CHAPTER 7: PERCENTS AND APPLICATIONS

Transcription

1 CHAPTER 7: PERCENTS AND APPLICATIONS Chapter 7 Contents 7. Introduction to Percents and Conversions Among Fractions, Decimals and Percents 7.2 Translating and Solving Percent Problems 7.3 Circle Graphs 7.4 Financial Applications of Percents 7.5 Application Problems 467

2 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages 7. Introduction to Percents and Conversions to Fractions and Decimals Percent comes from per centum which means by the hundred. A percent is a representation of how many out of one hundred. This means that an 85% on a test represents 85 out of 00 points on a test. In order to write a percent as a fraction, remember it is out of one hundred (over one hundred). Once the percent is written as a fraction, it needs to be reduced, if possible, to lowest terms. Example : Write 78% as a fraction % Write percent as a fraction by dividing by Simplify by first writing numerator and denominator in factored 2 50 form Then, divide out common factors in the numerator and denominator. Practice : Write 65% as a fraction. Answer: Watch It: Example 2: Write 3% as a fraction. 3 3% Write percent as a fraction by dividing by Note: The fraction 3 00 cannot be simplified. Practice 2: Write 7% as a fraction. Answer: 7 00 Watch It: 468

3 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Example 3: Write 5% as a fraction. 5 5% Write percent as a fraction by dividing by Simplify by first writing numerator and denominator in factored 5 20 form. 5 Then, divide out common factors in the numerator and denominator. Practice 3: Write 20% as a fraction. Answer: 5 Watch It: Example 4: Write 372% as a fraction % Write percent as a fraction by dividing by Simplify by first, writing numerator and denominator in factored form Then, divide out common factors in the numerator and denominator. 93 This is the answer as an improper fraction This is the answer as a mixed number. 25 Practice 4: Write 30% as a fraction. Answer: Watch It: 469

4 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages In some cases, the percent that is given involves a fractional part. In this case, it is easier to multiply the percent by, which, mathematically is the same as dividing by 00. As we 00 learned in Chapter 3, we can divide fractions by multiplying by the reciprocal of the divisor. % 00 % 00 % % TWO WAYS OF CONVERTING PERCENTS TO FRACTIONS Divide Percent by 00 Multiply Percent by 00 Percent 00 Fraction Form Percent Fraction Form 00 Looking back at the previous examples, in each case it could have been presented instead as multiplying the number by to determine the answer in fractional form Example 5: Write 6 % as a fraction. 3 In this example, we have a mixed number for the percent. Therefore using the method of multiplying the percent by will be the easier method to convert this percent to a fraction % % 3 3 First, convert the mixed number into an improper fraction. 20 Now multiply the percent as an improper fraction by Simplify by dividing the numerator and denominator by Practice 5: Watch It: Write 5 % 3 as a fraction. Answer:

5 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Example 6: Write 7 5 % 8 as a fraction % % 8 8 First, convert the mixed number into an improper fraction. 47 Now multiply the percent as an improper fraction by Multiply This fraction cannot be simplified. Thus this is the answer. Practice 6: Watch It: Write 5 2 % 6 as a fraction. Answer: In order to change a percent to a fraction, we multiplied by. This means that a percent 00 times 00 equals a fraction. To reverse this we would need to divide by (which is 00 equivalent to multiplying by 00). CONVERTING FRACTIONS TO PERCENTS Multiply Fraction by 00% Fraction Form 00% Percent Example 7: Write 3 as a percent % Multiply the fraction by % 75% 25 % Simplify by dividing the numerator and denominator by 4. 47

6 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Practice 7: Write 4 as a percent. Answer: 80% 5 Watch It: Example 8: Write 5 as a percent % Multiply the fraction by 00% % Simplify by dividing the numerator and denominator by % 2 This is the answer as an improper fraction percent. 62 % 2 This is the answer as a mixed number percent. = 62.5% This is the answer as a decimal percent. Practice 8: Write 3 as a percent. Answer: 60% 5 Watch It: Example 9: Write 4 as a percent Rewrite the mixed number as an improper fraction % Multiply the improper fraction by 00% % 25% % Simplify by dividing the numerator and denominator by 4. Practice 9: Write 3 5 as a percent. Answer: 60% Watch It: 472

7 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Example 0: Write as a percent. 00% 00% Multiply the whole number by 00%. Practice 0: Write 6 as a percent. Answer: 600% Watch It: When converting a percent to a number as in Examples, 2, 3, and 4, we wrote the number over one hundred. This represents dividing the number by 00. We will follow this same process when using decimals. To convert from a percent to a decimal, divide by 00. CONVERTING PERCENTS TO DECIMALS Divide the Percent by 00 Percent 00 Decimal Form The decimal point in the percent moves 2 places to the left. Example : Write 82% as a decimal. When converting from a percent, divide by 00: 82% = = 0.82 Practice : Write 68% as a decimal. Answer: 0.68 Watch It: Example 2: Write 45% as a decimal. When converting from a percent, divide by 00: 45% = = 0.45 Practice 2: Write 32% as a decimal. Answer: 0.32 Watch It: 473

8 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Example 3: Write 9% as a decimal. When converting from a percent, divide by 00: 9% = 9 00 = 0.09 Practice 3: Write 4% as a decimal. Answer: 0.04 Watch It: Example 4: Write 235% as a decimal. When converting from a percent, divide by 00: 235% = = 2.35 Practice 4: Write 26% as a decimal. Answer:.26 Watch It: When converting a number to a percent as in Examples 7, 8, 9, and 0, we multiplied by 00%. We will follow this same process when using decimals. To convert a number to a percent, multiply by 00%. CONVERTING DECIMALS TO PERCENTS Multiply the Decimal by 00% Decimal Form 00% % The decimal point moves two places to the right. Example 5: Write 0.29 as a percent. When converting to a percent, multiply by 00%: % = 29% Practice 5: Write 0.49 as a percent. Answer: 49% Watch It: 474

9 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages Example 6: Write 0.9 as a percent. When converting to a percent, multiply by 00%: % = 90% Practice 6: Write 0.3 as a percent. Answer: 30% Watch It: Example 7: Write as a percent. When converting to a percent, multiply by 00%: % = 5.8% Practice 7: Write as a percent. Answer: 2.5% Watch It: Example 8: Write 7 as a percent. When converting to a percent, multiply by 00%: 7 00% = 700% Practice 8: Write 5 as a percent. Answer: 500% Watch It: Watch All: 475

10 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages 7. Introduction to Percents Exercises. Convert to a fraction: 25% 2. Convert to a fraction: 56% 3. Convert to a fraction: 3% 4. Convert to a fraction: 7 % 2 5. Convert to a fraction: 90% 6. Convert to a fraction: 5% 7. Convert to a fraction: 25% 8. Convert to a fraction: 62% 9. Convert to a fraction: 33 % 3 0. Convert to a fraction: 300%. Convert to a fraction: 2. Convert to a percent: 3 3. Convert to a percent: Convert to a percent: Convert to a percent: 2 66 % Convert to a percent: Convert to a percent: 2 8. Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: 23. Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent: Convert to a percent:. 3. Convert to a percent: Convert to a decimal: 29% 33. Convert to a decimal: 24.6% 34. Convert to a decimal: 5.75% 35. Convert to a decimal: 2% 36. Convert to a decimal: 320% 37. Convert to a decimal: 2.4% 38. Convert to a decimal: 0.75% Convert to a decimal: 25 % Convert to a decimal: 62% 4. Convert to a decimal: 0.03% 42. Convert to a decimal: 37% 43. Convert to a decimal: 800% 476

11 CCBC Math 08 Introduction to Percents and Conversions Among Section 7. Fractions Decimals and Percents pages 7. Introduction to Percents Exercise Answers % 3 83 % % % % 7. 50% 8. 60% % 3 37 % % % % % % % or 2.5% % % % 30. % 3. 3%

