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7th Grade Math Percents 2012-12-07 www.njctl.org

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Table of Contents Relating Fractions, Decimals and Percents 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 Return to table 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  12.5%  0.15

1 Find the lowest value A 5% B 1/2 C .5% D .05 Answer: C

Find the greatest value 2 Find the greatest value A 120% B 1.02 C .2% D 1.19 Answer: A

Find the greatest value 3 Find the greatest value A 6% B .6 C 60 D 6 Answer: C

4 Find the lowest value A 2% B .2 C .02 D .2% Answer: D

5 Find the lowest value A 50% B 500% C 50.0 D 50.01 Answer: A

Express each decimal or percent as a fraction in lowest terms: 1) 18%  2) 0.85  3) 4)   5) 5.008  6) 0.0001 Click to Reveal Click to Reveal Click to Reveal Click to Reveal Click to Reveal Click to Reveal

Express each fraction as a percent: 1)   2)  3) Click to Reveal Click to Reveal Click to Reveal

6 Express as a fraction. Answer: 37/200

7 Express as a decimal. Answer: 0.167

8 Express as a percent. Answer: 34.5%

9 Express as a decimal. Answer: 0.075

10 Express as a percent. Answer: 81.25%

11 Express as a percent. Answer: 72.727%

Three Types of Percent Problems Return to table 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...

Write a mathematical sentence Examples: Find 40% of 60 Write a mathematical sentence Click .40 60 = 24 20% of 90 Write a mathematical sentence Click .20 90 = 18

Write a mathematical sentence X = .10 88 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? Answer: .12 x 70 = 8.4 X = .40 x 28 X = 11.20

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 2. Substitute given values into the proportion. 3. Solve the proportion. is of % 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.

Click on each box to see if you substituted correctly. Example: What is 25% of 400? Steps 1. Set up the proportion. 2. Substitute. 3. Solve. is of % 100 = ? 25 400 100 Click on each box to see if you substituted correctly. What is 25% of 400? Click 400 x 25 = 100w 10,000 = 100w 10,000/100 = w 100 = w

Click on each box to see if you substituted correctly. Example: What is 32% of 300? Steps 1. Set up the proportion. 2. Substitute. 3. Solve. is of % 100 = ? 32 300 100 Click on each box to see if you substituted correctly. What is 32% of 300? 300 x 32 = 100w 9,600 = 100w 9600/100 = w 96 = w Click

= Try it: What is 20% of 180? Steps 1. Set up the proportion. % is 2. Substitute. 3. Solve. is of % 100 = Answer: 36

12 Find 30% of 45 Answer: 13.5

13 What is 15% of 90? Answer: 13.5

14 Find the greater value. A 20% of 16 B 10% of 90 C 25% of 40 D Answer: C

15 Find the greater value. A 2% of 1000 B 5% of 500 C 10% of 300 D Answer: C

Identify any values that are equal. 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 Answer: A, C, D

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 x 100 = x .20 x = 500

17 56 is 70% of what? Answer: 80

18 12% of what number is 6? Answer: 50

19 65% of what number is 10? Answer: 15.385

20 27 is 150% of what number? Answer: 18

21 1% of what number is 12? Answer: 1,200

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? Answer: 400%

23 30 is what percent of 36? Answer: 83.3%

24 What percent of 18 is 180? Answer: 1000%

25 2 is what percent of 1? Answer: 200%

26 What percent of 25 is 20? Answer: 80%

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? Answers: 60 5% 13.5

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? Answer: D

Find the greatest percentage value. 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? Answer: D

29 Find 20% of 78. Answer: 15.6

30 8 is what percent of 28? Answer: 28.571%

31 What number is 3% of 17? Answer: 0.51

32 Find 27% of 54. Answer: 14.58

33 23 is what percent of 200? Answer: 11.5%

34 What percent is 35 of 20? Answer: 175%

35 56% of what number is 40? Answer: 71.429

36 45 is 30% of what number? Answer: 150

37 62% of 40 is what number? A 24.8 B .0155 C 24.8% D 15.5 Answer: A

Percent of Change Return to table 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  Original amount: 40 New amount: 30  New amount: 10 Percent of change=   Percent of change=

Identify the percent of change as an increase or decrease 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 Answer: Increase 66.7% Decrease 58% Increase 29.3%

A CD's original price was $12.99. It is now on sale for $10.99. Try This! A CD's original price was $12.99. It is now on sale for $10.99. What is the percent of change? Answer: 15.4% decrease

A student's first test grade was 60. The second test grade was an 85. Try This! A student's first test grade was 60. The second test grade was an 85. What was the percent of change? Answer: 41.7% increase

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? Answer: 33.7% increase

Find the percent of change. 39 Original Amount: 500 New: 700 Find the percent of change. Answer: 40% increase

