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Progressive Mathematics Initiative
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7th Grade Math
Percents
2012-12-07
www.njctl.org
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Table of Contents
Click on the topic to go to that section
Relating Fractions, Decimals and Percents
Three Types of Percent Problems
Percent of Change
Representing Percent Equations Algebraically
Applied Percent of Decrease
Applied Percent of Increase
Real-life Application Problems
Common Core: 7.RP.3, 7.EE.2, 7.EE.3
Relating Fractions,
Decimals & Percents
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of contents
Helping you remember...
Fill in each box below with an example of the
process described.
% to a fraction
% to a decimal
fraction to a %
decimal to a %
Order the numbers from least to greatest.
0.15
12.5%
0.095
In order to do this, they must all be in the same form.
Let's turn them all into percents:
15%
12.5%
16%
9.5%
So least to greatest:
9.5%
12.5%
15%
16%
0.095
0.15
12.5%
1
Find the lowest value
A
5%
B
1/2
C
.5%
D
.05
2
Find the greatest value
A
120%
B
1.02
C
.2%
D
1.19
3
Find the greatest value
A
6%
B
.6
C
60
D
6
4
Find the lowest value
A
2%
B
.2
C
.02
D
.2%
5
Find the lowest value
A
50%
B
500%
C
50.0
D
50.01
Express each decimal or percent as a fraction in lowest
terms:
1) 18%
Click to Reveal
4)
Click to Reveal
2) 0.85
Click to Reveal
5) 5.008
Click to Reveal
3)
Click to Reveal
6) 0.0001
Click to Reveal
Express each fraction as a percent:
1)
Click to Reveal
2)
3)
Click
to
Reveal
Click
to
Reveal
6
Express as a fraction.
7
Express as a decimal.
8
Express as a percent.
9
Express as a decimal.
10 Express as a percent.
11
Express as a percent.
Three Types of Percent
Problems
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of contents
Remember, percents are "parts of a whole".
The part is the numerator and the whole is the
denominator.
17% means 17 parts per 100 or
We are going to solve problems involving percents.
There are 3 types of problems:
1. Find the part What number is 54% of 34?
2. Find the whole 4 is 60% of what number?
3. Find the percent 18 is what percent of 28?
Two words that will occur in these types of problems
are "is" and "of".
These words have specific meanings in math.
"Is" means equals (=)
"Of" means multiply
To solve a percent problem, translate the words into an
equation.
Change the following:
1. Percent into a decimal
2. "is" to "="
3. "of" to " "
4. Unknown to "x"
Then, solve the equation.
Finding the Part...
Examples:
Find
Write a mathematical sentence
40% of 60
.40
60 = 24
Click
20% of 90
.20
90 =Click
18
Write a mathematical sentence
What is 10% of 88?
Write a mathematical
sentence
X = .10
88
X = 8.8
Try these:
Find 12% of 70
What is 40% of 28?
Another Method:
You can also solve percent problems by setting up a
proportion.
Since percents are parts of a whole, you can create the
following proportion:
When figuring out which is the "part" and which is the
"whole", remember that you take a percent of the whole
and the answer is the part. In other words, the whole is
with the word "of" and the part is with the word "is".
Proportion Method
Steps
1. Set up the proportion as shown. is
of
=
%
100
Note: You can use this box to solve many problems
involving percents!
2. Substitute given values into the proportion.
Note: Try to find the numbers that are attached to the
words/symbols: is, of, or percent.
3. Solve the proportion.
Example: What is 25% of 400?
is
Steps
1. Set up the proportion.
of
2. Substitute.
What is 25% of 400?
3. Solve.
400 x 25 = 100w
10,000 = 100w
10,000/100 = w
100 = w
Click
?
400
=
25
%
100
100
Click on each box to see if
you substituted correctly.
Example: What is 32% of 300?
Steps
is
1. Set up the proportion.
of
?
=
300
32
100
Click on each box to see if
you substituted correctly.
2. Substitute.
What is 32% of 300?