12 CCBC Math 08 Translating and Solving Percent Problems Section pages 7.2 Translating and Solving Percent Problems In this section we will study how to solve percent problems. Percent problems usually have a phrase in the question in one of the following forms: Percent of Whole is Part Part is Percent of Whole We will solve these types of problems by translating these questions into an algebraic equation as we did in Section 5.7. We will look for key words and use them to create an equation which we solve. The two most important key words for this section are is and of: is means equal (=), and of means multiply (). Also, the word what is an indicator of an unknown quantity, which we will represent with a variable. Then, we will use solving methods that we first examined in Chapter 5. PARTS OF A PERCENT PROBLEM PERCENT WHOLE PART Rate or portion of 00 The total amount The portion of the whole The % symbol follows the number P will be used as the variable for Percent Follows the word of W will be used as the variable for Whole Will be found on the other side of the word is T will be used as the variable for ParT Note: When solving percent problems algebraically, you will need to change the percent to its decimal form. Recall from the last section that you change a percent to a decimal by dividing by 00. If the percent is the missing element from the problem, the algebraic answer to the equation that is solved will be in decimal form. Thus, you will need to change the decimal form into a percent in your final answer. Recall from the last section that you change a decimal to a percent by multiplying by 00%. In the next few examples, the unknown value will be the percent. When we solve the equation, the value will be given as a decimal and we will need to change it to a percent for our final answer. 478

13 CCBC Math 08 Translating and Solving Percent Problems Section pages Example : What percent of 20 is 5? We need to determine the Percent. Use P to represent the percent. Statement: What percent of 20 is 5? Equation: P 20 = 5 Next, solve this equation for P. P20 5 P P 0.25 P % 25% Divide both sides by 20. This is the decimal form. Change to a percent by multiplying by 00%. Practice : What percent of 72 is 36? Answer: 50% Watch It: Example 2: What percent of 320 is 64? We need to determine the Percent. Use P to represent the percent. Statement: What percent of 320 is 64? Equation: P 320 = 64 Next, solve this equation for P. P P P 0.2 P % 20% Divide both sides by 320. This is the decimal form. Change to a percent by multiplying by 00%. Practice 2: What percent of 200 is 50? Answer: 25% Watch It: 479

14 CCBC Math 08 Translating and Solving Percent Problems Section pages Example 3: 20 is what percent of 80? We need to determine the Percent. Use P to represent the percent. Statement: 20 is what percent of 80 Equation: 20 = P 80 Next, solve this equation for P. 20 P80 20 P P P.500% 50% Divide both sides by 80. This is the decimal form. Change to a percent by multiplying by 00%. Practice 3: 42 is what percent of 68? Answer: 25% Watch It: Example 4: What percent of 96 is 80? We need to find the Percent. Use P to represent the percent. Statement: What percent of 96 is 80? Equation: P 96 = 80 Next, solve this equation for P. P96 80 P96 80 Divide both sides by The decimal form of this fraction gives a repeating decimal. Therefore, we simplify this fraction and then change the fraction into a percent P Factor the numerator and denominator P Divide out common factors % P Change to a percent by multiplying by 00% P % Simplify by dividing out common factors P 83 % This is the percent as a mixed number

15 CCBC Math 08 Translating and Solving Percent Problems Section pages Practice 4: What percent of 20 is 90? Answer: 75% Watch It: In the next few problems the unknown value will be the part. We must remember to change the percent value that is given to a decimal before we solve the equation. Example 5: 8% of 220 is what number? We need to determine the ParT. Use T to represent the part. Statement: 8% of 220 is what? Equation: = T Next, solve this equation for T T 39.6 T Multiply to solve. Practice 5: 5% of 60 is what number? Answer: 9 Watch It: Example 6: What is 20% of 35? We need to determine the ParT. Use T to represent the part. Statement: What is 20% of 35? Equation: T = Next, solve this equation for T. T T Multiply to solve. Practice 6: What is 30% of 90? Answer: 27 Watch It: 48

16 CCBC Math 08 Translating and Solving Percent Problems Section pages Example 7: What is 300% of 7? We need to determine the ParT. Use T to represent the part. Statement: What is 300% of 7? Equation: T = 3 7 Next, solve this equation for T. T T 37 5 Multiply to solve. Practice 7: What is 45% of 60? Answer: 87 Watch It: Example 8: What is 5 % 2 of 248? When given the percent we must convert it to a decimal. 5 % % We need to determine the ParT. Use T to represent the part. Statement: What is 5 % 2 of 248? Equation: T = Next, solve this equation for T. T T Multiply to solve. Practice 8: Watch It: What is 3 % 5 of 325? Answer:

17 CCBC Math 08 Translating and Solving Percent Problems Section pages In the next few problems the unknown value will be the whole. We must remember to change the percent value that is given to a decimal before we solve the equation. Example 9: 30 is 5% of what? We need to determine the Whole. Use W to represent the whole. Statement: 30 is 5% of What? Equation: 30 = 0.5 W Next, solve this equation for W W W W Divide both sides by 0.5. Practice 9: 60 is 5% of what number? Answer: 200 Watch It: Example 0: 80% of what number is 8? We need to determine the Whole. Use W to represent the whole. Statement: 80% of what is 8 Equation:.8 W = 8 Next, solve this equation for W..8W 8.8W W 45 Divide both sides by.8. Practice 0: 200% of what number is 90? Answer: 45 Watch It: 483

18 CCBC Math 08 Translating and Solving Percent Problems Section pages Example : 45 is 30% of what? We need to determine the Whole. Use W to represent the whole. Statement: 45 is 30% of What? Equation: 45 = 0.3 W Next, solve this equation for W W W W Divide both sides by 0.3. Practice : 32 is 20% of what number? Answer: 60 Watch It: We can also solve percent problems as we did proportion ratio problems in Chapter 6. With this method you will not need to convert percents to decimals or decimals to percents. Every problem is set up the same no matter what the unknown value is. SOLVING PERCENT PROBLEMS USING PROPORTIONS Percent 00 Part Whole Example 2: What percent of 20 is 5? We need to determine the Percent. Whole = 20 Part = 5 Fill in the proportion with the known information and solve for the unknown. P Cross multiply. 20P P P 25% Divide. The answer is in percent form. Practice 2: What percent of 72 is 36? Answer: 50% Watch It: 484

19 CCBC Math 08 Translating and Solving Percent Problems Section pages Example 3: What is 8% of 220? We need to determine the ParT. Whole = 220 Percent = 8 Fill in the proportion with the known information and solve for the unknown. 8 T T T T 39.6 Cross multiply. Divide. Practice 3: 5% of 60 is what number? Answer: 9 Watch It: Example 4: 30 is 5% of what? We need to determine the Whole. Part = 30 Percent = 5% Fill in the proportion with the known information and solve for the unknown W 5W W W 200 Cross multiply. Divide. Practice 4: 60 is 5% of what number? Answer: 200 Watch It: Watch All: 485

20 CCBC Math 08 Translating and Solving Percent Problems Section pages 7.2 Translating and Solving Percent Problems Exercises. What is 30% of 300? 2. What is 8.9% of 0? 3. 20% of 4000 is what? 4. 5% of 90 is what? is 25% of what? is 80% of what? 7. What percent of 25 is 25? 8. What percent of 00 is 50? is what percent of 400? is what percent of 60?. What is 25% of 225? 2. What is 68% of 80? 3. What is 33 % of 25? 4. 25% of 200 is what? % of 86.5 is what? 6. 5 is 25% of what? is 80% of what? is 20% of what? is % of what? is.5% of what? 2. 0 is what percent of 50? is what percent of 00? is what percent of 260? is what percent of 20? is what percent of 20,000? 486

21 CCBC Math 08 Translating and Solving Percent Problems Section pages 7.2 Translating and Solving Percent Problems Exercise Answers % 8. 50% 9. 0% 0. 40% or or or , , % 22. 2% % % or 2 % 25..0% or % 487

22 CHAPTER 7 Mid-Chapter Review. Convert to a fraction: 25% 2. Convert to a fraction: 93% 3. Convert to a fraction: 4. Convert to a percent: 5. Convert to a percent: % Convert to a percent: Convert to a percent: Convert to a decimal: 5.9% 9. Convert to a decimal: 8.5% 0. Convert to a decimal: 6%. What percent of 300 is 90? 2. What percent of 62 is 58.9? 3. What is 25% of 200? 4. What is 8% of 420? is 20% of what? is 88% of what? 7. What percent of 45 is 9? 8. 99% of 90 is what? is what percent of 425? is what percent of 60? 2. 47% of 70 is what number? 22. What is 25% of 60? % of what number is 550? 24. What is 83 % 3 of 96? 25. What percent of 320 is 64? 26. What percent of 200 is 230? 27. What percent of 20 is 2? % of what number is 642? % of what number is 96? 30. What is 82% of 00? 488

