This document contains notes and practice problems for a math lesson on percents. It includes examples of finding percent increases and decreases, percent discounts and markups, and solving proportions involving percents. Warm-up questions ask students to find 70% of 1/2 and change fractions and percents to decimals. The document then provides examples and practice problems for students to work through related to percents.

1. TGIF, December 12, 2014 Today:
 Warm-Up
 Percent Proportion
 Review for Test Tuesday
 Complete Classwork 2.7

2. 1. What percent of 28 is
ퟐ
ퟕ
?
2. You have $100 and are playing poker. On the first hand you
bet 50% of your money, and lose 100% of the bet.
On the second hand, you bet 100% of the remaining money
and win 100% of your bet back plus 50%. (150% of your bet)
After the first two hands, how much money do you have?
2a. What is 70% of ½?

3. 3. Five of 34 students were absent yesterday.
What percent of students were present?
5. On a test with 80 questions, you answered 68 correctly.
What was your percent score?

4. Mentis Arithmetica
2. What is 45% of 80? 3. 36 is what % of 30?
Try using mental math first:
1) 25% of 80 2) 120% of 50 3) 15% of 500
4) 300% of 9
5) 5% of 1500 6) What % of 35 is 7?

5. Test Review
Fill in the missing numbers:
1. 0.6 =
? ?
5
2. 0.14 =
50
? 4. 0.012 =
250
3. 0.016 =
125
?
Write as a decimal:
5. One-tenth of a dollar 6. One-tenth of ninety cents
7. One-tenth of two dollars and seventy cents.

8. Vocabulary :
Compliments:
Two numbers are compliments when their sum is equal to
100%. In other words, the sum of the two parts equals the
whole.
Every number has it’s own compliment.
Ratio: A comparison of two numbers by division expressed
in the following ways:
ퟏ
ퟐ, 1 to 2, 1:2
Ratio’s Must use the same Unit of Measurement

9. Finding Percent Increase & Decrease
A percent change is an increase or decrease
given as a percent of the original amount.
Percent increase describes an amount that
has grown and percent decrease describes
an amount that has be reduced.

10. Finding Price Before Increase or Decrease
This formula is used to find the original price of an item
when the price is known after increases such as taxes, tips,
and markups.
The formula is also used when the original price is not
known after decreases such as markdowns and discounts.
Original Price: Price after increase or decrease
1 + percent or
The cost of lunch after a 15% tip was $24.15. What was the
cost of the lunch alone? $24.15
1.15 1 +15%
= $21.00

11. Applying Percent Changes
Common percent changes are discounts and
markups.
A discount is an
amount by which an
original price is
reduced.
discount = % of original price
final price = original price – discount
A markup is an
amount by which a
wholesale price is
increased.
markup = % of wholesale cost
final price = wholesale cost + markup

12. Class Notes & Practice Problems:

13. Ex. 1A: Percent Increase and Decrease
Find each percent change. Tell whether it is a
percent increase or dFercormea 8s teo. 10
Simplify the fraction.
Change to a decimal.
= 0.25
= 25% Write the answer as a percent.
8 to 10 is an increase, so a change from 8 to 10 is a 25%
increase.

14. Ex. 1B: Finding Percent Increase and Decrease
Find the percent change. Tell whether it is a percent
increase or decrease.
From 75 to 30
Simplify the fraction.
Simplify the numerator.
Write as a decimal
= 0.6
= 60% Write the answer as a percent.
75 to 30 is a decrease, so a change from 75 to 30 is a
60% decrease.

15. Practice 1: Percent Increase and Decrease
Find the percent change. Tell whether it is a percent
increase or decrease.
1. From 200 to 110
= 0.45
Simplify the numerator.
Simplify the fraction.
= 45% Write the answer as a percent.
200 to 110 is an decrease, so a
change from 200 to 110 is a 45%
decrease.
Write as a decimal

16. Practice 2: Percent Increase and Decrease
Find each percent change. Tell whether it is a
percent increase or decrease.
2. From 25 to 30
Simplify the numerator.
Simplify the fraction.
= 0.20
Write as a decimal
= 20% Write the answer as a percent.
25 to 30 is an increase, so a change from 25 to
30 is a 20% increase.

17. Example 1: Percent Increase and Decrease
A. Find the result when 12 is increased by 50%.
0.50(12) = 6 Find 50% of 12. This is the amount of
increase.
12 + 6 =18
It is a percent increase, so add 6
to the
12 increased by 50% iso 1r8ig.inal amount.
B. Find the result when 55 is decreased by
60%.
0.60(55) = 33 Find 60% of 55. This is the amount of decrease.
55 – 33 = 22 It is a percent decrease so subtract 33 from
the original amount.
55 decreased by 60% is 22.

