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7th Grade
Percents
2015-11-30 www.njctl.org
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Three Types of Percent Problems Percent of Change Applied Percent of Decrease Real-life Application Problems Relating Fractions, Decimals and Percents Applied Percent of Increase Representing Percent Equations Algebraically
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Glossary
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SLIDE 2 Relating Fractions, Decimals & Percents
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% to a fraction % to a decimal fraction to a % decimal to a %
Helping you remember...
Fill in each box below with an example of the process described.
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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 12.5% 0.15
Ordering Slide 6 / 157
SLIDE 3
1 Find the lowest value. A 5% B 1/2 C .5% D .05
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2 Find the greatest value. A 120% B 1.02 C .2% D 1.19
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3 Find the greatest value. A 6% B .6 C 60 D 6
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SLIDE 4
4 Find the lowest value. A 2% B .2 C .02 D .2%
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5 Find the lowest value. A 50% B 500% C 50.0 D 50.01
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SLIDE 5 Express each fraction as a percent: 1) 2) 3)
click click click
Examples Slide 13 / 157
6 Express as a fraction.
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7 Express as a decimal.
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SLIDE 6
8 Express as a percent.
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9 Express as a decimal.
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10 Express as a percent.
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SLIDE 7 11 Express as a percent.
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Three Types of Percent Problems
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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:
What number is 54% of 34?
4 is 60% of what number?
18 is what percent of 28?
Types of Percent Problems Slide 21 / 157
SLIDE 8 Two words that will occur in these types of problems are: "is" "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.
Types of Percent Problems Slide 22 / 157 Finding the Part... Slide 23 / 157
Examples: Find 40% of 60 .40 60 = 24 20% of 90 .20 90 = 18 Write a mathematical sentence Write a mathematical sentence
Click Click
Find the Part Slide 24 / 157
SLIDE 9
What is 10% of 88? X = .10 88 X = 8.8 Try these: Find 12% of 70 What is 40% of 28?
Write a Mathematical Sentence Slide 25 / 157 Slide 26 / 157 Slide 27 / 157
SLIDE 10 Steps
- 1. Set up the proportion as shown.
- 2. Substitute given values into the proportion.
- 3. Solve the proportion.
is
% 100
=
Note: You can use this box to solve many problems involving percents! Note: Try to find the numbers that are attached to the words/symbols: is, of, or percent.
Proportion Method Slide 28 / 157
Example: What is 25% of 400? Steps
- 1. Set up the proportion.
- 2. Substitute.
- 3. Solve.
is
% 100
=
What is 25% of 400? 25 100 400 ?
Click on each box to see if you substituted correctly.
400 x 25 = 100w 10,000 = 100w 10,000/100 = w 100 = w
click
Proportion Method Example Slide 29 / 157
Example: What is 32% of 300? Steps
- 1. Set up the proportion.
- 2. Substitute.
- 3. Solve.
is
% 100
=
32 100 300 ?
Click on each box to see if you substituted correctly.
300 x 32 = 100w 9,600 = 100w 9600/100 = w 96 = w What is 32% of 300?
click
Proportion Method Example Slide 30 / 157
SLIDE 11 Try it: What is 20% of 180? Steps
- 1. Set up the proportion.
- 2. Substitute.
- 3. Solve.
is
% 100
=
Proportion Method - Try It Slide 31 / 157
12 Find 30% of 45.
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13 What is 15% of 90?
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SLIDE 12
14 Find the greater value. A 20% of 16 B 10% of 90 C 25% of 40 D 100% of 7
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15 Find the greater value. A 2% of 1000 B 5% of 500 C 10% of 300 D 15% of 100
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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
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SLIDE 13 Finding the Whole... Slide 37 / 157
X = 50 .40 X = 125 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
Finding the Whole Slide 38 / 157
100 = x .20 x = 500 Try This: 100 is 20% of what number? 100 = .20 x
Finding the Whole Slide 39 / 157
SLIDE 14
17 56 is 70% of what?
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18 12% of what number is 6?
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19 65% of what number is 10?
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SLIDE 15
20 27 is 150% of what number?
