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Fractions, Decimals, and Percents

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1. ? ? ? ? ? ? ? Fractions, Decimals, and Percents LESSON 5-1 Problem of the Day Replace the question marks with the correct digits. a. 8 9 + 6. = 15.96 b. 13. 0 – . 4 2 = 4.122 9, 9, 7 6, 4, 9, 8 5-1

2. 90 100 80 100 35 100 25 100 Fractions, Decimals, and Percents LESSON 5-1 Check Skills You’ll Need (For help, go to Lesson 2-2.) • 1. Vocabulary Review A rational number is a number that • can be written in the form ? . • Write each fraction in simplest form. • 3. • 4. 5. Check Skills You’ll Need 5-1

3. 90 ÷ 10 100 ÷ 10 9 10 35 ÷ 5 100 ÷ 5 720 a b 80 ÷ 20 100 ÷ 20 4 5 25 ÷ 25 100 ÷ 25 1 4 Fractions, Decimals, and Percents LESSON 5-1 Check Skills You’ll Need Solutions 1. , b0 2. = 3. = 4. = 5. = 5-1

4. What you think I can write as an equivalent fraction with a denominator of 100.  4 12 100 12 100 = I can rewrite as 12%.  4 3 25 3 25 3 25 3 25 3 • 4 25 • 4 = Multiply the numerator and denominator by 4. 12 100 = Simplify. = 12% Write the fraction as a percent. Fractions, Decimals, and Percents LESSON 5-1 Additional Examples Use mental math to write as a percent. Why it works Quick Check 5-1

5. 25 10 2.5 = Write the decimal as a fraction. 25 • 10 10 • 10 250 100 Write as an equivalent fraction with a denominator of 100. = = = 250% Write the fraction as a percent. 3 1,000 0.003 = Write the decimal as a fraction. 3 ÷ 10 1,000 ÷ 10 0.3 100 Write as an equivalent fraction with a denominator of 100. = = = 0.3% Write the fraction as a percent. Fractions, Decimals, and Percents LESSON 5-1 Additional Examples Write each decimal as a percent. a. 2.5 b. 0.003 Quick Check 5-1

6. 1 2 1 2 1 2 Write the percent as a fraction with a denominator of 100. 7 100 7 % = Rewrite the fraction as division. Write the mixed number as an improper fraction. ÷ 100 = 15 2 15 2 1 100 x = Multiply by the reciprocal of 100. 3 40 Simplify. = Fractions, Decimals, and Percents LESSON 5-1 Additional Examples A brand of cereal supplied 7% of the RDA of sodium. Write this portion of the RDA as a fraction. Quick Check 5-1

7. 27% = 0.27 0.24=0.24 =0.20 Answer: < 0.24 < 27% 1 5 1 5 1 5 Fractions, Decimals, and Percents LESSON 5-1 Additional Examples Order 27%, 0.24, and from least to greatest. Quick Check 5-1

8. 5 8 1 4 1 4 1 4 1 80 5 8 0.58 < 60% < Fractions, Decimals, and Percents LESSON 5-1 Lesson Quiz 1. A cereal supplies 1 % of the RDA for calcium. Write 1 % as a fraction. 2. Write as a percent. 3. Write 1.5 as a percent. 4. Order 60%, 0.58, and . 7 20 35% 150% 5-1

9. Estimating With Percents LESSON 5-2 Problem of the Day The Jackson Country Bird Sanctuary has three times as many owls as hawks. It has 40 hawks and owls in all. How many of each are in the sanctuary? 30 owls, 10 hawks 5-2

10. 2 3 9 10 5 9 3 4 3 7 Estimating With Percents LESSON 5-2 Check Skills You’ll Need (For help, go to Lesson 2-5.) 1. Vocabulary Review The multiplicative inverse of is Find each product. 2. 36 • 3. • 12 4. • 60 5. 81 • Check Skills You’ll Need 5-2

11. 12 1 2 • 12 3 • 1 2 • 12 3 • 1 7 3 2 3 9 10 60 1 9 • 60 10 • 1 9 • 60 10 • 1 36 • 3 1 • 4 36 1 3 4 36 • 3 1 • 4 9 1 Estimating With Percents LESSON 5-2 Check Skills You’ll Need Solutions 1. 2. • = = = 27 3. • = = = 8 4. • = = = 54 5. • = = = 45 4 6 1 1 9 81 1 5 9 81 • 5 1 • 9 81 • 5 1 • 9 1 5-2

12. 74% 0.75 Use a decimal that is close to 74%. Round 158 to a number that is compatible with 0.75. 158 160 74% of 158 0.75 of 160 = 0.75 • 160 Multiply to find 0.75 of 160. = 120 Simplify. Estimating With Percents LESSON 5-2 Additional Examples Estimate 74% of 158 using decimals. Quick Check 5-2