18. Example 2: Percent Increase and Decrease
A. Find the result when 72 is increased by 25%.
0.25(72) = 18 Find 25% of 72. This is the amount of
increase.
72 + 18 =90 It is a percent increase, so add 18
to the original amount.
72 increased by 25% is 90.
B. Find the result when 10 is decreased
by 40%.
0.40(10) = 4 Find 40% of 10. This is the amount of
decrease.
10 – 4 = 6 It is a percent decrease so subtract 4
from the original amount.
10 decreased by 40% is 6.

19. Applying Percent Changes
Common percent changes are discounts and
markups.
A discount is an
amount by which an
original price is
reduced.
discount = % of original price
final price = original price – discount
A markup is an
amount by which a
wholesale price is
increased.
markup = % of wholesale cost
final price = wholesale cost + markup

20. Practice 1: Percent Discounts
The entrance fee at an amusement park is $35.
People over the age of 65 receive a 20% discount.
What is the amount of the discount? How much do
people over 65 pay?
Method 1: A discount is a percent decrease. So find
$35 decreased by 20%.
0.20(35) = 7 Find 20% of 35. This is the
amount of the discount.
35 – 7 = 28 Subtract 7 from 35. This is the
entrance fee for people over
the age of 65.

21. Practice 2: Percent Discounts
Method 2: Subtract the percent discount from
100%.
100% – 20% = 80% People over the age of 65 pay 80% of
the regular price, $35.
0.80(35) = 28 Find 80% of 35. This is the entrance
fee for people over the age of 65.
35 – 28 = 7 Subtract 28 from 35. This is the
amount of the discount.
By either method, the discount is $7. People over the
age of 65 pay $28.00.

22. Practice 3: Percent Discounts
A $220 bicycle was on sale for 60% off. Find the sale
price.
Use Method 2:
100% – 60% = 40% The bicycle was 60% off of 100% .
0.40(220) = 88 Find 40% of 220.
By this method, the sale price is
$88.

23. Practice 1: Percent Markups
The wholesale cost of a DVD is $7. The markup is
85%. What is the amount of the markup? What is the
selling price?
Method 2
Add percent markup to 100%
Method 1
A markup is a percent increase. So find $7 increased by 85%.
0.85(100% 7) = + 5.95
85% = 185%
7 + 5.95 = 12.95
Find 85% of 7. This is the amount of the
The selling price is 185% of the
markup.
wholesale price, 7.
Add to 7. This is the selling price.
1.85(7) = 12.95 Find 185% of 7. This is the selling price.
Subtract from 12.95. This is the
12.95 ÷ 7 = 5.95 amount of the markup.
By either method, the amount of the markup is
$5.95. The selling price is $12.95.

24. Practice 2: Percent Markups
A video game has a 70% markup. The wholesale cost
is $9. What is the selling price?
Method 1
A markup is a percent increase. So find $9 increased
by 70%.
0.70(9) = 6.30 Find 70% of 9. This is the amount of
the markup.
9 + 6.30 = 15.30 Add to 9. This is the selling price.
The amount of the markup is $6.30. The selling price is
$15.30.

25. Find each percent change. Tell whether it is a
percent increase or decrease.
1. from 20 to 28.
40% increase
2. from 80 to 62.
22.5% decrease
3. from 500 to 100.
80% decrease
4. find the result when 120 is increased by 40%.
5. find the result when 70 is decreased by 20%.
168
56
Lesson Quiz: Part I

26. Find each percent change. Tell whether it is a percent
increase or decrease.
6. A movie ticket costs $9. On Mondays, tickets are
20% off. What is the amount of discount? How
much would a ticket cost on a Monday?
7. A bike helmet cost $24. The wholesale cost was
$15. What was the percent of markup?
$1.80; $7.20
60%
Lesson Quiz: Part II

27. Example 2: Measurement Application
A flagpole casts a shadow that is 75 ft long at the
same time a 6-foot-tall man casts a shadow that is 9 ft
long. Write and solve a proportion to find the height
of the flag pole.
Since h is multiplied by 9, divide both sides
by 9 to undo the multiplication.
The flagpole is 50 feet tall.

30. Percents
Warm Up What is 70% of ½?
Change each percent to a decimal.
1. 73% 0.73
2. 112%
3. 0.6% 0.006
4. 1%
Change each fraction or mixed number to a
decimal.
5. 6. 0.3
7. 8.
Solve each proportion.
9. 10.
0.8
1.12
1.2
0.01
0.5
12 4.2

31. Percents
Example 3B: Finding the Whole
20 is 0.4% of what number?
Method 2 Use an equation.
20 = 0.4% of x
20 = 0.004 • x
5000 = x
Write an equation. Let x represent
the whole.
Write the percent as a decimal.
Since x is multiplied by 0.004,
divide both sides by 0.004 to
undo the multiplication.
20 is 0.4% of 5000.

32. Percents
90 is 120% of what number?
Method 1 Use a proportion.
Use the percent proportion.
120x = 9000
x = 75
120% of 75 is 90.
Let x represent the whole.
Find the cross products.
Since x is multiplied by 120, divide
both sides by 120 to undo the
multiplication.