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21 1% of what number is 12?
Slide 44 / 157 Finding the Percent... Slide 45 / 157
SLIDE 16 What percent of 80 is 24?
x 80 = 24
X = 24 80 X = .30 X = 30% Remember, you can solve this by:
- 1. Translating into an equation
- 2. Setting up a proportion
Finding the Percent Slide 46 / 157
60 is what percent of 15?
60 = x 15
60 = X 15 4 = X 400% = X
Finding the Percent Slide 47 / 157
22 What percent of 3 is 12?
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SLIDE 17
23 30 is what percent of 36?
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24 What percent of 18 is 180?
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25 2 is what percent of 1?
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SLIDE 18
26 What percent of 25 is 20?
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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?
Percent Problems Slide 53 / 157
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?
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SLIDE 19
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?
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29 Find 20% of 78.
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30 Eight is what percent of 28?
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SLIDE 20
31 What number is 3% of 17?
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32 Find 27% of 54.
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33 23 is what percent of 200?
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SLIDE 21
34 What percent is 35 of 20?
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35 Fifty six percent of what number is 40?
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36 Forty five is 30% of what number?
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SLIDE 22 37 Sixty two percent of 40 is what number? A 24.8 B .0155 C 24.8% D 15.5
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Percent of Change
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Percent of Change is 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
- riginal 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
Percent of Change Slide 66 / 157
SLIDE 23 Find the percent of change (be sure to label your answer as an increase or decrease). Examples: Original amount: 20 Original amount: 40 New amount: 30 New amount: 10 Percent of change= Percent of change=
Percent of Change Slide 67 / 157
Identify the percent of change as an increase or
- decrease. Then find the percent of change.
- 1. Original: 45
New: 75
New: 42
New: 75
Percent of Change Slide 68 / 157
Try This! A CD's original price was $12.99. It is now on sale for $10.99. What is the percent of change?
Percent of Change Slide 69 / 157
SLIDE 24
Try This! A student's first test grade was 60. The second test grade was an 85. What was the percent of change?
Percent of Change Slide 70 / 157
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?
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39 Original Amount: 500 New: 700 Find the percent of change.
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SLIDE 25 40 Original Amount: 52 New: 17 Find the percent of change.
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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?
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42 Find the percent of change. Original price: $120 Sale price: $75
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SLIDE 26 43 Find the percent of change. Original price: $80 Sale price: $50
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44 A stereo, originally priced at $360, is on sale for $200. What is the percent of change?
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Representing Percent Equations Algebraically
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SLIDE 27 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.
Representing Percent Equations Algebraically Slide 79 / 157 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
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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.
Representing Percent Equations Algebraically Slide 81 / 157
SLIDE 28 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
Representing Percent Equations Algebraically Slide 82 / 157
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! 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 Click
Representing Percent Equations Algebraically Slide 83 / 157
You Try.
- 1. A smart phone is on sale for $299, or 18% off. What was the
- riginal 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
Representing Percent Equations Algebraically Slide 84 / 157
SLIDE 29 45 Write an equation to represent the problem, then
- solve. Be prepared to show 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?
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46 Choose the equation that represents the situation. The population of a town increased by 1%. A x + 0.01x = 1.01x B x + 0.1x = 1.1x C x - 0.1x = 0.9x D x - 0.01x = 0.99x
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47 Write an equation to represent the problem, then solve. Be prepared to show 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?
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SLIDE 30 48 Choose the equation that represents the situation. A 15% discount. A x + 0.15x = 0.85x B x + 1.5x = 2.5x C x - 0.015x = 0.985x D x - 0.15x = 0.85x
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49 Write an equation to represent the problem, then
- solve. Be prepared to show 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?
Slide 89 / 157 Simple Interest Formula
Larry invests $100 in a savings plan. The plan pays 4.5% interest each year on his $100 account balance. The following chart shows the balance on his account after each year for the next 5 years. He did not make any deposits of withdrawals during this time. Time (in years) Balance (in dollars) 1 104.50 2 109.00 3 113.50 4 118.00 5 122.50 · What pattern(s) do you notice from the table? · What is simple interest? · How is it calculated? · Can you create a formula to represent the pattern(s) you notice?