Find the percent of change. 40 Original Amount: 52 New: 17 Find the percent of change. Answer: 67.3% decrease

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? Answer: 1.8% decrease

Find the percent of change. Original price: $120 Sale price: $75 42 Find the percent of change. Original price: $120 Sale price: $75 Answer: 37.5% decrease

Find the percent of change. Original price: $80 Sale price: $50 43 Find the percent of change. Original price: $80 Sale price: $50 Answer: 37.5% decrease

44 A stereo, originally priced at $360, is on sale for $200. What is the percent of change? Answer: 44.4% decrease

Representing Percent Equations Algebraically Return to table 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

This could mean increase m by 15% or multiply m by 1.15. Click 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

A smart phone is on sale for $299, or 18% off. What You Try. 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 Answer: 1. 1x - .18x = 299 $364.63 2. 2% increase 3. 10% decrease

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? Answer: $23.58 1x + .06x = 25 x = $23.58

Choose the equation that represents the situation. 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 Answer: A

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? Answer: 19 = 1x - .12x about 22 students

Choose the equation that represents the situation. 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 Answer: D

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? Answer: 1x + .24x = 55.80 $45

Applied Percent of Decrease Return to table 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. 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. Method 2: Subtract the percent from 100% (percent you are paying) Find the percent of the original price. (Discount) (Sale price) (Percent you pay) (Sale price)

Using either method, the answer is $17.50 A manager wants to provide a 30% discount for everything in his store. Find the sale price of a $25 sweater. Using either method, the answer is $17.50 Click to view Method 1 Click to view Method 2 (Percent you pay) (Discount) (Sale price) (Sale price) Click to view answer

The pants are on sale for $29.25 The manager has pants, priced at $45, that he needs to mark down 35%. What will be the sale price of the pants? The pants are on sale for $29.25 Click to view Method 1 Click to view Method 2 (Discount) (Percent you pay) (Sale price) (Sale price) Click to view answer

The stereo's original cost is $425. 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? Click to view Method 1 Click to view Method 2 (Discount) (Percent you pay) (Sale price) (Sale price) Click to view answer He is willing to pay $297.50 for the stereo.

50 Decrease 400 by 10% Answer: 360

A $710 computer is to be discounted 30%. What will be the sale price? 51 A $710 computer is to be discounted 30%. What will be the sale price? Answer: $497

52 A necklace, priced at $120, is to be marked down 
15%. What will be the sale price? Answer: $102

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? Answer: 1,336.65 About 1,337 students

54 The store is having a 40% off sale. What percent will the customers pay? Answer: 60%

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? Answer: $76.80

Applied Percent of Increase Return to table 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. 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) (Percent you pay) (New cost) (New cost)

Find the new amount Increase 60 by 10% Increase 68 by 12% Answers: 66 76.16

56 Increase 36 by 25%. Answer: 45

57 Increase 40 by 15% Answer: 46

Tip: An amount added to a bill for services provided 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? Answers: $15 $90 The waitress will earn $11.25 less.

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: 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?

What is the total cost of a $250 stereo in the state of NJ? 58 What is the total cost of a $250 stereo in the state of NJ? Answer: $267.50

Calculate the sales tax on a $125 bicycle. 59 Calculate the sales tax on a $125 bicycle. Answer: $8.75

60 Mike wants to leave a 20% tip. His bill is $35.50. How much is the tip? Answer: $7.10

61 A $65 restaurant tab is put on the table. The couple plans on leaving an 18% tip. How much should be left altogether? Answer: $76.70

What is the total cost of a $123 ipod, including tax? 62 What is the total cost of a $123 ipod, including tax? Answer: $131.61

Real-Life Application Problems Return to table 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. Answer: 12 + 12 x 1.50 = 30

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? Answers: 12 = .80 x X 15 = x 20% of what is 18? .20 x X = 18 $90

Determine the sales tax. (*Note: You never tax on the tip) 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. Answer: Tip: $9.30 Tax: $4.34 Cost: $18.91

What is the new sale price of the laptop? 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. Answer: $199.82 $234 $227.50 30% off gives the best price. Taking 
20% off, after a 10% discount, is 
giving 20% off a lesser price.

Wholesale price: $56 Markup percent: 50% New price ? 63 Wholesale price: $56 Markup percent: 50% New price ? Answer: $84

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? Answer: 28.57%

65 560 people were surveyed. 25% said they prefer Coke. How many people prefer Coke? Answer: 140 people

Increase 50 by 25%. What is the new amount? 66 Increase 50 by 25%. What is the new amount? Answer: 62.50

67 What is the original price on a pair of boots that sell for $72 after a 25% discount? Answer: $96

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? Answer: $150.66