7.1 Percent, Decimal and Fractions

Transcription

1 Chapter 7. Percent 7.1 Percent, Decimal and Fractions 7.2 Solving Percent Problems Using Percent Equations and Proportions 7.3 Application of Percent 7.4 Estimation with Percent 7.5 Interest 1

2 7.1 Percent, Decimal and Fractions 1. Explain the meaning of percent 2. Write percents as fractions 3. Write percents as decimals 4. Write decimals as percents 5. Write fractions as percents 2

3 7.1 Percent, Decimal and Fractions 1. Explain the meaning of percent (review) Percent Percent means parts per one hundred. Percent is a fraction with denominator equals to 100. The symbol for percent is %. So 93 percent is written as 93%. And 93% is nothing but a fraction: 93 % =

4 7.1 Percent, Decimal and Fractions 1. Explain the meaning of percent (review) e.g.1 In 2009, 17 of the 100 U.S. Senators were women. a. What percent of the senators were women? b. What percent of the senators were men? Ans. 17% Ans. 83% 2. Write percents as fractions (review) (drop the % symbol, write the given number over 100) e.g.2 ANATOMY The human body is 65% water. Write this percent as a fraction in simplest form. Ans

5 7.1 Percent, Decimal and Fractions 2. Write percents as fractions (review) e.g.3 UNIONS In 1983, 20.1% of the U.S. labor force belonged to a union. Write this percent as a fraction in simplest form. Ans , e.g.4 Write 16 % as a fraction in simplest form. Ans

6 7.1 Percent, Decimal and Fractions 2. Write percents as fractions (review) e.g.5 a. Write 150% as a fraction in simplest form. Ans. 3 2 b. Write 0.42% as a fraction in simplest form. Ans. 21 5,000 6

7 7.1 Percent, Decimal and Fractions 3. Write percents as decimals (review) Drop the %, divide the given number by 100. e.g.6 MARKET SHARE BY VISITORS (%) For the week ended May 23, 2009, the market shares by Dancing with the Stars and by America s Most Wanted are 10.42% and 3.49% respectively. a. Write the percent of market share for the Dancing with the Stars as a decimal. Ans b. Write the percent of market share for the America s Most Wanted as a decimal. Ans

8 7.1 Percent, Decimal and Fractions 3. Write percents as decimals (review) e.g.7 POPULATION The population of the state of 1 Virginia is approximately 2 % of the population of 2 the U.S. Write this percent as a decimal. Ans e.g.8 (moving decimal point to the left by two places) a. Write 540% as a decimal. Ans. 5.4 b. Write 0.1% as a decimal. Ans

9 7.1 Percent, Decimal and Fractions 4. Write decimals as percents (review) Write Decimals as Percents To write a decimal as a percent, multiply the decimal by 100 by moving the decimal point 2 places to the right, and then insert a % symbol. e.g.9 Write as a percent. Ans. 45.7% 9

10 7.1 Percent, Decimal and Fractions 5. Write fractions as percents (review) Write fractions as Percents 1. Write the fraction as a decimal by dividing its numerator by its denominator. 2. Write the resulted decimal as a percent. e.g.10 ADVERTISING There is a commercial that says, Four out of five dentists recommended sugarless gum for their patients who chew gum. Express this recommendation as a percent. Ans. 80% 10

11 7.1 Percent, Decimal and Fractions 5. Write fractions as percents (review) 15 4 e.g.11 Write as a percent. Ans. 375% What if division does NOT terminate? 1 e.g.12 Write as a percent. Give the exact answer and an 3 approximation to the nearest tenth of one percent. Ans % 33.3% 11

12 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 1. Translate percent sentences to percent equations 2. Solve percent equations to find the amount 3. Solve percent equations to find the percent 4. Solve percent equations to find the base 12

13 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 1. Translate percent sentences to percent equations e.g.1 Translate each percent sentence into a percent eq. a. What number is 25% of 37? Ans. x = 25% 37 b. What percent of 300 is 12? (be careful on this) c. 110% of what percent is 341? Ans. x 300 = 12 Ans. 110 % x =

14 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 2. Solve percent equations to find the amount Consider a example: 420 is 84% 500. amount (part) percent base (whole) 14

15 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 2. Solve percent equations to find the amount Percent Equation (Formula) Any percent sentence can be translated to a percent equation that has the form: amount = percent base or part = percent base e.g.3 What number is 340% of 28.4? x = 340% 28.4 Ans. x =

16 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 3. Solve percent equations to find the percent e.g.4 34 is what percent of 40? 34 = x 40 Ans. x = 0.85 = 85% Comment: convert the decimal to percent. e.g.6 What percent of 7 is 9.1? (reverse the sentence) x 7 = 9.1 Ans. x = 1.3 = 130% 16