Derived from
( (
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SLIDE 31 To find simple interest, use:
Derived from
( (
Interest = Principal x Rate x Time I = P x r x t I = Prt · r is the percent of the principal that is paid over a period of time (usually per year). · t is the time. · r and t must be compatible. For example, if r is an annual interest rate, then t must be written in years.
Simple Interest Formula Slide 91 / 157
Can Money Grow? A Look at Simple Interest Larry invests $100 in a savings plan. The plan pays 4 1/2% interest each year on his $100 account balance. How much money will Larry earn in interest after 3 years? 5 years? 3 years: 5 years: I = Prt
Derived from
( (
I = Prt I = 100(0.045)(3) I = 13.50 I = 100(0.045)(5) I = 22.50 Larry will earn $22.50 in interest after 5 years.
Simple Interest Formula Slide 92 / 157
How can you find the balance of Larry's account at the end of 5 years? Answer: Add the interest earned after 5 years to the beginning balance. $22.50 + $100 = $122.50 Click
Derived from
( (
Simple Interest Formula Slide 93 / 157
SLIDE 32 50 A $1,000 savings bond earns simple interest at the rate
- f 3% each year. The interest is paid at the end of every
- month. How much interest will the bond have earned
after 3 months?
Problem derived from
( (
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51 Mr. Williams wants to know how long it will take an investment of $450 to earn $200 in interest if the yearly interest rate is 6.5%, paid at the end of each year.
Problem derived from
( (
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52 Find the amount of simple interest, A, earned on a $600 investment after 1 1/2 years if the semi-annual (6 month) interest rate is 2%.
Problem derived from
( (
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SLIDE 33 53 A $1,500 loan has an annual interest rate of 4 1/4% on the amount borrowed. How much time has elapsed if the interest is not $127.50?
Problem derived from
( (
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Applied Percent of Decrease
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There are situations when the percent of change is going to be a
· Discounts · Sales · Reduction in Population
Applied Percent of Decrease Slide 99 / 157
SLIDE 34
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.
Applied Percent of Decrease Slide 100 / 157 Slide 101 / 157 Slide 102 / 157
SLIDE 35
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54 Decrease 400 by 10%
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SLIDE 36
55 A $710 computer is to be discounted 30%. What will be the sale price?
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56 A necklace, priced at $120, is to be marked down 15%. What will be the sale price?
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57 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?
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SLIDE 37 58 The store is having a 40% off sale. What percent will the customers pay?
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59 $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?
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Applied Percent of Increase
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SLIDE 38
There are situations when the percent of change is going to be an increase . Examples are: · Tips · Sales Tax · Increase in Population
Applied Percent of Increase Slide 112 / 157
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.
Applied Percent of Increase Slide 113 / 157 Slide 114 / 157
SLIDE 39
Find the new amount Increase 60 by 10% Increase 68 by 12%
Applied Percent of Increase Slide 115 / 157
60 Increase 36 by 25%.
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61 Increase 40 by 15%
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SLIDE 40 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
Tip Slide 118 / 157
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?
Tip Slide 119 / 157
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.
Sales Tax Slide 120 / 157
SLIDE 41 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(1.07) = 25,145
Sales Tax Slide 121 / 157 Discuss
How are tips and sales tax alike?
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62 What is the total cost of a $250 stereo in the state
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SLIDE 42
63 Calculate the sales tax on a $125 bicycle.
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64 Mike wants to leave a 20% tip. His bill is $35.50. How much is the tip?
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65 A $65 restaurant tab is put on the table. The couple plans on leaving an 18% tip. How much should be left altogether?
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SLIDE 43 Real Life Application Problems
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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.
Application Problems Slide 128 / 157
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?
Application Problems Slide 129 / 157
SLIDE 44
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?
Application Problems Slide 130 / 157
A couple left their waiter a 20% tip in the amount of $18. What was the cost of their meal?
Application Problems Slide 131 / 157
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.
Application Problems Slide 132 / 157
SLIDE 45 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?
Application Problems Slide 133 / 157
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.