23 CCBC Math 08 Mid-Chapter 7 Review Sections 7. to 7.2 M i d - C h a p t e r 7 R e v i e w A n s w e r s % % % % % 2. 95% % % % % 26. 5% %

24 CCBC Math 08 Circle Graphs Section pages 7.3 Circle Graphs Statistical information is often presented in table form or in the form of a line graph, bar graph, or circle graph. These graphs are a way of presenting or visualizing statistical data for easy comparisons. For circle graphs, also known as pie charts, the information is presented in the form of a circle divided into sectors, each sector representing a part of a whole or a percent. The total of items in the pie chart must always come to 00%. Example : A class of 25 students took a mathematics test. The results are presented in the circle graph below. Test Results A, 20% B, 32% C, 40% F, 8% Use the circle graph above to answer the following questions. a. How many students earned a C grade? Answer: 40% of the students earned a C. Therefore, we calculate 40% of the 25 students who took the test. 40% of 25 = = 0 students. b. How many students earned an A or a B on the test? Answer: 20% of the students earned an A, and 32% earned a B. 20% + 32% = 52% 52% of 25 = = 3 students. c. What percent of the students failed the test? How many failed the test? Answer: 8% of the students failed test, earning an F. 8% of 25 = = 2. Two students failed the test. d. How many more students earned a C grade than earned a B grade? Answer: 40% of 25 = = 0 C students 32% of 25 = = 8 B students 0 8 = 2 more students earned a C than a B. e. What percent of the entire class passed the test? Answer: Subtract the 8% who failed from 00%. 00% 8% = 92% passed the test. Or, 20% + 32% + 40% = 92% passed the test. 490

25 CCBC Math 08 Circle Graphs Section pages Practice : A class of 40 students took a mathematics test. The results are presented in the circle graph below. Test Results A, 5% B, 30% C, 45% F, 0% Use the circle graph above to answer the following questions. a. How many students earned a C grade? b. How many students earned an A or a B on the test? c. What percent of the students failed the test? How many failed the test? d. How many more students earned a C grade than earned a B grade? e. What percent of the entire class passed the test? How many students passed? Answers: a. 8 students, b. 8 students, c. 0% or 4 students, d.6 students, e. 90% or 36 students Watch It: 49

26 CCBC Math 08 Circle Graphs Section pages Example 2: A student researched expenses to live on campus at a four year university in Maryland to compare costs to a community college expense for one semester. Costs per Term to stay on campus 4-year University in Maryland food, $2,24.00 tuition, $2, room, $2, parking, $74.00 Use the circle graph above to answer the following questions. a. What percent of the costs shown is for tuition? Round to the nearest tenth of a percent. Answer: Determine the total of the costs provided = 873 Tuition is what percent of total cost 2830 = P(873) = P = 34.6% (Rounded to the nearest tenth of a percent)) b. If the student has financial aid that covers 45% of their total cost for the semester, how much will the student need to pay? Answer: Financial aid is 45% of total cost Financial aid = (0.45)(873) Financial aid = Student pays total cost financial aid = = $4,

27 CCBC Math 08 Circle Graphs Section pages Practice 2: A student research the cost to attend the Community College of Baltimore County for one semester. The following chart shows the breakdown of the costs. Fees $20 Cost per Term at CCBC Books $400 Tuition $650 Use the circle graph above to answer the following questions. a. What percent of the costs provided is for tuition? Round your answer to the nearest tenth of a percent. b. If the student has financial aid that covers 45% of their total cost for the semester, how much will the student need to pay? Answers: a. 76.0%, b. $,93.50 y Watch It: 493

28 CCBC Math 08 Circle Graphs Section pages Example 3: There were 205,87 crimes in the State of Maryland in 200 broken down by category in the circle graph below. motor vehicle thefts, 8, Crime Statistics - Maryland homicides, 426 robberies,,053 rapes,,228 aggravated assaults, 8,898 larceny/thefts, 8,853 breaking & entering, 36,700 Use the circle graph above to answer the following questions. a. What category accounts for the largest number of crimes? What percent does it represent? Round to the nearest tenth of a percent. Answer: Larceny/theft is the largest category. Larceny is what percent of crimes? 8853 = P (20587) = P 57.9% of the crime is larceny/thefts. b. What percent of the crimes are rapes? Round to the nearest tenth of a percent. Answer: P (20587) = 228 P = P = 0.6% of the crimes are rapes. 494

29 CCBC Math 08 Circle Graphs Section pages Practice 3: There were 37,97 crimes in the City of Baltimore, Maryland in 200 broken down by category in the circle graph below. 200 Crime Statistics - Baltimore Maryland Murder, 223 Rape, 265 Arson, 32 Robbery, 3,336 Larceny and Theft, 6,298 Motor Vehicle Theft, 4,409 Aggravated Assault, 5,492 Burglary, 7,573 Use the circle graph above to answer the following questions. Make certain that answers are rounded to the nearest whole number or the nearest tenth of a percent for the final answer. a. What category accounts for the most amount of crimes? What percent does it represent? b. What percent of the crimes are burglaries? Answers: a. larceny and theft 43.0%, b. 20.0% Watch It: Watch All: 495

30 CCBC Math 08 Circle Graphs Section pages 7.3 Circle Graphs Exercises Questions 4 Ms. Blossom is a school teacher. She earns $3,500 per month for her hard work as a teacher. Her monthly expenses are represented by the circle graph below. Savings, 0% Budget Utilities, 2% Food, 25% Transportation, 8% Rent, 45%. How much does Ms. Blossom spend on rent? 2. How much does she save every month? 3. What percent of her monthly income does Ms. Blossom spend on utilities and transportation? What is the amount? 4. What is Ms. Blossom s total expenditure other than rent? 496

31 CCBC Math 08 Circle Graphs Section pages Questions 5 9 A company called Compassion International employs people of many different ages, from twenty-year-olds to seventy-year-olds. The circle graph below shows the number of employees in each age group. 70 yr olds, 6 Ages 20 yr olds, 2 60 yr olds, 5 50 yr olds, 0 30 yr olds, yr olds, 8 5. Find the total number of employees at this company. 6. What age group has the greatest number of employees? 7. What percent of employees are in their sixties or seventies? How many are they? 8. How many employees are under fifty years of age? 9. What age group makes up 6% of the total number of employees? 497

32 CCBC Math 08 Circle Graphs Section pages Questions 0 6 A certain country has a population of 24 million. This country is divided into six ethnic groups: Akan, European, Ewe, Ga, Gurma, and Moshi. This information is presented in the circle graph below. Ethnic Groups Akan, 44% Moshi, 6% Ga, 8% Ewe, 3% Gurma, 3% European, 6% 0. Which is the predominant ethnic group?. What percent is the largest ethnic group? 2. How many people are in the largest ethnic group? 3. Which ethnic groups have the same percentage of people? 4. How many people are in each group of people with the same percent? 5. The Ewe group is larger than the Ga group by what percent? 6. How many more people belong to the Ewe group than the Ga group? 498

33 CCBC Math 08 Circle Graphs Section pages Questions 7 20 A survey asked 60 mathematics students what they thought were the most pertinent factors that contributed to their success in mathematics education. Among the factors considered were the role of the instructor, taking responsibility for their homework assignments, study skills, being tutored on a one-on-one basis, and study groups. The results of the survey are presented in the circle graph below. Math Students Study Groups, 0% Study Skills, 5% Tutoring, 5% Doing Homework, 45% Instructor, 25% 7. What did students consider most important in successfully learning math? 8. How many students thought instructors play a major role in getting them to be successful in mathematics? 9. What percent of the students believed that study skills are very important in their mathematics education? 20. How many students chose study skills? 499

34 CCBC Math 08 Circle Graphs Section pages Questions 2 25 Crime Statistics for 200 for Baltimore County were recorded and are a matter of public record. The statistics are presented in the circle graph below. 200 Crime Statistics - Balto. County motor vehicle thefts,,880 homicides, 20 breaking & entering, 4,9 aggravated assaults, 2,862 rapes, 26 larceny/thefts, 8,29 robberies,,34 2. What was the total number of crimes in Baltimore County for 200? 22. What percentage of the crimes were larceny/thefts? Is this consistent with the percentage of larceny/thefts for the State of Maryland (see Example 3)? Round to the nearest tenth of a percent if necessary. 23. What is the total for breaking & entering, aggravated assaults, and rapes? 24. What percentage of the total crime is accounted for in those listed in #23? Round to the nearest tenth of a percent, if necessary. 25. What percent of the crimes are robberies? Round to the nearest tenth of a percent, if necessary. 500

