Section 6.5 Applications Involving Percents

Section 6.5 Applications Involving Percents The focus of this section is to show how to set up a proportion to solve word problems involving real-life applications of percent. If the student needs a review of proportions and percent, Sections 6.2 and 6.3 should be read before continuing. Note: For problems involving a percent, we set up a proportion where one side is reserved for the percent, and the other side is reserved for two numbers. We let a variable represent the percent or amount we are trying to find. Number Percent = Original amount Since represents the fractional form of the percent, will always equal. We only need to assign values to the variables, and. This is illustrated in the following examples. Example 6.5.1 A jacket was originally priced at $145. It is now on sale at 30% of the original price. How much does the jacket cost now? First, we will set up a proportion to find 30% of $145, and then we will subtract this amount from the original price of $145. The original price of $145 represents the whole or initial amount, and should be written in the denominator of the right Copyright 2014 Luis Soto-Ortiz 480

side of the proportion (it s the in the figure above). We also know the percent, which we write as (it s the in the figure above). There is only one spot remaining (the ) which in this case is the amount that will be subtracted from the original price. This gives the following proportion: = Answer: 30 = 145 30 145 = 4350 = 43.50 = 1 43.50 = The original price of the jacket will be reduced by $43.50. Thus, its discounted price is $145 - $43.50 = $101.50. Example 6.5.2 Wendy went to a store and bought an LED TV for $1,590. If there is a 6% sales tax, how much will she pay in total? First, we will set up a proportion to find 6% of $1,590, and then we will add this amount to the original price of $1,590. The original price of $1,590 represents the whole or initial amount, and should be written in the denominator of the right Copyright 2014 Luis Soto-Ortiz 481

side of the proportion (it s the ). We also know the percent, which we write as (it s the ). There is only one spot remaining (the ) which in this case is the amount in taxes that Wendy will have to pay. This gives the following proportion: = Answer: 6 = 1590 6 1590 = 9540 = 9540 = 95.40 = 1 95.40 = The cost of the LED TV, including the 6% tax, will be $1,590 + $95.40 = $1685.40. Example 6.5.3 A professional basketball team won 56 out of 82 games this season. What was the winning percentage of this team? In other words, what percentage of the total numbers of games played did the team win? Also, what percent of the 82 games did the team lose? Round your answer to the nearest tenth. Copyright 2014 Luis Soto-Ortiz 482

In this question, 82 represents the entire or total number of games played, and should be written in the denominator of the right side of the proportion (it s the ). We also know the number of games won (the ) out of the total games played. We do not know the percent won, which we write as (it s the ). This gives the following proportion: = Answer: = 56 82 82 = 56 82 = 5600 82 82 = 5600 82 1 = 68.29268 = 68.3 Rounding the answer to the nearest tenth, the winning percentage of this basketball team was 68.3%. This means that the team lost % 68.3% = 31.7% of the 82 games they played. Copyright 2014 Luis Soto-Ortiz 483

Example 6.5.4 A car dealer sold 1150 vehicles last year. Of this number, 38% were trucks. How many vehicles sold last year were trucks? In this question, 1150 represents the entire or total number of vehicles sold, and should be written in the denominator of the right side of the proportion (it s the ). We also know the percent of trucks sold, which we write as (it s the ). We do not know the number of trucks sold (the ) out of the total number of vehicles sold. This gives the following proportion: = Answer: 38 = 1150 38 1150 = 43700 = 43700 = 437 = 1 437 = Last year, the car dealer sold 437 trucks. Copyright 2014 Luis Soto-Ortiz 484

Example 6.5.5 In a company, there are 632 employees, 280 of which are female. A. What percent of the employees are female? B. What percent of the employees are male? In this question, the total number of employees is 632 and should be written in the denominator of the right side of the proportion (it s the ). We also know the number of female employees (the ). We do not know the percent of trucks sold, which we write as (it s the ). This gives the following proportion: = Answer: = 280 632 632 = 280 632 = 28000 632 632 = 28000 632 1 = 44.3 = 44.3 Of the 632 workers that the company employs, 44.3% are female. Since % 44.3% = 55.7%, this means that 55.7% of the employees are male. Copyright 2014 Luis Soto-Ortiz 485

Example 6.5.6 In a small district, 2,530 people turned out to vote for proposition A. Forty percent of the voters voted against it. A. What percent of the electorate voted in favor of Proposition A? B. How many people voted in favor of Proposition A? In this question, the total number of voters was 2,530 and should be written in the denominator of the right side of the proportion (it s the ). We also know that the percent of the electorate who voted against Proposition A was 40%. This means that % 40% = 60% voted in favor of it. We write this percent as (it s the ). We do not know the number of people who voted in favor of Proposition A (the ). This gives the following proportion: = Answer: 60 = 2530 60 2530 = 151800 = 151800 = 1518 = 1 1518 = Of the 2530 people who voted, 1518 voted in favor of Proposition A. Copyright 2014 Luis Soto-Ortiz 486