17 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 4. Solve percent equations to find the base e.g.7 7 is 35% of what number? Ans. 20 Comment: change percent to decimal in the equation. e.g is 33 % of what number? Comment: keep the fractions 1 3 Ans

18 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 1: PERCENT EQUATIONS (review) 4. Solve percent equations to find the base e.g.9 CLOSED CLASSES On the first day of classes, 36 of the algebra sections offered by a college mathematics department were closed. If this represented 72% of all sections of algebra, what is the total number of algebra sections offered that semester? Ans. 50 sections 18

19 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 1. Write percent proportions 2. Solve percent proportion to find the amount 3. Solve percent proportion to find the percent 4. Solve percent proportion to find the base 5. Read circle graphs 19

20 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 1. Write percent proportions 1 Consider an example: we know that = 50%, re-write: 2 The amount-to-base ratio 1 = The percent ratio Percent Proportion To translate a percent sentence to a percent proportion, use the following form: amount is to base as percent is to 100 part is to whole as percent is to 100 amount base = percent 100 or part whole = percent

21 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 1. Write percent proportions e.g.1 Translate each percent sentence to a percent proportion. a. What number is 25% of 37? x 37 = b. What percent of 300 is 12? c. 110% of what number is 34? 12 x = = x Comment. Need to identify part and whole. 21

22 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 2. Solve percent proportion to find the amount e.g.2 What number is 28% of 300? e.g.3 What number is 340% of 28.4? Ans

23 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 3. Solve percent proportion to find the percent e.g.4 34 is what percent of 40? Ans. 85% e.g.6 what percent of 7 is 9.1? Comment: reverse the sentence. Ans. 130% 23

24 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 4. Solve percent proportion to find the base e.g.7 7 is 35% of what number? Ans. 20 e.g is % of what number? 33 Ans

25 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 4. Solve percent proportion to find the base e.g.9 CLOSED CLASSES On the first day of classes, 36 of the algebra sections offered by a college mathematics department were closed. If this represented 72% of all sections of algebra, what is the total number of algebra sections offered that semester? Ans. 50 sections 25

26 7.2 Solving Percent Problems Using Percent Equations and Proportions Method 2: PERCENT PROPORTIONS (review) 5. Read circle graphs e.g.10 HOUSEING At the end of 2007, approximately 110 million housing units in the U.S. were occupied. Use the information in the following graph to find the number of millions of units that were owner occupied in Ans millions of units Renter Occupied 31% Owner Occupied 69% 26

27 7.3 Application of Percent 1. Calculate sales taxes, total cost, and tax rates 2. Calculate commissions and commission rates 3. Find the percent of increase or decrease 4. Calculate the amount of discount, the sale price and the discount rate 27

28 7.3 Application of Percent 1. Calculate sales taxes, total cost, and tax rates Finding the Sales Tax sales tax = sales tax rate purchase price e.g.1 Find the sale tax on a purchase of $64.50 if the sales tax rate is 7%. Ans. $4.52 Finding the total cost total cost = purchase price + sales tax e.g.2 Find the total cost of a $ refrigerator if the sales tax rate is 2.8%. Ans. $

29 7.3 Application of Percent 1. Calculate sales taxes, total cost, and tax rates e.g.3 CITY INCOME TAX For the year 2008, a resident of Detroit, Michigan, paid $550 in city income tax on taxable earning of $22,000. What is the Detroit city income tax rate? Ans. Tax rate = = 2.5% 29

30 7.3 Application of Percent 2. Calculate commissions and commission rates Find the Commission commission = commission rate sales e.g.4 SELLING REAL ESTATE A real estate agent earns a 3% commission on all sales. Find her commission if she sells a house for the listing price of $212,000. Ans. $6,360 30

31 7.3 Application of Percent 2. Calculate commissions and commission rates Find the Commission commission = commission rate sales e.g.5 JEWELRY SALES A jewelry salesperson earned a commission of $78 for selling a pearl necklace priced at $1,300. Find the commission rate. Ans. Commission rate = 0.06 = 6% 31

32 7.3 Application of Percent 3. Find the percent of increase or decrease e.g.6 TALK RADIO Refer to the table. Find the percent increase in the number of news/talk radio stations from 2004 to year number of stations , ,360 First find increase = 1,360 1,280 = 80 Next find the percent of increase = increase original amount = = 6.25% 32

33 7.3 Application of Percent 3. Find the percent of increase or decrease Finding the Percent of Increase or Decrease 1. Subtract the smaller number from the larger number to find the increase or decrease. 2. Find the percent of increase or decrease by dividing the number in step 1 by the original number. e.g.7 INSURANCE COST A college student s good grades earned her a student discount on her car insurance premium. Find the percent of decrease if her annual premium was lowered from $1,050 to $ % 33