13. 1 5 19% Use a fraction that is close to 19%. 297 300 Round to a number that is compatible with 5. 1 5 of 300 19% of 297 60 1 5 300 1 1 5 = • = 60 Multiply to find of 300. 1 Estimating With Percents LESSON 5-2 Additional Examples A video store rented 297 videos. The customers returned 19% of the videos late. Estimate, using fractions, how many videos were returned late. Quick Check About 60 videos were returned late. 5-2

14. What you think The coat costs about \$60. I know 50% of 60 equals of 60, or 30. Since 5% is one-tenth of 50%, then 5% of 60 is one-tenth of 30, which is 3. So, the amount Dion has saved is about \$30 minus \$3, which is \$27. 1 2 Estimating With Percents LESSON 5-2 Additional Examples Dion is saving for a coat that costs \$59.95. She has saved 45% of the cost. Estimate how much she has saved. 5-2

15. To find 50% of a number, multiply the number by 0.5. 50% of 60 = 0.5 • 60 = 30 Simplify. 5% of a number is one-tenth of 50% of that number. 1 10 5% of 60 = (50% of 60) 1 10 = (30) Substitute. = 3 Simplify. 45% = 50% – 5% 45% of 60 = 30 – 3 = 27 Simplify. Estimating With Percents LESSON 5-2 Additional Examples (continued) Why it works Quick Check So, 45% of \$59.95 is about \$27. 5-2

16. 3 4 • 40 = 30 3 5 (30) = 18 students or 0.6 • 30 = 18 Estimating With Percents LESSON 5-2 Lesson Quiz Show the numbers you use to estimate. Numbers used may vary, Samples are given. 1. Use decimals to estimate 54% of 29. 2. Use fractions to estimate 74% of 38. 3. Estimate a 15% tip on \$8.15. 4. About 60% of 27 students are in the play. 0.5 • 30 = 15 0.1(8) + 0.1(4) = 0.8 + 0.4 = \$1.20 5-2

17. Percents and Proportions LESSON 5-3 Problem of the Day Use graph paper. Design and draw a diagram to determine which has the greater area—a square with sides of 10 cm or a circle with a diameter of 10 cm. Draw a circle within the square to prove that the square has the greater area. 5-3

18. 4 b 20 100 8 12 e 100 240 n 12 5 s 4 75 100 6 y 24 100 Percents and Proportions LESSON 5-3 Check Skills You’ll Need (For help, go to Lesson 4-3.) 1. Vocabulary Review Two equal ratios form a Solve each proportion. 2. = 3. = 4. = 5. = 6. = Check Skills You’ll Need 5-3

19. 2 3 Percents and Proportions LESSON 5-3 Check Skills You’ll Need Solutions 1. proportion 2. 20b = 400; b = 20 3. 12e = 800; e = 66 4. 12n = 1,200; n = 100 5. 100s = 300; s = 3 6. 24y = 600; y = 25 5-3

20. n 240 32 100 = Write a proportion. Write the cross products. 100n = 240 • 32 100n = 7,680 Simplify. 100n 100 7,680 100 = Divide each side by 100. n = 76.8 Simplify. Percents and Proportions LESSON 5-3 Additional Examples Find 32% of 240. Quick Check 5-3

21. n 24 160 100 = Write a proportion. A diagram can help you understand the problem. Write the cross products. 100n = 24 • 160 100n = 3,840 Simplify. Percents and Proportions LESSON 5-3 Additional Examples Brenda saw a blender for \$24 in a bargain store. In a second store, the same blender was 160% of the cost of the blender in the bargain store. Find 160% of \$24. 5-3

22. 100n 100 3,840 100 = Divide each side by 100. n = 38.4 Simplify. Percents and Proportions LESSON 5-3 Additional Examples (continued) The price of the blender in the second store was \$38.40. Quick Check 5-3

23. A diagram can help you understand the problem. 11,500 w 14 100 Write a proportion. = 11,550 • 100 = 14w Write the cross products. 1,155,000 = 14w Simplify. 1,155,000 14 14w 14 = Divide each side by 14. 82,500 = w Use a calculator. Percents and Proportions LESSON 5-3 Additional Examples Suppose 11,550 elementary students make up 14% of a city’s population. What is the population of the city? 5-3

24. Check for Reasonableness 82,500 is about 80,000. Since 14% of 80,000 is 11,200, which is close to 11,155, the answer is reasonable. Percents and Proportions LESSON 5-3 Additional Examples (continued) The population of the city is 82,500 people. Quick Check 5-3