Example 6.5.7 At Tech-O University there are currently 360 male faculty. If this number represents 45% of the total number of faculty, how many faculty currently work at Tech-O University? In this question, we know the percent of male faculty working at the university, 45%, which we write as (it s the ). We also know that the number of male faculty is 360 (this is the ). We do not know the total number of faculty working at Techo University (the ). Therefore, the proportion can be set up as: = Answer: 45 = 360 45 = 360 45 = 36000 45 36000 = 45 45 1 = 800 = 800 Tech-O University currently employs 800 faculty in total. Copyright 2014 Luis Soto-Ortiz 487

Example 6.5.8 The current selling price of a 3-bedroom house is $218,000. If the price of the house decreased by 12% from a year ago, what was the price of the house a year ago? Round your answer to the nearest dollar. Since the price of the house decreased by 12% compared to the price it had a year ago, this means that its current price of $218,000 represents % 12% = 88% of the price it had a year ago. Hence, we know the percent, 88%, which we write as (it s the ). We also know the current price which is $218,000 (this is the ). We do not know the original price of the house a year ago (the ). Therefore, the proportion can be set up as: = Answer: 88 = 218000 88 = 218000 88 = 21,800,000 88 21,800,000 = 88 88 1 = 247,727.2727 = 247,727.2727 Rounded to the nearest dollar, the price of the house a year ago was $247,727. Copyright 2014 Luis Soto-Ortiz 488

Applications of Percent Involving Simple Interest An important application of percent occurs when one deposits, invests or borrows an amount of money and a monthly or yearly simple interest rate determines the total balance after a certain number of months or years. The following definitions are used in simple interest rate problems: Principal (P): original amount deposited, invested or borrowed. Simple Interest Rate (r): percent of the original amount that was invested or borrowed and that is added to the balance each month (for monthly interest rates) or yearly (for yearly interest rates). Interest (I): The amount of money that is added to the original amount invested or borrowed each month or each year based on the simple interest rate. Therefore, interest represents a change in money. Time (t): Length of time that the original amount (principal) is invested or borrowed. One way to solve simple interest word problems is to set up the following proportion for simple interest: = We substitute the quantities that are known into the variables, and then solve the proportion by the cross multiply and divide method to determine the value of the remaining variable. Note that in the above proportion, the simple interest rate is a percent that is written as a fraction interest rate is 3%, we let = 3 and write simple interest rate is 7.12%, we let = 7.12 and write.. For example, if the simple in the proportion. Similarly, if the in the proportion. We will now illustrate the use of the simple interest proportion with some examples. Copyright 2014 Luis Soto-Ortiz 489

Example 6.5.9 Patricia deposited $1,500 in a savings account that has an annual simple interest of 5%. How much interest will she earn in 8 years? In this example, we know the values of the following variables: P = $1,500 r = 5% t = 8 years and we are asked to find I, which is the amount of interest that Patricia will earn in 8 years. = Answer: 1,500 = 5 8, = = 1,500 5 8 = 60,000 60,000 = 1 = 600 = $600 Patricia will earn $600 in simple interest in 8 years if she deposits $1,500 at an annual rate of 5%. Copyright 2014 Luis Soto-Ortiz 490

Example 6.5.10 John wants to have an interest income of $3,000 in one year. How much must he invest for one year at 6% simple interest? In this example, we know the values of the following variables: r = 6% I = $3,000 t = 1 year and we are asked to find P, which is the original amount that John must invest. = Answer: 3,000 = 6 1, = 3,000 = 6 1 300,000 = 6 300,000 6 = 6 6 50,000 = 1 = $50,000 John must invest $50,000 for one year at a yearly simple interest of 6% to have an interest income of $3,000 in one year. Copyright 2014 Luis Soto-Ortiz 491

Example 6.5.11 A student borrowed some money from his father at 2% simple annual interest to buy a car. He paid his father $360 in interest after 3 years, how much did he borrow? In this example, we know the values of the following variables: r = 2% I = $360 t = 3 year and we are asked to find P, which is the original amount that the student borrowed from his father. = Answer: 360 = 2 3 = 360 = 2 3 36,000 = 6 36,000 6 = 6 6 6,000 = 1 = $6,000 The student borrowed $6,000 from his father at an annual simple interest of 2% for 3 years. Copyright 2014 Luis Soto-Ortiz 492