35 CCBC Math 08 Circle Graphs Section pages 7.3 Circle Graph Exercise Answers. $, $ %, $ $, year-olds 7. 28%, year-olds 0. Akan. 44% 2. 0,560, European and Moshi 4. 3,840, % 6.,200, Taking responsibility in doing homework assignments % , %, this is larger than the State of Maryland s percentage of 57.9% , % % 50

36 CCBC Math 08 Financial Applications of Percents Section pages 7.4 Financial Applications of Percents Interest To understand the financial applications in this section dealing with interest, we must first define some terms. Then we will learn how to using these ideas to solve problems. Principal: The principal is the original amount of money borrowed on which interest is calculated. Interest: Interest is the fee charged by a lender to borrow money. Interest rate: An interest rate is the rate, stated as a percent, charged for the amount borrowed per period of time, usually one year. Interest can be divided into two categories: simple interest and compound interest. Simple interest: Simple interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods. Simple interest is normally used for a single period of less than a year, such as 30 or 60 days. Compound interest: Compound interest arises when interest is added to the principal; the interest that has been added also earns interest. We need a complicated formula to calculate compound interest, so we will only study simple interest problems in this section. To show the difference between these two types of interest, suppose you deposit $500 each into two accounts, one with simple interest and one with compound interest. The simple interest will be calculated on the $500 deposited in the account every time it is calculated. However the compound interest will be calculated on the account balance (of the deposit amount plus any interest already earned). Simple interest account Compound interest account Interest Balance Interest Balance $5.00 $55.00 $5.00 $55.00 $5.00 $ $5.45 $ $5.00 $ $5.9 $ $5.00 $ $6.39 $ $5.00 $ $6.88 $ How do we calculate simple interest on a loan (or a deposit)? The solution is in the formula I = Prt, where I is the interest, P is the principal (amount loaned or deposited), r is the interest rate, and t is the time measured in years. In the simple interest formula, you can solve for any of the four quantities (I, p, r, or t) if the other three quantities are known. 502

37 CCBC Math 08 Financial Applications of Percents Section pages CALCULATING SIMPLE INTEREST I = Prt I = Interest P = Principal r = rate t = time Note: The rate is given in percent form and must be changed to decimal form. Example : What is the simple interest on a loan of $6,000 for one year if the interest rate is 8.25%? Loan of $6,000 = P one year = t interest rate of 8.25% = = r I = Prt I = (6000)(.0825)() I = 495 Thus the interest is $495. Practice : What is the simple interest on a loan of $30,000 for one year if the interest rate is 2.25%? Watch It: Answer: $675 Example 2: If Barry borrows $2,000 for 3 years at a 3% annual simple interest rate, how much interest will Barry pay? Borrows $2,000 = P for 3 years = t at a 3% annual simple interest rate = 0.3 = r I = Prt I = (2000)(.3)(3) I = 780 Thus the interest is $780. To pay off the loan, Barry would have to pay back the $2,000 that he borrowed plus the additional $780 in interest: $2,000 + $780 = $2,780. Practice 2: If Sarah borrows $,000 for 4 years at a 3% annual simple interest rate, how much interest will Sarah pay? Watch It: Answer: $

38 CCBC Math 08 Financial Applications of Percents Section pages Example 3: What is the interest gained in a $,500 investment at an 8% annual simple interest rate for 9 months? Notice that the time is given in months, and since we are given an annual interest rate, we must convert the 9 months to years. Since there are 2 months in a year, 9 months represents 9 of a year. So we divide 9 2 = 0.75 = t 2 I = Prt I = (500)(.08)(.75) I = 90 The interest is $90. The value of the investment at the end of 9 months would be the original investment amount plus the interest earned: $,500 + $90 = $,590. Practice 3: What is the interest earned on an account with an initial investment of $250 at a 4.5% annual simple interest rate for 6 months? Watch It: Answer: $28.3 Example 4: What simple annual rate is necessary for an investment to grow from $200 to $500 in two years? P = $200 t = two years Since the investment is to grow from $200 to $500, there is an increase based on the interest added to the account. Interest = = 300 = I I P r t r r 0.75 r % r 75% r Divide both sides by 400. Convert to a percent. Thus the interest rate is 75%. 504

39 CCBC Math 08 Financial Applications of Percents Section pages Practice 4: Sarah needs $5,000 for college in 6 years. If she invests $5000 now, what simple annual rate is necessary for the investment to grow to $5,000 in 6 years? Watch It: Answer: 33.3% Example 5: How much time does it take an investment of $3,000 to triple in value if the annual simple interest rate is 20%? P = $3,000 r = 20% = 0.20 If the investment triples from $3000, the final balance is 3 times 3000 = $9000. The difference from investment to ending balance is interest. I = $9,000 $3,000 = $6,000. I P r t t t 0 t It will take ten years for the investment to triple in value. Practice 5: If the annual simple interest rate at a bank is 0%, how many years does Maria need to leave her $4000 in the bank so that she doubles the value? Watch It: Answer: 0 years 505

40 CCBC Math 08 Financial Applications of Percents Section pages Total Cost: If interest is charged, the total cost is the principal and the interest added together. We use the following variables and formula to compute the total cost in simple interest problems. CALCULATING TOTAL COST WITH SIMPLE INTEREST A = P + I or A= P + Prt A = Total Cost I = Interest P = Principal r = rate t = time Example 6: A television costs $900. The store gives Ray credit calculated with simple interest at 30% over 2 years. How much does Ray pay for his television? The television costs $900, so P = 900 The interest rate is 30%, so r =.30 The length of the loan is two years, so t = 2. Substitute these values into the formula A = P + Prt. A = P + Prt A = (900)(.30)(2) A = A = 440 So, the total amount that Ray would pay is $,440. Practice 6: Leroy buys a $200 computer. The store gives him credit using simple interest at 20% over 2 years. How much does Leroy pay for his computer? Watch It: Answer: $

41 CCBC Math 08 Financial Applications of Percents Section pages Credit Cards and Simple Interest Credit cards are often used instead of cash to purchase goods and services. The material that follows explains some of the terminology found in credit card transactions and use agreements. Annual Percentage Rate (APR): The Annual Percentage Rate is the annual interest rate associated with the credit card. Credit Limit: The Credit limit is the maximum amount the cardholder is allowed to charge on a credit card. Transaction Fee: Transaction Fees are fees charged to the customer for late payments, exceeding the credit limit, or other non-interest fees. Period: The period is the length of time used to calculate the balance (usually 365 days). Periodic Rate: The Periodic Rate is the Annual Percentage Rate divided by the period. Purchases: Purchases are charges on a credit card from purchasing goods and services. Cash Advances: Cash advances are charges on a credit card for cash received. Balance Transfers: Balance transfers are charges on a credit card from transfers from other accounts. Finance Charge: The finance charge is the total amount to be paid in transaction fees and interest. Grace Period: The grace period is the allotted period of time by which a credit card balance must be paid in order to avoid finance charges. 507

42 CCBC Math 08 Financial Applications of Percents Section pages How is the interest on a credit card calculated? Most widely used credit cards (Visa, MasterCard, American Express, Discover, etc.) use the Average Daily Balance approach to calculate the amount of interest on purchases during a period of time. In this section, we will explain how to calculate the Average Daily Balance. Consider the following credit card statement. Tingling National Bank Credit Card Covering the Period April 5 through May 4 APR = 22.9% Beginning Balance: $ DATE TRANSACTION AMOUNT 4/8 Pleasant Pastures Gas $0.00 4/23 Blue Mollusk Seafood Restaurant $ /0 Payment Received $ /0 New Army Clothing Store $65.00 What is the amount of interest for this period? The first thing we must find is the Average Daily Balance. From April 5 through April 7, the balance is $ (3 days) From April 8 through April 22, the balance is $ $0.00 = $ (5 days) From April 23 through April 30, the balance is $ $35.00 = $ (8 days) From May through May 9, the balance is $ $00.00 = $85.00 (9 days) From May 0 through May 4, the balance is $ $65.00 = $ (5 days) To find the Average Daily Balance, we multiply each balance by the respective number of days, add those products together, and divide by 30 (since the statement covers 30 days.) Avg. Daily Bal. = ($ $ $ $ $250 5) 30 = ( ) 30 = (765) 30 = $ The amount of interest is obtained using the Simple Interest Equation I = Prt. The principal is the average daily balance, the rate is the annual percentage rate, and the time is in years (the number of days divided by 365). I = Prt I = (0.229) (30/365) I = 4.50 Therefore, the interest for this account for this period is $