25. A diagram can help you understand the problem. 26 80 p 100 Write a proportion. = 26 • 100 = 80p Write the cross products. 2,600 = 80p Simplify. 2,600 80 80p 80 = Divide each side by 80. 32.5% = p Simplify and insert a percent sign. Percents and Proportions LESSON 5-3 Additional Examples 26 is what percent of 80? 5-3

26. Percents and Proportions LESSON 5-3 Lesson Quiz 1. Find 25% of 160. 2. The price of a music CD is \$12. If the store raises the price to 125% of its current price, what will be the new price of the CD? 3. So far, the sixth grade class has sold 32 tickets to their play. The number represents 20% of the tickets that are available. How many tickets are available? 4. 98 is what percent of 56? 40 \$15 160 175% 5-3

27. Percents and Equations LESSON 5-4 Problem of the Day Write the prime factorization of 364. 22 7  13 5-4

28. Percents and Equations LESSON 5-4 Check Skills You’ll Need (For help, go to Lesson 1-7.) 1. Vocabulary Review Is 2 • 8 = 16 an equation or an expression? Explain. Solve each equation. 2. 0.25p = 10 3. 12.25 = 9.8x 4. 24 = 1.6s 5. 0.64k = 0.02 Check Skills You’ll Need 5-4

29. 0.25p 0.25 10 0.25 12.25 9.8 9.8x 9.8 24 1.6 1.6s 1.6 0.64k 0.64 0.02 0.64 Percents and Equations LESSON 5-4 Check Skills You’ll Need Solutions 1. Equation; it contains an = sign. 2. = ; p = 40 3. = ; x = 1.25 4. = ; s =15 5. = ; k = 0.03125 5-4

30. Words number of problems correct is 84% of 25 Let = the number of problems correct. c Equation c = 84% of 25 c = 21 Simplify. Check for Reasonableness 84% of 25 80% of 25. Since 80% of 25 is 20, which is close to 21, the answer is reasonable. Percents and Equations LESSON 5-4 Additional Examples Quick Check Misha got 84% correct on a 25 problem test. How many did he answer correctly? c = 0.84 • 25 5-4

31. 12 = 0.08 • w Write a percent equation. 150 = w Simplify. 12 0.08 0.08w 0.08 = Divide each side by 0.08. Percents and Equations LESSON 5-4 Additional Examples Use an equation. 12 is 8% of what number? Quick Check 5-4

32. Percents and Equations LESSON 5-4 1. Find 81% of 110. 2. You buy a book for \$17.80. Sales tax is 8%. What is the sales tax cost of the book? 3. 45 is 75% of what number? 4. Find what percent 68 is of 80. Lesson Quiz 89.1 \$1.42 60 85% 5-4

33. Percent of Change LESSON 5-5 Problem of the Day Dan, Susan, Monica, and Jose want to talk on the phone once to each of the others. How many telephone calls will be made? 6 calls 5-5

34. 9 8 6 22 4 15 11 3 Percent of Change LESSON 5-5 Check Skills You’ll Need (For help, go to Lesson 5-1.) 1. Vocabulary Review A is a ratio that compares a number to 100. Write each fraction as a percent. Round to the nearest tenth of a percent. 2. 3. 4. 5. Check Skills You’ll Need 5-5

35. 9 8 6 22 4 15 11 3 Percent of Change LESSON 5-5 Check Skills You’ll Need Solutions 1. percent 2. = 1.125; 112.5% 3. = 0.27; 27.3% 4. = 0.26; 26.7% 5. = 3.6; 366.7% 5-5

36. 10.50 2.50 amount of change P = original amount Use a calculator to divide. 10.50 2.50 4.2 = 420% Write the decimal as a percent. Check for Reasonableness 420% of 2.5 400% of 3. Since 400% of 3 = 12, which is close to 13, the answer is reasonable. Percent of Change LESSON 5-5 Additional Examples Ten years ago, Max’s comic book was worth \$2.50. Now it is worth \$13. Find the percent of increase in value. amount of change = 13 – 2.50 = 10.50 The percent of increase in value is 420%. Quick Check 5-5

37. 8 ft 4 in. = 8 • 12 + 4 = 100 in. 9 ft 2 in. = 9 • 12 + 2 = 110 in. Write measures in the same units. 10 100 amount of change P = original amount = 0.1 Simplify. = 10% Write the decimal as a percent. Percent of Change LESSON 5-5 Additional Examples Andre changed the height of his basketball hoop from 8 ft 4 in. to 9 ft 2 in. Find the percent of increase. amount of change = 110 – 100 = 10 Quick Check The height of the basketball hoop increased by 10%. 5-5