Example 6.5.12 Richard deposits $5,400 and has a balance of $6000 after a year. Determine the simple annual interest rate of the account. Round your answer to the nearest tenth of a percent. In this example, we know the values of the following variables: P = $5,400 I = $6,000 $5,400 = $600 t = 1 year and we are asked to find r, which is the simple annual interest rate of the account. Answer: = 600 1 = 5,400 600 = 5,400 1 $ $, = 60,000 = 5,400 60,000 5,400 = 5,400 5,400 11.111 = 1 11.1% The account in which Richard deposited the $5,400 has an annual simple interest rate of approximately 11.1%. Copyright 2014 Luis Soto-Ortiz 493

This video summarizes the use of proportions to solve real-life applications involving percent: http://www.youtube.com/watch?v=yl0rb6t09vm Classwork 6.5 1. In January of this year, in a northeastern city, it snowed a total of 6 days. What percent of the total days of January did it snow in this city? Round your answer to the nearest whole percent. 2. If 40% of all the students in a history class are males and there are 14 male students enrolled, how many students in total are enrolled in the history class. 3. A laptop was originally priced at $799. After a promotional discount, it is now on sale for $599. What was the % discount? Round your answer to the nearest whole percent. 4. The value of a classic car has increased by 20% in the last ten years. If it was valued at $25,000 ten years ago, what is its current value? 5. Five years ago, George used to earn $12,000. He now earns $15,000 annually. What is the percent increase in his salary compared to five years ago? 6. Carmen and her family had dinner at a restaurant. The bill, without the tip, was $83.40. If she leaves a tip of 10% of the bill, how much will she pay in total? Copyright 2014 Luis Soto-Ortiz 494

7. A baseball team lost 70 of its 162 games. What was its winning percentage? Round you answer to the nearest tenth of a percent. 8. Marleen answered 41 out of 50 questions correctly. What percent of all the questions did Marleen answer correctly? What percent of all the questions did she get wrong? 9. Rachel bought a sweater whose original price was $85. If the sweater is now on sale at a 30% discount, how much is the discount and what is the discounted price of the sweater? 10. Jackie was late paying her monthly bank statement. Her balance was $2,460 but now will have to pay an additional 9% interest in addition to the $2,460. How much does Jackie now owe to the bank? 11. Ben bought 2 new tires for his truck. If each tire cost $109 and he paid 8% sales tax, how much did Ben pay in total? 12. Anthony made an investment in which he originally invested $5,000. If his gains amounted to 4.8% of the amount he invested, how much money did he gain from the investment? 13. Diana made an investment in which she originally invested $10,000. If she actually lost 2.6% of the amount she invested, how much money did she lose from the investment and how much money does she currently have now? 14. A house was selling for 240,000 five years ago. It s current price is only $180,000. What was the percent drop in price? Copyright 2014 Luis Soto-Ortiz 495

15. Tatiana earns 8% in commission for her monthly sales. If last month Tatiana sold $6,327 in merchandise, how much money did she earn in commission alone? 16. A community college experienced an increase in student enrollment this semester. Last semester there were 18,278 students enrolled, but this semester there are 22,307 students enrolled. What was the percent increase in enrollment? Round your answer to the nearest whole percent. 17. At an Interior Design Store each salesperson receives a 7.5% commission on sales. What would a salesperson earn if she sold $1,750 in home furnishings? 18. The cost of annual tuition at a university increased from 10,500 to $11,300. What is the percent increase in tuition to the nearest tenth of a percent? 19. If you purchase a DVR player that costs $320, how much sales tax will you pay to the nearest cent if the rate is 9.25%? 20. Greg and Fred ran in a school election where 1,600 students voted for one of the two candidates. If Greg received 760 votes and Fred received the rest of the votes, what percent of all the votes did Fred receive? By what percent did Fred win the election? 21. Norma deposited $2,000 in a savings account at the simple interest rate of 4% per year. How much simple interest will she earn in 5 years? 22. Lorraine borrowed $4,000 from her cousin Suzy at the rate of 8% per annum. If Lorraine repaid everything after two years, how much did she pay Suzy in total? Copyright 2014 Luis Soto-Ortiz 496

23. David put $3,500 into an investment yielding 4.5% annual simple interest. How much interest would he gain if he makes an 8-year investment? 24. Vivian invested $2,500 at an annual rate of 5%. How long will it take until Vivian earns $1,125 in interest? 25. Jeremy invested $1,500 in an account that paid him 8.25% simple interest. What will the total balance of his account be after 6 years? Copyright 2014 Luis Soto-Ortiz 497

CW 6.5 Solutions: 1) 31, 19%. 2) 35 (14 21 ). 3) 25%. 4) $30,000. 5) 25%. 6) $91.74. 7) 92 162. 56.8%. 8) 82% 18%. 9) $25.50. $59.50 10) $2,681.40. 11) $235.44. 12) $240. 13) $260. $9,740. 14) 25%. 15) $506.16. 16) 22%. 17) $131.25. 18) 7.6%. 19) $29.60. 20) 52.5%. 5%. 21) $400 22) $4,640 23) $1,260 24) 9 years 25) $2,242.50 Copyright 2014 Luis Soto-Ortiz 498