3. Solve.
300 x 32 =
100w
9,600 = 100w
9600/100 = w
Click
96 = w
%
100
Try it: What is 20% of 180?
is
Steps
1. Set up the proportion.
of
2. Substitute.
3. Solve.
=
%
100
12 Find 30% of 45
13 What is 15% of 90?
14 Find the greater value.
A
20% of 16
B
10% of 90
C
25% of 40
D
100% of 7
15 Find the greater value.
A
2% of 1000
B
5% of 500
C
10% of 300
D
15% of 100
16 Identify any values that are equal.
A
What is 40% of 80?
B
60% of 70
C
25% of 128
D
200% of 16
Finding the Whole...
Remember, you can solve this by:
1. Translating into an equation
2. Setting up a proportion
40% of what number is 50?
.40
X = 50
X = 50
.40
X = 125
Try This:
100 is 20% of what number?
100 = .20
100 = x
.20
x = 500
x
17 56 is 70% of what?
18 12% of what number is 6?
19 65% of what number is 10?
20 27 is 150% of what number?
21 1% of what number is 12?
Finding the Percent...
Remember, you can solve this by:
1. Translating into an equation
2. Setting up a proportion
What percent of 80 is 24?
x 80 = 24
X = 24
80
X = .30
X = 30%
60 is what percent of 15?
60 = X 15
60 = X
15
4=X
400% = X
22 What percent of 3 is 12?
23 30 is what percent of 36?
24 What percent of 18 is 180?
25 2 is what percent of 1?
26 What percent of 25 is 20?
You have just studied three different types of percent
problems.
Try all 3 types:
24 is 40% of what number?
42 is what percent of 840?
What is 30% of 45?
27 Find the largest value.
A
What is 50% of 50?
B
What number is 45% of 60?
C
30 is 60% of what number?
D
25% of what number is 150?
28 Find the greatest percentage value.
A
What percent of 30 is 18?
B
60 is what percent of 90?
C
What percent of 70 is 210?
D
1,000 is what percent of 100?
29 Find 20% of 78.
30 8 is what percent of 28?
31 What number is 3% of 17?
32 Find 27% of 54.
33 23 is what percent of 200?
34 What percent is 35 of 20?
35 56% of what number is 40?
36 45 is 30% of what number?
37 62% of 40 is what number?
A
24.8
B
.0155
C
24.8%
D
15.5
Percent of Change
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of contents
Percent of Change:
The ratio of the amount of increase or decrease to the
original amount
It is an increase when the new amount is larger than the
original and a decrease when the new amount is smaller
than the original.
To find the percent of change, use the following
proportion:
Percent of change: Amount of increase or decrease = %
Original Amount
100
Find the percent of change
(be sure to label your answer as an increase or decrease).
Examples:
Original amount: 20
New amount: 30
Original amount: 40
New amount: 10
Percent of change=
Percent of change=
Identify the percent of change as an increase or
decrease. Then find the percent of change.
1. Original: 45
New: 75
2. Original: 100
New: 42
3. Original: 58
New: 75
Try This!
A CD's original price was $12.99.
It is now on sale for $10.99.
What is the percent of change?
Try This!
A student's first test grade was 60.
The second test grade was an 85.
What was the percent of change?
38 In 2005, the price of a McDonald's
hamburger was $0.89. In 2010, the price of a
McDonald's hamburger was $1.19. What was
the percent of change?
39 Original Amount: 500
New: 700
Find the percent of change.
40 Original Amount: 52
New: 17
Find the percent of change.
41 The number of students who attended FHS
in 2010 was 1405. In 2011, 1380 students
attended FHS. What was the percent of
change in student enrollment?
42 Find the percent of change.
Original price: $120
Sale price: $75
43 Find the percent of change.
Original price: $80
Sale price: $50
44 A stereo, originally priced at $360, is on sale
for $200. What is the percent of change?
Representing Percent
Equations
Algebraically
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of contents
You have already begun translating percent problems
into equations.
Remember...
To solve a percent problem, translate the words into an
equation.
Change:
1. Percent into a decimal
2. "is" to "="
3. "of" to " "
4. Unknown to "x"
Then, solve the equation.
Think about this...
100% + 5% = 105%
What does that equation look like in decimal form?
1 + 0.05 = 1.05
So, if you increase the price of a shirt 5%, the new price is
105% of the original price. To represent that algebraically,
you would write it this way:
Let s = the original price of the shirt
1s + 0.05s = 1.05s
Example:
You sell a shirt for $15.50. This price represents a 5%
increase from the price you paid for the shirt. How much
did it cost you to purchase the shirt?
