to the
left of the 2.
Example: Write 3.95 as a percent.
3.95 = 3.95. = 395%
7 of 60
8 of 60
CQ5-03. Write 0.615 as a percent.
1.
2.
3.
4.
10
61.5%
0.615%
0.00615%
0%
1.
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
10
1. 1%
2. 10%
3. 0.1%
4. 0.001%
615%
0%
1
CQ5-04. Write 0.001 as a percent.
0%
4.
9 of 60
17
0%
1.
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
4.
10 of 60
17
CQ5-06. In Minnesota, 0.06 of the state is
CQ5-05. Write 2.467 as a percent.
covered by water, more than any other state.
What percent is covered by water?
1. 0.02467%
2. 246.7%
3. 24.67%
4. 2467%
10
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
16
10
6%
0.06%
60%
0.6%
0%
3.
15
1.
2.
3.
4.
0%
4.
17
0%
1.
11 of 60
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
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MAT001 – Chapter 5 - Percent
Percents to Fractions
Section 5.2
Any percent can be written as a fraction whose
denominator is 100. The fraction should be
simplified if possible.
Changing Between
Percents, Decimals,
and Fractions
Example: Convert 59.6% into a fraction in
simplest form.
59.6% 0.596 Write as a decimal.
596
1000
149
250
Write as a fraction.
Simplify.
13 of 60
14 of 60
CQ5-07. Write 135% as a fraction
Percents to Fractions
in simplified form.
1.
2.
3.
4.
Example: Convert 6 4 % into a fraction in simplest
5
form.
6
4
%
5
4
6
5
100
6
4
5
34
5
17
250
Write as a fraction.
100
1
100
Write the division horizontally.
Multiply by the reciprocal.
10
29
20
5
4
27
20
0%
0%
1.
0%
2.
0%
3.
4.
Simplify.
15 of 60
1
2
Fractions to Percents
4
5
6
7
8
9
10
11
written as 87 1 %.
13
17
80
14
15
16
16 of 60
17
to a percent.
34%
17.80%
21.25%
42.5%
0%
0%
1.
Write as a percent.
17 of 60
12
10
1.
2.
3.
4.
Example: Write 7 as a percent.
8
7
7 8
Divide.
8
This could also be
0.875
Write as a decimal.
87.5%
3
CQ5-08. Change
A convenient way to write a fraction as a percent is
to write the fraction in decimal form first and then
convert the decimal into a percent.
2
13.5
100
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
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MAT001 – Chapter 5 - Percent
CQ5-09. Change 5
1.
2.
3.
4.
510%
5.1%
3
40
CQ5-10. The brain represents
1
to a percent.
10
average person’s weight. What percent is this?
10
10
1.
2.
3.
4.
51%
0.51%
0%
0%
1.
0%
2.
75%
7.5%
0.075%
13.3%
0%
3.
0%
0%
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
19 of 60
17
CQ5-11. LeBron scored a basket 72 times out of
150 shots. What percent of his shots did he
score?
1
2
3
4
5
6
7
8
9
10
11
12
2.
13
14
3.
0%
15
16
17
4.
20 of 60
Equivalent Forms
Fractions, percents, and decimals are three different
forms for the same number.
10
1.
2.
3.
4.
0%
4.
1.
1
of an
42%
50%
Example: Complete the following table of
equivalent notations.
48%
63%
Fraction Decimal Percent
9
10
0%
0%
1.
0%
2.
0%
3.
0.56
4.
3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
%
8
Example continues.
22 of 60
21 of 60
Equivalent Forms
Section 5.3A
Fraction Decimal Percent
9
10
14
25
1
32
0.9
90%
56%
0.56
3
0.03125
1
%
8
Solving Percent
Problems Using
Equations
The fraction is changed to a decimal is changed to a percent.
9
10
0.56
3
1
%
8
9
10
56%
25
8
100
0.9
90%
56
100
0.03125
14
25
3125
100000
1
32
23 of 60
24 of 60
4 of 10
MAT001 – Chapter 5 - Percent
Percent Problems into Equations
To solve a percent problem, we express it as an
equation with an unknown quantity.
The following table is helpful when translating from
a percent problem to an equation.
Word
Mathematical Symbol
of
is
Multiplication symbol:
=
what
Any letter, for example, n
find
n=
Percent Problems into Equations
Example:
Translate into an equation.
What is 9% of 65?
n
= 9%
65
Example:
Translate into an equation.
24 is what percent of 144?
or ( ) or ·
24 =
n
144
25 of 60
Solving a Percent Problem
26 of 60
Solving a Percent Problem: Amount Unknown
A percent problem has three different parts.
amount = percent
amount = percent
base
base
Example:
What is 9% of 65?
Any one of the three quantities may be unknown.
1. When we do not know the amount:
n = 10% 500
2. When we do not know the base:
50 = 10% n
3. When we do not know the percent:
50 = n 500
n
= 9%
n
= (0.09) (65)
65
n
= 5.85
5.85 is 9% of 65.
27 of 60
28 of 60
Solving a Percent Problem: Base Unknown
CQ5-12. What is 23% of 256?
amount = percent
1.