33. Percents
Lesson Quiz: Part 1
Find each value. Round to the nearest tenth
if necessary.
1. Find 20% of 80.
16
2. What percent of 160 is 20?
3. 35% of what number is 40?
4. 120 is what percent of 80?
5. Find 320% of 8.
6. 65 is 0.5% of what number?
12.5%
114.3
150%
25.6
13,000

34. 1. Order from least to greatest: 2/8, 2.8%, 8/2, .28
3. 20 is 40% of what number?
4. 36 is what percent of 30?
5. What is the total cost of a $21.00 lunch and 15%
tip?
6. Which fraction must have more than two decimal
places?
A.) ¼ B.) 2/5 C.) 12/50 D.) 5/6 E.)
None
Warm-Up:

35. * A certain item used to sell for seventy-five cents a pound, you see that it's
been marked up to eighty-one cents a pound. What is the percent increase?
First, I have to find the increase: 81 – 75 = 6
The price has gone up six cents. Now I can find the percentage increase over the
original price.
Note this language, "increase/decrease over the original", and use it to your
advantage: it will remind you to put the increase or decrease over the original value,
and then divide.
This percentage increase is the relative change:
6/75 = 0.08 or an 8% increase in price per pound.
An important category of percentage exercises is markup and markdown problems.
For these, you calculate the markup or markdown in absolute terms (you find by how
much the quantity changed), and then you calculate the percent change relative to
the original value. So they're really just another form of "increase - decrease"
exercises.
* A computer software retailer used a markup rate of 40%. Find the selling
price of a computer game that cost the retailer $25.
The markup is 40% of the $25 cost, so the markup is: (0.40)(25) = 10

36. A golf shop pays its wholesaler $40 for a certain club, and then sells it to a golfer for $75. What is
the markup rate? First, I'll calculate the markup in absolute terms: 75 – 40 = 35
Then I'll find the relative markup over the original price, or the markup rate: ($35) is (some
percent) of ($40), or: 35 = (x)(40) so the relative markup over the original price is: 35 ˜ 40 = x =0.875
Since x stands for a percentage, I need to remember to convert this decimal
value to the corresponding percentage. The markup rate is 87.5%.
* A shoe store uses a 40% markup on cost. Find the cost of a pair of shoes that
sells for $63.
This problem is somewhat backwards. We have the selling price, which is cost
plus markup, and they gave me the markup rate, but they didn't tell me the actual
cost or markup. So I have to be clever to solve this.
I will let "x" be the cost. Then the markup, being 40% of the cost, is 0.40x. And
the selling price of $63 is the sum of the cost and markup, so:
63 = x + 0.40x 63 = 1x + 0.40x 63 = 1.40x 63 ˜ 1.40 = x= 45
The shoes cost the store $45.

37. * An item originally priced at $55 is marked 25% off. What is the sale price?
First, find the markdown. The markdown is 25% of the price of $55, so: x =
(0.25)(55) = 13.75
By subtracting this markdown from the original price, find the sale price:
55 – 13.75 = 41.25 The sale price is $41.25.
* An item that regularly sells for $425 is marked down to $318.75. What is the
discount rate?
First, I'll find the amount of the markdown: 425 – 318.75 = 106.25
Then I'll calculate "the markdown over the original price", or the markdown
rate: ($106.25) is (some percent) of ($425), so: 106.25 = (x)(425)
...and the relative markdown over the original price is: x = 106.25 ˜ 425 =
0.25
Since the "x" stands for a percentage, I need to remember to convert this
decimal to percentage form. The markdown rate is 25%.

38. * An item is marked down 15%; the sale price is $127.46. What was the original
price?
This problem is backwards. They gave me the sale price ($127.46) and the
markdown rate (15%), but neither the markdown amount nor the original price. I
will let "x" stand for the original price. Then the markdown, being 15% of this
price, was 0.15x. And the sale price is the original price, less the markdown, so I
get: x – 0.15x = 127.46 1x – 0.15x = 127.46
0.85x = 127.46 x = 127.46 ˜ 0.85 = 149.952941176...
This problem didn't state how to round the final answer, but dollars-and-cents
is always written with two decimal places, so: The original price was $149.95.
Note in this last problem that I ended up, in the third line of calculations, with an
equation that said "eighty-five percent of the original price is $127.46". You can
save yourself some time if you think of discounts in this way: if the price is 15%
off, then you're only actually paying 85%. Similarly, if the price is 25% off, then
you're paying 75%; if the price is 30% off, then you're paying 70%; and so on.
While the values below do not refer to money, the procedures used to solve these
problems are otherwise identical to the markup - markdown examples on the
previous page.
* Your friend diets and goes from 125 pounds to 110 pounds. What was her
percentage weight loss? First, I'll find the absolute weight loss: 125 – 110 = 15
This fifteen-pound decrease is some percentage of the original, since the rate
of change is always with respect to the original value. So the percentage is
"change over original", or: 15 = (x)(125)
15 ˜ 125 = x (The change, 15, is over the original, 125.) 15 ˜ 125 = 0.12