Homework 6.5 1. If Roxanne gives a tip at a restaurant of 8% of the bill and the bill without the tip is 42.75, how much will she end up paying in total? 2. You bought a painting for $15,000 three years ago. It is now valued at $20,000. What was the percent increase in its value during the last three years? 3. A car stereo was originally priced at $200. After a promotional discount, it is now on sale for $125. What was the % discount? 4. At a community college, 45 math courses are being offered this semester. If 20% of all the courses offered by the college are math courses, how many courses in total are being offered this semester? 5. Marie used to earn $18,000 annually five years ago. She now earns $36,000 annually. What was the percent increase in her salary compared to five years ago? 6. Michael took a psychology exam and he answered 37 out of the 40 questions correctly. What percent of the total number of questions did Michael got correct? 7. Kimberly earned $300 this week working at a bookstore. Of the $300, she put $65 in her savings account. What percent of her salary did she save? Round your answer to the nearest tenth of a percent. Copyright 2014 Luis Soto-Ortiz 499

8. A professional basketball player made 11 out of 16 free throws in a game. What percent of all the free throw shots he attempted did he make? Round the percent to the nearest whole number. 9. Anne saves 15% of her monthly income for college expenses. If she earned $1,328 this month, how much money did she save for college expenses? 10. In a country of 80 million people who are eligible to work, if the unemployment rate is approximately 6.4%, approximately how many people are unemployed? 11. Judith earns $82,693 per year working for a pharmaceutical company. If 23% of her salary is set aside to pay the house mortgage, how much does she pay for mortgage each year? 12. Sharon is a graduate student working towards her Master s in Teaching Education. If 32% of her monthly salary of $2,075 goes towards paying her rent, how much does Sharon pay in rent each month? 13. In his last basketball game, Dirk made 14 out of 20 field goals. What was Dirk s field goal percentage in his last game? 14. All the teachers working at a particular school district will receive an increase in their current annual salary of 15%. After this agreement, what will be the new annual salary of a teacher whose current annual salary is $52,600 ignoring any other increases to her salary? 15. A local cable television company currently charges $36 per month. It plans an increase in its monthly charge of 9%. What will the new monthly rate be? Copyright 2014 Luis Soto-Ortiz 500

16. In 1980, the median age of U.S. residents was 30 years. By 1996, the median age had increased by about 15.3%. What was the median age in 1996? Round your answer to the nearest tenth. 17. A worker s weekly take-home pay was $480 after deductions totaling 40%. What is the worker s weekly gross pay? 18. A cereal company advertises that its 16-ounce cereal box represents 25% more cereal than before. What was the original amount of cereal in a box? 19. A volleyball team won 90% of 80 games played. How many games did they win? 20.At Markham Middle School, 846 students ride the bus to school. If this number is 60% of school enrollment, then how many students are enrolled at the school? 21. Ralph wrote a check for $7,820 to pay off a loan, which was given to him at a rate of 5% simple interest for 3 years. How much money did he borrow originally? 22. If $3,840 is invested in an account at 5% annual simple interest, how long will it take the account balance to grow to $4,800? 23. Calculate the interest earned in an investment where the Principal (initial amount invested) was $1,500, the annual simple interest rate was 7% and the time of investment was 8 years. 24. Selena deposited $1,400 in her bank account. After 3 years, the account is worth $1,694. Determine the simple interest rate of the account. Write your answer as a percent. Copyright 2014 Luis Soto-Ortiz 501

25. Calculate the monthly simple interest rate (as a percent), in an investment where the Principal (initial amount invested) was $360, the interest earned was $17.55 and the time of investment was 9 months. The following website has additional word problems with solutions involving percent. You may set up a proportion to solve these problems, just like we did in this section. You can then compare your answers to the posted solutions. http://cnx.org/content/m35007/latest/?collection=col10615/latest Copyright 2014 Luis Soto-Ortiz 502

HW 6.5 Solutions: 1) $46.17. 2) 33%. 3) 37.5%. 4) 225. 5) %. 6) 92.5%. 7) 21.7% $300. 8) 69%. 9) $199.20. 10) 5,120,000. 11) $19,019.39. 12) $664. 13) 70%. 14) $52,600 $60,490. 15) $39.24. 16) 1996 34.6. 17) $800. 18) 12.8. 19) 72 80. 20) 1,410. 21) $6,800 22) 5 years 23) $840 24) 7% 25) 6.5% Copyright 2014 Luis Soto-Ortiz 503