2.
3.
4.
base
Example:
36 is 6% of what?
58.88
5.89
1113.04
36 = 6%
10
11.13
n
36 = 0.06n
0%
0%
1.
0%
2.
36
0.06n
=
0.06
0.06
600 = n
0%
3.
4.
36 is 6% of 600.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
29 of 60
30 of 60
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MAT001 – Chapter 5 - Percent
CQ5-13.
1. 4.35
2. 52.2
3. 69.6
4. 34.8
0%
1.
2
3
4
5
Government elections. This was 32% of the students
enrolled. How many students were enrolled?
1.
2.
3.
4.
10
0%
1
CQ5-14. A total of 2,480 students voted in the Student
17.4 is 25% of what number?.
6
7
8
9
10
11
12
0%
2.
13
14
16
0%
17
24 = 144n
24
144n
=
144
144
0.16 = n
2
16 % = n
3
0%
1.
31 of 60
1
2
3
4
5
6
7
8
9
10
11
12
1.
2.
3.
4.
144
13
0%
3.
14
15
16
4.
32 of 60
17
1.18%
84.71%
118%
10
91.46%
0%
0%
1.
2
24 is 16 % of 144.
3
33 of 60
0%
2.
CQ5-15. What percent of 85 is 72?
amount = percent base
Example:
24 is what percent of 144?
n
10
4.
Solving a Percent Problem: Percent Unknown
24 =
7,750
0%
3.
15
794
7,440
6,200
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
34 of 60
The Parts of a Percent Proportion
Section 5.3B
Proportions are another way to solve a percent
problem.
amount
percent number
=
base
100
Solving Percent
Problems Using
Proportions
IS
OF
%
100
To use this proportion, we need to find the amount,
base, and percent number in a word problem.
35 of 60
36 of 60
6 of 10
MAT001 – Chapter 5 - Percent
The Parts of a Percent Proportion
IS
OF
amount
percent number
=
base
100
The Parts of a Percent Proportion
Example:
%
100
Identify the amount, base, and percent number.
What is 9% of 65?
10% of 500 is 50.
p is the percent
number.
The amount, a, is the part
compared to the whole.
The value of p is 9.
p=9
The base, b, is the
entire quantity.
50
500
The base usually follows
the word of. b = 65
The amount, a, is unknown.
a = the amount (unknown)
24 is what percent of 144?
The base usually follows
the word of. b = 144
The amount is 24.
a = 24
10
100
The value of p is unknown.
p = the percent (unknown)
37 of 60
38 of 60
Solving a Percent Problem: Amount Unknown
amount
percent number
=
base
100
a
b
IS
p
100 OF
CQ5-16. What is 23% of 256?
%
100
1. 58.88
2. 5.89
3. 1113.04
4. 11.13
Example:
p=9
What is 9% of 65? a = unknown
a
b
p
100
a
65
100a
100a
100
a
b = 65
9
100
585
585
100
5.85
Cross multiply.
10
Divide both sides by 100.
0%
Simplify.
0%
1.
0%
2.
0%
3.
4.
5.85 is 9% of 65.
39 of 60
1
2
Solving a Percent Problem: Base Unknown
amount
percent number
=
base
100
a
b
IS
p
100 OF
p
100
36
b
6b
6b
6
b
5
6
7
8
9
10
11
12
13
14
15
16
40 of 60
17
%
100
1.
2.
3.
4.
b = unknown
6
100
3600
3600
6
600
4
CQ5-17. 15% of what number is 75?
Example:
p=6
36 is 6% of what? a = 36
a
b
3
Cross multiply.
475
11.25
500
10
375
Divide both sides by 6.
0%
Simplify.
0%
1.
0%
2.
0%
3.
4.
36 is 6% of 600.
41 of 60
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
42 of 60
7 of 10
MAT001 – Chapter 5 - Percent
Solving a Percent Problem: Percent Unknown
amount
percent number
=
base
100
a
b
IS
p
100 OF
CQ5-18. What percent of 85 is 72?
%
100
Example:
p = unknown
24 is what percent of 144? a = 24
a
b
p
100
24
144
p
100
144 p
2400
1.
2.
3.
4.
b = 144
Cross multiply.
144 p
144
b
2400
144 Divide both sides by 144.
2
0.16 16 % Simplify.
3
2
24 is 16 % of 144.
3
43 of 60
1.18%
84.71%
118%
10
91.46%
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
11
12
0%
2.
13
14
0%
3.
15
16
4.
44 of 60
17
Solving General Percent Problems
Example:
Mary received a raise of 8% of her monthly salary.
The amount of her raise was $48.16 per month.
What was her monthly salary before her raise?
Section 5.4
Solving Applied
Percent Problems
This problem can be solved using either the
equation method or the proportion method.
a
b
48.16 is 8% of n
p
100
48.16
b
IS
OF
The unknown quantity is the base.
45 of 60
Solving General Percent Problems
Example:
Mary received a raise of 8% of her monthly salary. The
amount of her raise was $48.16 per month. What was her
monthly salary before her raise?