Application Problems Slide 134 / 157
66 Wholesale price: $56 Markup percent: 50% New price ?
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SLIDE 46
67 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?
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68 Five hundred sixty people were surveyed. 25% said they prefer Coke. How many people prefer Coke?
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69 Increase 50 by 25%. What is the new amount?
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SLIDE 47
70 What is the original price on a pair of boots that sell for $72 after a 25% discount?
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71 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?
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72 What is the total cost of a $123 ipod, including tax?
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SLIDE 48 A teacher survey students in four classes to determine the location for a field trip. Each student chose only one
- location. The table shows the number of students from
each class who chose each location. (Use this table for the next two questions.)
From PARCC PBA sample test calculator #4
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Part A Determine the percent of students in each class who chose the museum. What is the order from greatest to least of the percents for each class? Drag and drop the classes into the correct order from greatest to least with the greatest at the top.
Class E Class G Class F Class H
From PARCC sample test
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73 Part B The total number of students who chose the zoo is how many times as great as the total number of students who chose the planetarium?
From PARCC sample test
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SLIDE 49 74 A store owner paid $15 for a book. She marked up the price of the book by 40% to determine its selling price. Part A What is the selling price of the book?
From PARCC EOY sample test calculator #10
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75 Part B A customer buys a different book that has an original selling price of $38. The book is discounted 25%. The customer must pay a 6% sales tax on the discounted price of the book. What is the total amount the customer pays for the discounted book?
From PARCC EOY sample test calculator #10
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76 The students in Noami's class sold calendars for a fund- raiser this year and last year. This year, the selling price
- f each calendar was $13.25. The price this year
represents 6% more than the selling price of each calendar last year. Part A What is the selling price of each calendar last year?
From PARCC EOY sample test calculator #6
continued...
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SLIDE 50 77 Part B The students in Naomi's class earned 20% of the selling price of each calendar sold this year and last year. · At last year's selling price, Naomi's class sold 650 calendars. · At this year's selling price, Naomi's class sold 600 calendars. Select a choice from each group to fill in the blanks. The students in Naomi's class earned more money from the fundraiser ________ by ________. A last year B this year C $20 D $25 E $35 F $50 G $60
From PARCC EOY sample test calculator #6
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78 Each bulleted statement describes how the amount of
income tax is determined for yearly incomes in different ranges. · Yearly incomes of 8,925 or less are taxed at a flat rate
· For yearly incomes from $8,926 to $36,250, the first $8,925 is taxed at 10% and any income beyond that is taxed at 15%. · For yearly incomes greater than $36.250, the first $8,925 is taxed at 10%, the next $27,325 is taxed at 15% and any income beyond $36,250 is taxed at 25%. Part A
- Mr. Vance's yearly taxable income is $35,675. What is the
dollar amount taken our for taxes based on Mr. Vance's income?
From PARCC EOY sample test calculator #9
continued...
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79 Part B
- Mr. Rivera's taxable income is $20 each hour before
taxes are taken out. Mr. Rivera worked a total of 40 hours each week for 50 weeks. What is the dollar amount taken
- ut for taxes based on Mr. Rivera's taxable income?
From PARCC EOY sample test calculator #9
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SLIDE 51
Glossary
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SLIDE 52 Slide 154 / 157
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Percent of Change
The ratio of the amount of increase or decrease to the original amount.
Increas e Decreas e
new amount
amount
>
new amount
amount
<
amount: 20 new amount: 30
Amount of increase or decrease = % Original Amount 100
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Back to Instruction
Tip
An amount added to a bill for services provided.
Customers traditionally tip 18 - 20% for good service in restaurants and salons. 20% tip on $45 bill: .20(45) = $9 tip How much money will you leave on the table? $45 bill + $9 = $54 total
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SLIDE 53 Back to Instruction
Sales Tax
An amount of money that is calculated by applying a percentage rate to the taxable price of a sale.
In NJ the sales tax rate is 7%. A car costs $23,500. What is the amount of the sales tax? $23,500(0.07) = $1,645 How much money will you pay total? $23,500 + $1,645 = $25,145
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