Let s = the original price of the shirt
1s + 0.05s = 15.50
1.05s = 15.50
s = $14.76
The shirt cost you $14.76.
Example:
The population of your school decreased by 13% from
last year to this year. If there are 957 students in the
school this year, how many were there last year?
2 students solved this differently. Who is correct? Why?
Is one method easier than the other?
Student 1: Student 2:
100% - 13% = 87% 1n - .13n = 957
87% of what is 957? 0.87n = 957
0.87n = 957 n = 1,100 students
n = 1,100 students
So, what does this mean?
m + 0.15m = 1.15m
This could mean increase m by 15% or multiply m by
1.15.
They mean the same thing!
Click
Likewise, what is the meaning of
w - 0.42w = 0.58w
This means both decrease w by 42% or multiply w by
0.58.
Click
You Try.
1. A smart phone is on sale for $299, or 18% off. What
was the original price of the phone? Write and solve
an equation to represent this situation.
2. What does this equation mean?
p + 0.02p = 1.02p
3. What does this equation mean?
h - 0.1h = 0.9h
45 Write an equation to represent the problem,
then solve. Be prepared to show me your
equation!
When you go shopping, you must pay an
additional 6% in sales tax. What is the price
of your items before taxes if your final price
is $25?
46 Choose the equation that represents the
situation.
The population of a town increased by 1%.
A
B
C
D
x + 0.01x = 1.01x
x + 0.1x = 1.1x
x - 0.1x = 0.9x
x - 0.01x = 0.99x
47 Write an equation to represent the problem,
then solve. Be prepared to show me your
equation!
The number of students in your class has
decreased by 12% since September. How
many students were there at the start if there
are currently 19 students?
48 Choose the equation that represents the
situation.
A 15% discount.
A
B
C
D
x + 0.15x = 0.85x
x + 1.5x = 2.5x
x - 0.015x = 0.985x
x - 0.15x = 0.85x
49 Write an equation to represent the problem,
then solve. Be prepared to show me your
equation!
When you paid your bill at a restaurant, you
included 24% more to cover tax and tip. If
you paid $55.80, what was the amount of the
original bill?
Applied Percent of
Decrease
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of contents
There are situations when the percent of change is
going to be a decrease. Examples are:
• Discounts
• Sales
• Reduction in Population
When finding a discount, there are two different
methods you can use.
Method 1:
Find the percent of the original price (discounted amount
in $)
Subtract the discount from the original price.
Method 2:
Subtract the percent from 100% (percent you are paying)
Find the percent of the original price.
Example: A $50 sweater is on sale for 20% off.
Calculate the sale price.
Method 1:
Find the percent of the original price (discounted amount
in $)
Subtract the discount from the original price.
(Discount)
(Sale price)
Method 2:
Subtract the percent from 100% (percent you are paying)
Find the percent of the original price.
(Percent you pay)
(Sale price)
A manager wants to provide a 30% discount for everything
in his store. Find the sale price of a $25 sweater.
(Discount)
(Sale price)
(Percent you pay)
(Sale price)
Click to view Method 1
Click to view Method 2
Using either method, the answer is $17.50
Click to view answer
The manager has pants, priced at $45, that he needs to
mark down 35%. What will be the sale price of the
pants?
(Discount)
(Sale price)
(Percent you pay)
(Sale price)
Click to view Method 1
Click to view Method 2
The pants are on sale for $29.25
Click to view answer
Mark wants to purchase a stereo that is on sale, if he is
saving at least 30%.
The stereo's original cost is $425.
What is the most that he is willing to pay for the stereo?
(Discount)
(Sale price)
Click to view Method 1
(Percent you pay)
(Sale price)
Click to view Method 2
He is willing to pay $297.50 for the
stereo.
Click
to view answer
50 Decrease 400 by 10%
51 A $710 computer is to be discounted 30%.
What will be the sale price?
52 A necklace, priced at $120, is to be marked
down 15%. What will be the sale price?
53 The student population of the high school
will decrease by 5% next year. The current
population is 1407 students. How many
students will attend next year?
54 The store is having a 40% off sale. What
percent will the customers pay?
55 $80 boots are on sale for 20% off. After the
sale, the manager raises the price 20%. What
will be the selling price of the boots after the
sale?