Equation method
Proportion method
48.16 is 8% of n
48.16 = 0.08n
48.16 0.08n
0.08
0.08
48.16
n
0.08
602 n
48.16
b
8b
8
100
4816
8b
8
b
4816
8
602
8
100
%
100
46 of 60
CQ5-19. An inspector found that 4 out of 116
parts were defective. What percent of the parts
were defective?
1. 3.45%
2. 29%
3. 26.1%
4. 3.03%
10
Seconds
Remaining
0%
0%
1.
0%
2.
0%
3.
4.
Mary’s monthly salary before the raise was $602.
47 of 60
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
48 of 60
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MAT001 – Chapter 5 - Percent
Solving Markup Problems
Solving Markup Problems
Example:
Mark is taking Peggy out to dinner. He has $66 to spend. If
he wants to tip the server 20%, how much can he afford to
spend on the meal?
Percents can be added if the base is the same. These
types of problems are called markup problems.
Original
Cost
Let n = the cost of the meal.
15% of
Original
Cost
+
Cost of meal n +
100% of n
+
tip of 20% of the cost =
20% of n
=
120% of n
=
1.2n 66
1.2n
66
1.2
1.2
n 55
The markup is 15%.
$66
$66
$66
Mark and Peggy can spend
up to $55 on the meal itself.
49 of 60
CQ5-20. Bill and Ellen have $65 total to go out to dinner.
Knowing they need to leave 15% of their bill as a tip for the waiter,
what is the maximum amount they can spend on dinner? Excluding
the sales tax.
1. $56.52
2. $65.00
3. $55.25
4. $74.75
50 of 60
CQ5-21. A hotel charges an 8% tax, as required
by law to pay the city. If Mr. And Mrs. Smith
paid $167.40, what was their bill before the tax
was added to it?
1.
2.
3.
4.
10 Seconds
Remaining
0%
0%
0%
$155.00
$154.00
$93.00
$133.92
10
Seconds
Remaining
0%
0%
0%
1.
1.
1
2
3
4
5
6
7
8
9
10
11
12
2.
13
14
15
3.
16
17
51 of 60
0%
3.
4.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
52 of 60
Solving Discount Problems
Solving Discount Problems
Example:
Julie bought a leather sofa that was on sale for 35% off the
original price of $1200. What was the discount? How much
did Julie pay for the sofa?
The amount of a discount is the product of the
discount rate and the list price.
SALE!
25% off
Discount = discount rate
= 35%
= 420
list price
1200
The discount was $420.
Amount paid = list price – discount
The discount rate is 25%.
Discount = discount rate
0%
2.
4.
= 1200 – 420
list price
= 780
53 of 60
Julie paid $780 for the sofa.
54 of 60
9 of 10
MAT001 – Chapter 5 - Percent
Solving Commission Problems
The amount of money a person makes that is a percentage of
the value of sales is called a commission. It is calculated by
multiplying the percentage (called the commission rate) by
the value of the sales.
Section 5.5
Solving Commission,
Percent of Increase or
Decrease, and Interest
Problems
Commission = commission rate
value of sales
Example:
A salesperson has a commission rate of 18.5%. He sells
$43,250 worth of goods. What is his commission?
Commission = commission rate value of sales
= 18.5% $43,250
= 0.18.5 43,250
= 8001.25
His commission is $8,001.25.
55 of 60
CQ5-22. A real estate agent sells a house for
$87,000. She gets a commission of 6% on the
sale. What is her commission?
1.
2.
3.
4.
3
We sometimes need to find the percent by which a number
increases or decreases.
Percent of increase =
$5,220.00
10
Seconds
Remaining
$14,583.33
4
5
6
7
8
9
10
11
12
Amount of increase = original amount – new amount
= 17,280 – 16,000 = 1280
0%
0%
2.
13
14
0%
3.
15
16
17
57 of 60
amount of decrease
original amount
The car’s cost
increased by 8%.
58 of 60
Solving Simple Interest Problems
Interest is money paid for the use of money. The principal
is the amount deposited or borrowed. The interest rate is
per year, unless otherwise stated. If the interest rate is in
years, the time must also be in years.
Example:
Patrick weighed 285 pounds two years ago. After dieting, he
reduced his weight to 171 pounds. What was the percent of
decrease in his weight?
Interest = principal
I=P R T
Amount of decrease = original amount – new amount
rate
time
Example:
Find the simple interest on a loan of $3600 borrowed at 6% for 8
years.
= 285 – 171 = 114
amount of decrease
original amount
114
= 0.4
=
285
amount of increase
original amount
1280
= 0.08
=
16000
Percent of increase =
4.
Solving Increase & Decrease Problems
Percent of decrease =
amount of increase
original amount
Example:
The cost of a certain car increased from $16,000 last year to
$17,280 this year. What was the percent of increase?
$4,395.00
1.
2
Solving Increase & Decrease Problems
$1,458.33
0%
1
56 of 60
I=P
Percent of decrease =
Patrick’s weight
decreased by 40%.
59 of 60
R
= 3600
= 1728
T
0.06
8
The interest earned is $1,728.
60 of 60
10 of 10