34 7.3 Application of Percent 4. Calculate the amount of discount, the sale price and the discount rate Finding the Discount amount of discount = original price sale price Finding the Discount amount of discount = discount rate original price Find the Sale Price sale price = original price discount 34

35 7.3 Application of Percent 4. Calculate the amount of discount, the sale price and the discount rate e.g.8 BOOT SALES Find the amount of the discount on a pair of hiking boots if they are normally priced at $124.60, but are currently on sale for 15% off. Then find the sale price. discount = $18.69, sale price = $ e.g.9 FURNITURE SALES Find the discount rate on a desk lamp regularly priced at $89.95 that is on sale for $ Round to the nearest one percent. discount rate = 11% 35

36 7.4 Estimation with Percent 1. Estimate answers to percent problems involving 1% and 10% 2. Estimate answers to percent problems involving 50%, 25%, 5% and 15% 3. Estimate answers to percent problems involving 200% 4. Using estimation to solve percent application problems 36

37 7.4 Estimation with Percent 1. Estimate answers to percent problems involving 1% and 10% (quick answers) e.g.1 What is 1% of 722.6? Find the exact answer and an estimate using front-end rounding. Exact: 7.226, estimate: 7 e.g.2 What is 10% of 5,844 miles? Find the exact answer and an estimate using front-end rounding. Exact: mi, estimate: 600 mi 37

38 7.4 Estimation with Percent 1. Estimate answers to percent problems involving 1% and 10% (quick answers) What about percent problem involving 20%, 30%, 40% e.g.3 Estimate the answer: what is 20% of 208. step 1. 10% of 208 = step 2. Double the result in step 1: 42 38

39 7.4 Estimation with Percent 2. Estimate answers to percent problems involving 50%, 25%, 5% and 15% (quick answers) 50% of a number = the number 2 25% of a number = the number 4 5% of a number = (10% of the number) 2 15% of a number = 10% of the number + 5% of the number e.g.4 estimate the answer: what is 50% of 5,811,249? 3,000,000 e.g.5 estimate the answer: what is 25% of 11.47? 3 e.g.6 estimate 5% of 8,350? 400 (or 420) 400 (or 420) e.g.7 estimate 15% of $38.6? $6 $6 39

40 7.4 Estimation with Percent 3. Estimate answers to percent problems involving 200% (quick answers) Double the number! e.g.8 Estimate the answer: what is 200% of ? 25 How about Percent problem involving 300%? Percent problem involving 400%? Triple the number Quadruple the number 40

41 7.4 Estimation with Percent 4. Using estimation to solve percent application problems e.g.9 DAY CARE Of the 250 children attending an elementary school, 26% of them are enrolled in afterschool day care. Estimate the number of children in day care. 62 children (or 60 children) 41

42 7.5 Interest 1. Calculate simple interest 2. Calculate compound interest 42

43 7.5 Interest 1. Calculate simple interest Simple Interest Formula Interest = Principal rate time or I = P r t P: principal amount, r: annual rate, t: in years e.g.1 If $5,000 is invested for 1 year at a rate of 6%, how much simple interest is earned? $300 Total Amount total amount = principal + interest 43

44 1. Calculate simple interest 7.5 Interest e.g.2 If $500 is invested at 3.5% for 6 years, what will be the total amount of money in the investment account at the end of the 6 years? $605 e.g.3 NAVIGATION SYSTEM A truck borrowed $480 at 8% for 3 months to purchase a GPS system. Find the simple interest that must be paid on the loan. (fraction of a year) $

45 1. Calculate simple interest 7.5 Interest e.g.4 ADVERTISING EXPENSESA business borrowed $15,500 for 90 days to pay for advertising on the Internet. If the simple interest rate is 16%, find the total amount that must be repaid at the end of 90-day loan. Convert 90 day to year: 90/365 45

46 7.5 Interest 2. Calculate compound interest e.g.5 COMPOUND INTEREST As a gift for their newborn son, the parents open a $5,000 saving account in the baby s name. The interest rate is 3.8%, compounded quarterly. Find the amount of money in the account on the child s first birthday. $5, principal interest amount Q Q Q Q

47 7.5 Interest 2. Calculate compound interest Compound Interest Formula The total amount A in an account is A = P 1 + r n nt Where: P = principal, r = annual rate, t = number of years, n = number of compounding in one year e.g.5 COMPOUNDING DAILY An investor deposited $60,000 in a long term account at 5.8% interest, compounded daily. How much money will she be able to withdraw in 10 years if the principal is to remain in the bank $47,