43 CCBC Math 08 Financial Applications of Percents Section pages Example 7: Let's look at another statement. Weripuoff National Bank Credit Card Covering the Period July 23 through August 24 APR = 9.8% Beginning Balance: $74.76 Average Daily Balance: $249. DATE TRANSACTION AMOUNT 7/25 McReynold's Restaurant $.45 7/28 Washington Merlins Basketball Tickets $ /3 Payment Received $ /2 High's Black Swamp Theatres $3.75 8/20 WEW Cooked Wrestling Tickets $50.00 What is the amount of interest for this period? The amount of interest is: I = Prt I = 249. (0.98) (33/365) I = 4.46 Therefore the interest for this account for this period is $4.46. Practice 7: Use the following credit card statement to determine the amount of interest for the given statement period. Statement from the IGOTU Bank Credit Card Statement Period: December 5 thru Jan 4 APR=23% Beginning Balance: $25.00 Average Daily Balance: $25.48 DATE TRANSACTION AMOUNT 2/6 Yummy Food Store $ /9 Holly Meadows Farms $ /30 Payment Received $ /05 Never Say Never Boutique $00.00 Watch It: Answer: $

44 CCBC Math 08 Financial Applications of Percents Section pages These two examples show a reasonable amount of interest for one month. Of course this is only one month's worth of interest and many people have higher balances than the balances in the two previous examples. Remember, the Average Daily Balance approach is used to determine the interest on purchases. Interest obtained from the use of cash advances and balance transfers may be obtained another way, usually with an entirely different Annual Percentage Rate. In fact, most cash advances use the Compound Interest Formula (compounded daily). Watch All: 50

45 CCBC Math 08 Financial Applications of Percents Section pages 7.4 Financial Applications of Percents Exercises. A bank charges 6% simple interest. How much must you pay on a loan of $2000 for three years? 2. A bank charges 5% simple interest. How much interest must you pay on a loan of $700 for 2 years? 3. A bank charges 8.75% simple interest. How much interest must you pay on a loan of $4000 for one year? 4. A bank charges 7.5% simple interest. How much interest must you pay on a loan of $8000 for one year? 5. What is the interest on a $650 simple interest loan with a 0% rate for 8 months? 6. What is the total amount in interest made on a $000 simple interest investment if the rate is 8% and the term is 3 years? 7. How much would Jerry pay in total if he borrows $2000 at 5% simple interest for 5 years? 8. A computer costs $00. If a company is charging 30% simple interest for 3 years, how much is paid in total for the computer? 9. How much interest is earned if you invest $25,000 for 3 months at 5.25% simple interest rate? 0. How much interest is earned if you invest $25,000 for 3 years at 5.25% simple interest rate?. What is the account balance after 0 months if you deposit $7000 into a simple interest account earning 2%? 2. What interest rate is needed in order to double an initial investment of $2000 for five years? 3. What amount is the principal in a savings account if $3000 is earned in interest after 6 years at 4% simple interest rate? 4. Approximately how long will it take for an initial investment of $825 to become $000 if it is invested at a 3.5% simple interest rate? 5. How many months does it take to turn $4800 into $6000 if it is invested at a 30% simple interest rate? 5

46 CCBC Math 08 Financial Applications of Percents Section pages 6. Find the amount of interest from purchases for the credit card account below. Tingling National Bank Credit Card Covering the Period January through February 0 APR = 5.5% Beginning Balance: $ Average Daily Balance: $ DATE TRANSACTION AMOUNT /7 Payment $80.00 /20 Purchase $ /0 Purchase $ /04 Purchase $ Find the amount of interest from purchases for the credit card account below. Tanglang National Bank Credit Card Covering the Period November 7 through December 7 APR = 6.9% Beginning Balance: $23.24 Average Daily Balance: $30.36 DATE TRANSACTION AMOUNT /25 Purchase $65.2 /30 Purchase $.48 2/6 Purchase $ /3 Payment $

47 CCBC Math 08 Financial Applications of Percents Section pages 8. Find the amount of interest from purchases for the credit card account below. Tonglong National Bank Credit Card Covering the Period October 0 through November 9 APR = 9.9% Beginning Balance: $569.8 Average Daily Balance: $ DATE TRANSACTION AMOUNT 0/9 Purchase $ /22 Purchase $ /3 Purchase $65.50 /2 Payment $20.00 /6 Purchase $ Find the amount of interest from purchases for the credit card account below. Tinglung National Bank Credit Card Covering the Period April through April 25 APR = 2.0% Beginning Balance: $0.00 Average Daily Balance: $82.00 DATE TRANSACTION AMOUNT 4/ Purchase $ /4 Purchase $ /20 Credit $

48 CCBC Math 08 Financial Applications of Percents Section pages 20. Find the amount of interest from purchases for the credit card account below. Tingbling National Bank Credit Card Covering the Period January through February APR = 9.9% Beginning Balance: $, Average Daily Balance: $ DATE TRANSACTION AMOUNT 0/02 Purchase $ /04 Payment $ /05 Purchase $ /07 Purchase $22.2 0/20 Payment $ Makenzie did not use her credit card this past month, but she carried a balance of $25. She just received her statement covering the period March 5 through April 4, and her average daily balance is the $25 carried over from last month. If the APR is 0.99%, how much interest was added to the balance? 22. Stewart purchased items for this semester s classes with his new credit card on August 0 th. The total was $4500 and it was his first charge on the credit card. His statement comes for the period August 0 through September 0 so his average daily balance is $4500. With an introductory APR of 8.99%, what is the interest amount? 23. Samuel only uses his credit card to purchase presents for family and friends. Recently he purchased quite a few gifts. He carried an average daily balance of $865 into this statement which runs from July 8 through August 7. With APR of 5.99%, what is the new balance, including the new interest added? 24. William has a balance of $2734 on his credit card that has an APR of 24%. This statement covers a period of 3 days. What is the interest for the period? 25. In problem 24, suppose William decides to only make the minimum payment of $08, what would the balance be for the next period? 54

49 CCBC Math 08 Financial Applications of Percents Section pages 7.4 Financial Applications of Percents Exercise Answers. The interest is $ The interest is $ The interest is $ The interest is $ The interest is $ The interest is $ The total amount paid is $3, The total amount paid for the computer was $2, The interest is $ The interest is $ The account balance is $ The interest rate is 20%. 3. $2,500 was deposited. 4. Just over 6 years 5. 0 months 6. The interest for the period is $ The interest for the period is $ The interest for the period is $ The interest for the period is $ The interest for the period is $ The interest for the period is $ The interest for the period is $ The interest for the period is $25.33 and the new balance is $ The interest for the period is $ The new balance is $ (Previous balance $ interest payment 08) 55

50 CCBC Math 08 Application Problems Section pages 7.5 Applications The percent problems solved in the previous section were the most basic type. Real world applications provide a context and meaning to the values that you are given and the value that you need to find. In these problems, there will still be three values where two are given and one needs to be found. Sales tax, commission rates, and tip amounts are all examples of real world applications of percent problems. While the meaning may vary from problem to problem, in each case the problem can be simplified to the basic problems covered in the previous sections. Example : Julia is going to buy a shirt for $2.00. How much tax will she pay for the shirt if the sales tax rate is 6%? Statement: Tax is 6% of Shirt Price Equation: T = Next, solve this equation for T. T Multiply to solve. T Tax is $0.72 Practice : Julio is going to buy a shirt for $4.00. How much tax will he pay for the item if the tax rate is 7%? Watch It: Answer: $0.98 Example 2: The television that Miguel wants to purchase costs $449. How much tax will he pay if the tax rate is 8%? How much is the total cost? Statement: Tax is 8% of Television Price Equation: T = Next, solve this equation for T. T Multiply to solve. T Tax is $ Total cost = television cost + tax Total cost = $449 + $35.92 Total cost = $ Practice 2: The mixer that Patti wants to purchase costs $229. How much sales tax will she pay if the sales tax rate is 6%? What will the final cost of the mixer be? Watch It: Answer: tax: $3.74 final cost: $