Applied Percent
of Increase
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of contents
There are situations when the percent of change is
going to be an increase. Examples are:
• Tips
• Sales Tax
• Increase in Population
When finding an increase, there are two different
methods you can use.
Method 1:
Find the percent of the original price (increased amount)
Add the increase to the original price.
Method 2:
Add the percent to 100% (percent you are paying)
Find the percent of the original price.
Finding a New Amount
Increase 55 by 20%
(Mark up)
(New cost)
(Percent you pay)
(New cost)
Find the new amount
Increase 60 by 10%
Increase 68 by 12%
56 Increase 36 by 25%.
57 Increase 40 by 15%
Tip: An amount added to a bill for services provided.
Customers traditionally tip 18 - 20% for good service in
restaurants and salons.
Example:
If the restaurant bill is $45 and you want to leave a 20%
tip, how much money should you leave?
45 + .20(45) = 54
or 45(1.20) = 54
The customer will leave $54 on the table. The waitress
will receive a $9 tip and the restaurant will receive $45.
To calculate the amount of the tip only:
.20(45) = 9
Calculate a 20% tip on a $75 bill.
What will the customer leave in total?
For poor service, my friend will leave a 5% tip.
How much less will this waitress earn than the waitress
above?
Sales tax: An amount of money that is calculated by
applying a percentage rate to the taxable price of a sale.
Sales taxes are collected by the buyer from the seller,
who turns it over to the government. In NJ the sales tax
rate is 7%.
To calculate Sales Tax alone find the percent (tax) of the
price. That is the amount that you owe in addition to the
cost of the item.
To find the total cost of an item, you must add the sales
tax to the cost. There are 2 ways to do this:
1. Find the percent of the item and add it to the original
amount.
2. Find 100% + tax% of the original amount.
A car costs $23,500. How much sales tax will the customer
pay?
23,500(0.07) = $1645
What will the customer pay altogether for the car?
23,500 + 1645 = $25,145
The total cost of the car, including tax, can be calculated
as follows:
23,500 + .07(23,500) = 25,145
or
23,500(1.20) = 25,145
Discuss:
How are tips and sales tax alike?
58 What is the total cost of a $250 stereo in the
state of NJ?
59 Calculate the sales tax on a $125 bicycle.
60 Mike wants to leave a 20% tip. His bill is
$35.50. How much is the tip?
61 A $65 restaurant tab is put on the table. The
couple plans on leaving an 18% tip. How
much should be left altogether?
62 What is the total cost of a $123 ipod,
including tax?
Real-Life
Application Problems
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of contents
A store owner pays $12 for a particular bracelet.
To cover expenses, the owner will mark up the price
by 150%. Find the selling price of the bracelet.
The store is having a 20% off sale on all CD's.
With the sale, you pay $12 for a CD.
What was the original price?
A couple left their waiter a 20% tip in the amount of $18.
What was the cost of their meal?
You and 3 friends had dinner at a restaurant. The cost of
their meals is $62. They want to leave a 15% tip. Calculate
the tip.
When they arrive at the register the cashier will calculate
the sales tax on the meal at a rate of 7%.
Determine the sales tax. (*Note: You never tax on the tip)
Calculate the total cost of the meal for each of you.
A store is having a 25% off sale on ipods. You want to
purchase an ipod with an original price of $249. The
sales tax is 7%, which will be applied to the sale price
of the ipod. What is the total cost of the ipod?
A computer is on sale for 10% off the original price
of $325. When it doesn't sell, the manager marks it
down another 20% off the sale price.
What is the new sale price of the laptop?
Is the new sale price the same as it would be had
the manager taken 30% off of the original price?
Explain.
63 Wholesale price: $56
Markup percent: 50%
New price ?
64 Tickets cost $7 at the door. If purchased in
advance, the tickets cost $5. What is the
percent of discount for purchasing tickets in
advance?
65 560 people were surveyed. 25% said they
prefer Coke. How many people prefer Coke?
66 Increase 50 by 25%. What is the new amount?
67 What is the original price on a pair of boots
that sell for $72 after a 25% discount?
68 An ipod costs $176. It is on sale for 20% off
and will be taxed at a rate of 7% on the sale
price. What will be the total cost of the ipod?