51 CCBC Math 08 Application Problems Section pages Example 3: Nathaniel wants to buy a science kit with a sticker price of $ After a 6% sales tax, what is the final cost? Statement: Tax is 6% of Kit Price Equation: T = Next, solve this equation for T. T Multiply to solve. T Tax is $2.25. Total cost = $ $2.25 Total cost = $39.75 Practice 3: Catherine wants to buy a couch with a sticker price of $535. After a 6% sales tax, what is the final cost? Watch It: Answer: $567.0 Example 4: Jada found a pair of jeans that she would like to purchase. She has $65 to spend. If the price is $62.75 and the tax rate is 8%, does Jada have enough money? Explain your answer. Statement: Tax is 8% of Jeans Price Equation: T = Next, solve this equation for T. T Multiply to solve. T 5.02 Tax is $5.02. Total cost = $ $5.02 Total cost = $67.77 Jada does not have enough money. The total cost for the jeans, including tax is $67.77, which is more than the $65 she has. Practice 4: Silas found a cell phone that he would like to purchase. He has $84 to spend. If the price is $79 and the tax rate is 6%, does Silas have enough money? Explain your answer. Watch It: Answer: yes, needs $

52 CCBC Math 08 Application Problems Section pages Example 5: Samuel bought a computer for $450 and paid $0.50 in sales tax. What is the tax rate? Statement: Tax is Percent of Computer Price Equation: 0.50 = p 450 Next, solve this equation for p p 450 Divide both sides by 450 to solve p Convert to a percent. Tax rate is 7% Practice 5: Sally bought a pair of earrings for $25 and paid $8.75 in sales tax. What is the tax rate for the state where she bought the item? Watch It: Answer: 7% Example 6: The restaurant bill was $02. Ingrid paid the bill and left a tip of $8.36 for the waiter. What percent of the bill was the tip Ingrid left? Statement: Tip Is Percent of Bill Equation: 8.36 = p 02 Next, solve this equation for p p 02 Divide both sides by 02 to solve. 0.8 p Convert to a percent. 8% of the bill was left for the tip Practice 6: The cost of a salon visit was $55 (not including tip). An $ tip was given to the stylist. What percent of the cost was the tip? Watch It: Answer: 20% 58

53 CCBC Math 08 Application Problems Section pages Example 7: Bridget applied to take classes at the local community college. Due to a program with her employer, she only had to pay 60% of the tuition. If a threecredit class cost $,0, how much did she have to pay? Statement: Payment is 60% of Tuition Equation: T = Next, solve this equation for T. T 0.60(0) Multiply to solve. T Bridget paid $ for a three-credit class. Practice 7: Bridget applied to take classes at the local community college. Due to a program with her employer, she only had to pay 75% of the tuition. If a threecredit course cost $,0, how much did she have to pay? Watch It: Answer: $ Example 8: Temperance is a real estate agent who earns a 3% commission on each house that she sells. If she sells a house for $225,000 what will her commission be? Statement: Commission is 3% of House Sales Equation: T = Next, solve this equation for T. T 0.03(225000) Multiply to solve. T 6750 Temperance earns $6,750 for her commission on this house sale. Practice 8: Delia sold a house to one of her clients for $25,000. Her commission rate is 5% of the selling price of the house. How much commission did she earn? Watch It: Answer: $6,250 59

54 CCBC Math 08 Application Problems Section pages Example 9: Seeley earns a commission of 2% on his sales. He received a commission check of $7,000 for the month of March. How much did he sell in March? Statement: Commission is 2% of Sales Equation: 7000 = 0.02 W Next, solve this equation for W W Divide to solve W Seeley s sales total for March was $350,000. Practice 9: Jamie earns a commission of 4% on his sales. He received a commission check of $,440 for the month of April. How much did he sell in April? Watch It: Answer: $36,000 Example 0: Angela sells cars for a living. In one week, she sold $37,500 worth of cars and earned a commission of $,875. What is her commission rate? Statement: Commission is Percent of Sales Equation: 875 = P Next, solve this equation for p. 875 p(37500) Divide to solve p Convert to a percent. Angela s commission rate is 5%. Practice 0: If Josh sold a house for $235,000 last month and got a commission check for $4,00, what was his commission rate? Watch It: Answer: 6% 520

55 CCBC Math 08 Application Problems Section pages Example : In the state of Maryland, the sales tax rate had been 5% until January, 2008 when it was raised to 6%. Some citizens felt that one percent was not a big deal. Brayden s car was involved in a wreck and he needed to get a new one. He did his research and finally settled on one that would cost $9900. Suppose he tried to get his paperwork done before the end of 2007, what would his tax be? Suppose the bank took some time to complete the loan work and it did not go through until January 5, 2008, what would he pay in taxes? 2007: Tax is 5% Statement: Tax is 5% Of Price Equation: T = Next, solve this equation for T. T 0.05(9900) Multiply to solve. T 495 Tax = $ : Tax is 6% of price (9900) Statement: Tax is 6% Of Price Equation: T = Next, solve this equation for T. T 0.06(9900) Multiply to solve. T 594 Tax = $594 Waiting until the New Year (and the new tax rate), costs $594 $495 = $99 more in taxes for Brayden s car. Practice : In the state of Maryland, the sales tax rate had been 5% until January, 2008 when it was raised to 6%. Some citizens felt that one percent was not a big deal. The Greens wanted to purchase a big screen television to watch the Super Bowl that year. The television cost $3000. How much would they have saved if they bought it before January, 2008? Watch It: Answer: $30 52

56 CCBC Math 08 Application Problems Section pages Example 2: Little Timmy has been sick all week. On Monday his fever was 02.4 degrees. By Friday it had decreased 2.7%. What was his temperature on Friday? Round your answer to the nearest tenth. The first question we must answer is, What is 2.7% of 02.4 degrees? This will tell us how many degrees his fever has gone down. We can do this problem in either method that we learned in Section 7.2. We show this problem using the second method that was discussed. We need to determine the ParT. Whole = 02.4 Percent = 2.7% Fill in the proportion with the known information and solve for the unknown. % ParT 00 Whole 2.7 T T T T Cross multiply. Divide. Rounding to the nearest tenth of a degree, Timmy s fever decreased by 2.8 degrees. To determine what his temperature is now, we will subtract the number of degrees his fever went down from the original temperature = 99.6 Timmy s temperature is now 99.6 degrees. Practice 2: The temperature outside was 75 degrees in the afternoon. It dropped 44% during the day. What temperature was it in the evening? Round the answer to the nearest degree. Watch It: Answer: 42 degrees Example 3: Glynis earns $43,500 this year. Due to a promotion, she will get a 5% raise for next year. What is the dollar amount of her raise? What will her salary be next year? Raise = 5% of salary Raise = 0.05(43500) Raise = $275 New salary = present salary + raise New salary = New salary = $45,

57 CCBC Math 08 Application Problems Section pages Practice 3: Cassidy earns $54,300 this year. Due to a promotion, she will get a 3% raise for next year. What is the dollar amount of her raise? What will her salary be next year? Watch It: Answer: $629, $55,929 Example 4: Gretchen finds that there will be a 20% off sale at her favorite store. She decides to wait to buy the outfit she likes as it costs $82 until the sale. How much will the outfit cost on sale? Option A to solve: Sale is 20% off the original price. 20% of 82 = = $6.40 Sale saves Gretchen $6.40 so it is taken off the original price. Sale price = Sale price = $65.60 Option B to solve: Sale is 20% off, so Gretchen only pays 80% of price. 80% of 82 = = $65.60 Practice 4: Heather finds that there will be a 25% off sale at her favorite store. She decides to wait to buy the outfit she likes as it costs $76 until the sale. How much will the outfit cost on sale? Watch It: Answer: $57 Example 5: Carin purchased an item from the electronics store for $60 since it was 25% off. What was the original price of the item? The item was on sale for 25% off. Carin paid 75% of original price. 75% of original price is $ (n) = 60 n = 80 Original price was $80.00 Practice 5: April purchased a Raven s sweatshirt for $45. This was 30% off the original price. What was the original price of the sweatshirt? Watch It: Answer: $64.29 Watch all: 523

58 CCBC Math 08 Application Problems Section pages 7.5 Applications Exercises. Alberto wants to buy a remote control car with a sales price of $4.50. If the tax rate is 6%, how much will he pay in tax? 2. Benjamin bought camping supplies for an upcoming trip and spent $32. If the tax rate was 6%, what was the total cost of the camping items? 3. Casper bought new furniture with a price tag of $,250.00, but his total bill was $, What was the amount of tax? What was the tax rate? 4. Dave sells $56,000 worth of advertising in one month. If his commission rate is 8%, how much is his commission check? 5. Eliza hopes to earn a commission bonus of $4500. If her commission rate is 5%, how much does she have to sell in order to earn that bonus? 6. Fran sold $4,258,000 worth of real estate last year. Her commission earnings were $63,870. What was her commission rate? 7. The amount of tax paid on a new bed was $37.0 in a state where the tax rate is 7%. What was the price of the bed? 8. What is the commission made on a sale if the commission rate is 3% and the amount of sales is $52,345? 9. The sale price of a dining room set was $250. If the amount of tax paid was $93.75, what was the tax rate? 0. Tammy Faye sold $78,000 worth of Maybelline products last month. Her commission check was $7,20. What is the percent that she earns for commission?. For a hotel stay, the tax rate is much higher than the regular sales tax, often around 2%. If a hotel charges $525 for a two-night stay, how much tax will be charged? 2. With a budget of $95, LaToya wants to purchase a pair of jeans and a shirt. The jeans cost $45 and the shirt costs $40. If the tax rate is 7%, does she have enough money to purchase the outfit? 3. Michelle sells computers in order to earn extra money. Her employer pays a commission of %. If she wants to earn $650 in order to landscape her back yard, what amount of computer sales will meet this goal? 4. Daniel answered 2 problems correctly of the 25 problems on the test. What percent of the problems did Daniel get correct? 5. The national average cost for gasoline last week was $3.75. There was a 2% increase this week. What is the average cost this week? 6. If you get a 3% raise for the year, what is the amount of your raise and next year s salary if your salary this year is $36,500? 7. This month you weigh 85 pounds. After dieting and exercising for a month, you weigh 70 pounds. What is your percent weight loss? 8. The Dow Jones was 560 at the end of week. At the end of week 2 it is 28. How much did it drop and what percent is that of the original? Round the answer to the nearest tenth of a percent. 9. The price tag on a dress is $0 but Crystal has a coupon for 5% off any item. How much will Crystal save if she uses her coupon on the dress? 20. Edward finds a refrigerator listed for $ that is marked 20% off. What is the sale price of the refrigerator? 524

59 CCBC Math 08 Application Problems Section pages 7.5 Applications Exercise Answers. $ $ $62.50; 5% 4. $4, $90, % 7. $ $, % 0. 4%. $63 2. Yes 3. $5, % 5. $ $095; $37, Approximately 8.% ; approximately 3.7% 9. $ $

60 CCBC Math 08 Chapter 7 Summary Section 7. CHAPTER 7 SUMMARY Percents and Applications Percent: the number of parts out of 00 parts The drawing to the right shows 30%. There are 30 shaded squares out of 00 squares. Converting Percents to Fractions P% P Divide the percent by Example: Write 35% as a fraction Express as a fraction with 00 as the denominator. Rewrite in factored form and cancel common factors. The answer is simplified. Converting Percents to Decimals P% P P00 Divide percent by Example: Write 6% as a decimal. Converting Fractions to Percents n % d 00% P Multiply the fraction by 00% 30 Example: Write 3 as a percent % Multiply fraction by 00%. 8 To simplify, divide numerator 25% and denominator by % Multiplying gives the answer as 2 8 an improper fraction percent. 75 OR % Convert answer to a mixed 2 number or decimal percent. 37 % 37. 5% 2 Converting Decimals to Percents Decimal 00% P% Multiply decimal by 00% Example: Write as a percent Divide percent by 00 OR Move decimal point 2 places to the left % 34% Multiply decimal by 00% OR Move decimal point 2 places to the right Section 7.2 What percent of 35 is 7? Percent Whole Part OR 7 is what percent of 35? Part Percent Whole Percent Part 00 Whole P P P 20% 2% of 50 is what number? Percent Whole ParT OR What is 2% of 50? ParT Percent Whole Percent ParT 00 Whole 2 T T T 6 25% of what number is 46? Percent Whole Part OR 46 is 25% of what number? Part Percent Whole Percent Part 00 Whole W 25W W

61 CCBC Math 08 Chapter 7 Summary Section 7.3 Section 7.4 Circle Graph useful for showing parts out of a whole each sector (pie-sliced wedge) is part of the whole the total of all sectors must equal 00% Example:. How many total students are in this class? What percent of students earned A s? P P P 20% Principal: (a) amount of money borrowed OR (b) amount of money deposited in an account Interest: (a) fee charged to borrow money OR (b) money earned on a deposit Interest Rate: (a) percent charged for money borrowed OR (b) percent earned on money deposited Simple Interest: interest calculated on original principal only Calculating Simple Interest: Interest = Principal Rate Time in Years OR I=Prt Example: What is the simple interest on a loan of $9000 for 4 years if the interest rate is 7.5%? Section 7.5 I = Interest =? P = Principal = $9000 r = Rate = 7.5% t = Time (in years) = 4 I = P r t I ($9000)(7.5%)(4) I ($9000)(0 075)(4) I $2700 Example: What is the interest on a credit card for the period from March 20 through April 2 if the Average Daily Balance during this period was $89.62 and the APR was 8.6%? I = Interest charged =? P = Average Daily Balance = $89.62 r = APR = Annual Percentage Rate = 8.6% t = Number of Days 365 = 33/365. I = P r t I ($89 62)(8 6%)(33 / 365) I ($89 62)(0 86)(33 / 365). I $ Applications: Sales Tax, Commission Rates, Tips Example: Jake is buying a shirt for $7. How much sales tax will he pay if the tax rate is 6%? Sales Tax = Rate Purchase T 6% $7.. T 0 06 $7 T $

62 . Convert to a percent: Convert to a decimal: 77 % Convert to a fraction: 2 % 6 4. Convert to a percent: is 28% of what? is what percent of 40? CHAPTER 7 Chapter Review 200 students are put into different rooms. The percentage of students in each room is described in the graph below. Use this graph to answer #7 9. Room E 22% Room A 8% Room D 9% Room C 7% Room B 24% 7. How many students are in room B? 8. How many students are in A and C? 9. What is the percent of students in rooms B, D, and E? 0. A bank charges 2% simple interest. How much interest must you pay on a loan of $5000 for 5 years?. If you want to triple the value of $5000 investment earning 0% simple interest, how long will it take? 2. How many years does it take to turn $5,000 into $8,600 if it is invested at 2% simple interest? 3. How much interest is earned from depositing $2500 at 2% simple interest for 8 months? 4. A car costs $2,500. If a company is charging 8% simple interest for 7 years, how much is paid in total for the car? 528

63 CCBC Math 08 Chapter 7 Review 5. Anton did not use his credit card this past month, but carried a balance of $45. He just received his statement covering the period March 5 through April 4, and his average daily balance is the $45 carried over from last month. If the APR is 4.59%, how much interest was added to the balance? 6. Carin bought a kayak for $32. The sales tax rate is 5% so how much did she pay in taxes? 7. Cora bought camping supplies for an upcoming trip and spent $030. If the tax rate was 8%, what was the total cost of the camping supplies? 8. The amount of tax paid on a television was $87.0 in a state where the sales tax rate is 4%. What was the price of the television? 9. Aaron answered 3 problems correctly of the 27 problems on the test. What percent of the problems did Aaron get correct? Round to the nearest whole percent. 20. Felipe felt the service at the restaurant was good and wanted to leave a 20% tip on his $22.00 bill. How much did he leave for the tip? 2. Colin has a job where he earns a 6% commission on his sales each month. What monthly sales would earn him $4000 for the month? 22. Last month you weighed 94 pounds. After dieting and exercising for a month, you now weigh 76 pounds. What is your percent weight loss? Round to the nearest tenth of a percent. 23. Lucy hopes to earn a commission bonus of $3500. If her commission rate is 8%, how much does she have to sell in order to earn that bonus? 24. The Black Friday sale will have 35% off all TVs so Sebastian decides to use that day to purchase the TV he wanted which was regularly $499. How much did he spend for the TV on Black Friday? Mixed Review (3x 5) (3. 5.6) x = x 5 = x 6x + 2 = (3x 4) Is this true? x w m 529

64 CCBC Math 08 Chapter 7 Review C h a p t e r 7 R e v i e w A n s w e r s. 380% % % % 0. $ years years 3. $ $9, $ $ $ $ % 20. $ $25, % 23. $43, $ x x = x = 35. x = x Yes, it is true 38. x = w = m =

65 Math 08 Final Exam Review. Expand and simplify: (Section.6) 2 a. 5 b. 2. Simplify: a. b. 2 ( 3) c. 3 ( 2) d. 4 ( 2) 3 (Section.2,.6,.7) (Section.3,.4,.7) 2 36 ( 3) ( 5) e (Section.5,.6,.7) (Section.3,.5) c. 5( 7 6 2) (Section.3,.5,.7) 3. Translate into a math expression and find its value: a. 9 more than the product of 6 and 2 (Section 2.) b. the difference of 24 and 2 cubed (Section 2.) 4. Determine the perimeter of a square if the length of each side is 5 km. (Section 2.2) 5. Determine the perimeter of the rectangle. (Section 2.2) 6. Determine the perimeter of the triangle. (Section 2.2) 3 cm 7. Determine the circumference of the circle. 22 Use. (Section 3.8) 7 42 m 3 m 2cm 4 cm 4 m 5 cm 9. Determine the area of the triangle. (Section 3.8) 4 in 5 in 0. Determine the area of the rectangle. (Section 3.8). Determine the area of the circle below. Use 3 4. (Section 4.6). 4.6 m 2. Determine the volume of the solid. (Section 2.2) 9 ft 3 in 3. Determine the volume of the cylinder. 22 Use. (Section 3.8) 7 7 cm 4. Determine the volume of the sphere. Use 22. (Section 3.8) 7 in 5 ft 4 ft 2 cm 4cm 8. Determine the area of a square if the length of each side is 5 cm. (Section 2.2) 5. Determine the volume of a cube if each side is 5 inches long. (Section 2.2) 53

66 6. Determine the mean of the following values: 5, 5, 6, 2, 9, 3 (Section 2.3) 7. The ages of the company employees are 26, 45, 22, 25, 24, 30, and 34. Determine the median. (Section 2.3) 8. The temperatures for the past several days were 2º, 6º, º, 3º, 2º, and 4º. Determine the mode. (Section 2.3) 9. Determine the mean of the values, 2, (Section 3.4) 20. The times (in minutes) for the top six runners were 2.38, 2.4, 2.33, 2.4, 2.45, and Determine the median. (Section 4.4) 2. Emily s yearly salary is $60,840. If Emily gets paid 36 times per year, how much does she get per pay period? (Section 2.4, 2.5) 22. Use the chart below to answer the questions that follow. (Section 2.4) a. What was the total number of visitors in 202? b. How many more adults visited the park in 203 than in 200? 23. Write the following as mixed numbers: (Section 3.) a. 32 b Write the following as an improper fraction: (Section 3.) a. 8 b Compute the following: (Section 3.2, 3.3) 8 5 a. 5 2 b. d e c. 3 3 f Compute the following: (Section 3.2) a Compute the following: (Section 3.4, 3.5) 4 5 a b. b g. c h. d i. e j. 28. Compute the following: (Section 3.6) a f. b

67 29. A tree is feet tall. When it was first planted it was 8 2 feet tall. How much 3 did the tree grow from when it was first planted? (Section 3.8) 30. If a doctor uses of a 600 ml bottle of 3 medicine, how much did she use? (Section 3.8) 3. There are 20 samples of bacteria each weighing 7 grams. What is the weight of 8 all 20 samples? (Section 3.8) 32. Convert the following: (Section 3.7, 4.5) a. 60 inches to feet d. 65 mm to hm b. 72 ounces to pounds e. 647 cm to m c. gallon to cups f kg to g 33. Write the following in simplest fraction form. (Section 4.2) a b Write 6 3 as a decimal and round to the hundredths place. (Section 4.2) 35. Compute the following: (Section 4.3) a. 5 3 ( 2 45) c. ( 7 94) ( 3 4) b d Compute: (Section 4.4) 37. Sausage costs $2.24 a pound. How much will 3.5 pounds cost? (Section 4.6) 38. A sweater regularly sells for $ You bought the sweater on sale for $3.65. How much money did you save? (Section 4.6) 39. How many 0.75 ounce bags of candy can I fill from a 5 ounce bag of candy? (Section 4.6) 40. Rewrite 5 x using the Commutative Property of Addition. (Section 5.2) 4. Rewrite 3 using the Associative Property of Multiplication. (Section 5.2) 42. What is the additive inverse of (-2)? (Section 5.2) 43. What is the reciprocal of 3 5? (Section 5.2) 44. Simplify the following: (Section 5.) a. 9x 4y 7 5y 6x b. 5 6a 8 9a 4 7b 6 3b 2 c. 45. Simplify: 7 2 a a (Section 5.3) a. 3(8x 4) b. (6m 5) c. 2 (6 x 5) Evaluate 2xy 3 when x 9and y. (Section 5.) 47. Is x 24 a solution of 9 x 8? (Section 5.) 48. Translate the following into an equation and then solve the equation. (Section 5.7) a. Five less than two times a number is 9. b. The sum of one-third of a number and 4 is

68 49. Solve the following equations: a. x 9 7 f. 2y (Section 5.4) (Section 5.6).. b. 4 9x 5 68 g. 3x5x 6 2 c. (Section 5.5) (Section 5.6) 3 x 3 h x (Section 5.5) (Section 5.6) z d. 3 i. 3(5x 3) 4 2 (Section 5.5) (Section 5.6) e. a 2 j. 4( x3) 8x 32 (Section 5.5) (Section 5.6) 50. Write the ratio of 28 to 32 in simplest fraction form. (Section 6.) 5. Write the ratio of 5:80 in simplest fraction form. (Section 6.) 52. Refer to the monthly budget shown below. In lowest terms, give the ratio of the amount budgeted for gas and electric to the total budget. (Section 6.) Item Amount Rent $400 Food $400 Gas and electric $00 Phone $200 Entertainment $ There are 2 women and 3 men in a room. What is the ratio of men to adults? (Section 6.) 54. There are 70 students and 34 computers at the school. In simplest fractional form, give the ratio of computers to students? (Section 6.) 55. Hanna hiked 40 km in 6 hours. What was her rate in simplest fractional form? (Section 6.2) 56. A painter earned $3.40 for 2 hours of work. What is the painter s unit rate of pay? (Section 6.3) 57. A 6-ounce box of cereal cost $3.49 and an 8-ounce box cost $3.78. Which is the better deal? (Section 6.5) 58. Solve the following proportions: (Section 6.4) a. b c. n 5 2 n n A solution calls for 5 ml of a drug for every 38 liters of water. How much water is needed for 4.5 ml of the drug? (Section 6.5) 60. If 2.5 pounds of carriage bolts costs $3.80, how much will 2.5 pounds cost? (Section 6.5) 6. When billing a household, a utility company charges for the kilowatt-hours used. A kilowatt-hour (kwh) is a standard measure of electricity. If the cost of kwh is $0.94, what is the electricity bill for a household using 873 kwh in a month? Round the answer to the nearest cent. (Section 6.5, 4.6) 62. Write 7 8 (Section 7.) as a percent. 63. Write the following in simplest fraction form: (Section 7.) a. 84% c. 50% b. 99% 534

69 64. Write 9% as a decimal. (Section 7.) 65. Write 0.65 as a percent. (Section 7.) is 5% of what number? (Section 7.2) 67. What percent of 50 is 27? (Section 7.2) 68. What is 35% of 400? (Section 7.2) 69. A family s total monthly expenses are $2600. The graph below shows the breakdown of the expenses. Use the graph to answer the questions that follow. (Section 7.3) 70. What is the simple interest on a loan of $30,000 for 5 years if the interest rate is 8.4%? (Section 7.4) 7. An automobile company is recalling 8% of its cars sold in Baltimore. If 350 cars were sold in Baltimore, how many cars are being recalled? (Section 7.5) 72. Alison bought a sofa for $239 and paid $4.34 in tax. What is the tax rate? (Section 7.5) 73. Andre is paid 2.5% commission on his sales. If he wants to earn $2500 in commission, what amount of sales must he make? (Section 7.5) 74. What simple annual interest rate is needed for an investment to grow from $3000 to $7500 in 6 years? (Section 7.4) a. What amount is spent on rent? b. What amount is spent on food and utilities combined? c. How much more is spent on utilities than on food? 75. A shirt costs $45 and a pair of pants costs $50. If sales tax is 6%, do I have enough to buy both with $00? Explain why or why not. (Section 7.5) 76. How much time does it take an investment of $2500 to double in value if the annual simple interest rate is 5%? (Section 7.5) 535