38. 60,989 557,927 amount of change P = original amount = 0.109313584 Use a calculator. Write the decimal as a percent. Round to the nearest tenth. 11% Percent of Change LESSON 5-5 Additional Examples In 1980, the population of a city was 557,927. In 1990, its population was 496,938. Find the percent of decrease. Round to the nearest tenth. amount of change = 557,927 – 496,938 = 60,989 Quick Check The population decreased by about 11%. 5-5

39. Percent of Change LESSON 5-5 Lesson Quiz Round to the nearest whole percent. 1. 81 people attended last year’s annual picnic, and 93 people attended this year’s picnic. What is the percent of increase? 2. The speed limit on a highway was 55 miles per hour last year. This year the speed limit was increased to 65 miles per hour. What is the percent increase in the speed limit? 3. The population in Arthur County, Nebraska dropped from 462 in 1990 to 444 in 2000. What was the percent of decrease? about 15% about 18% about 4% 5-5

40. Markup and Discount LESSON 5-6 Problem of the Day Round to the underlined place. a. 0.09972 b. 0.109 c. 17.51 d. 0.998 0.0997 0.1 18 1.00 5-6

41. Markup and Discount LESSON 5-6 Check Skills You’ll Need (For help, go to Lesson 5-4.) 1. Vocabulary Review A relates a part to the whole. Use an equation to solve each problem. 2. What number is 16% of 25? 3. Find 80% of 250. 4. 33 is 3% of what number? 5. 0.55% of what number is 77? Check Skills You’ll Need 5-6

42. Markup and Discount LESSON 5-6 Check Skills You’ll Need Solutions 1. percent 2. 0.16 • 25 = n; n = 4 3. 0.80 • 250 = n; n = 200 4. 0.03 • w = 33; w = 1,100 5. 0.0055 • w = 77; w = 14,000 5-6

43. = \$3.36 – \$2.10 Substitute. = \$1.26 Subtract. 1.26 2.10 markup percent of markup = store’s cost = 0.6 Write the fraction as a decimal. = 60% Write the decimal as a percent. Markup and Discount LESSON 5-6 Additional Examples Find the percent of markup for a stapler costing the school store \$2.10 and selling for \$3.36. markup = selling price – store’s cost Quick Check 5-6

44. 25% of \$40 equals the markup. 0.25 • 40 = 10 Multiply to find the markup. = \$10 \$40 + 10 = \$50 store’s cost + markup = selling price The store sells the skirt for \$50. Markup and Discount LESSON 5-6 Additional Examples A store sells a skirt that costs the store \$40 and marks up the price 25%. What is the selling price for this skirt? Method 1 Find the markup first. Then find the selling price. 5-6

45. 125% of \$40 equals the selling price. 1.25 • 40 = 50 Multiply to find the selling price. = \$50 Markup and Discount LESSON 5-6 Additional Examples (continued) Method 2 Find the selling price directly. The selling price equals 100% of the store’s cost plus a markup of 25% of the store’s cost. So, the selling price of the skirt is 100% + 25%, or 125%, of \$40. The store sells the skirt for \$50. Quick Check 5-6

46. 0.35 • 94.99 = 33.2465 Multiply to find the discount. Round to the nearest cent. = \$33.25 94.99 – 33.25 = 61.74 regular price – discount = sale price Markup and Discount LESSON 5-6 Additional Examples A shoe store advertises a 35%-off sale. What is the sale price of shoes that regularly cost \$94.99? Method 1 Find the discount first. Then find the sale price. 35% of \$94.99 equals the discount. The sale price is \$61.74. 5-6

47. 65% of \$94.99 equals the sale price. 0.65 • 94.99 = 61.744 Multiply to find the sale price. = \$61.74 Round to the nearest cent. Markup and Discount LESSON 5-6 Additional Examples (continued) Method 2 Find the sale price directly. The sale price equals 100% of the regular price minus a discount of 35% of the regular price. The sale price is 100% – 35%, or 65%, of \$94.99. The sale price is \$61.74. Quick Check 5-6

48. r –(0.25 • r) = 6 Substitute. Write the percent as a decimal. 0.75r = 6 Combine like terms: r – 0.25r = 0.75r. 0.75r 0.75 6 0.75 Divide each side by 0.75. = r = 8 Simplify. Markup and Discount LESSON 5-6 Additional Examples You buy a CD at the sale price of \$6. This is 25% off the regular price. Find the regular price of the CD. regular price – 25% of regular price = sale price Let r = the regular price. Quick Check The regular price of the CD is \$8. 5-6

49. Markup and Discount LESSON 5-6 Lesson Quiz 1. A pair of shoes costs the store \$40. The store sells them for \$65. What is the percent markup? 62.5% 2. A school service club sells calendars. Each calendar costs the club \$5.50. The club marks up the price 80%. What is the selling price of each calendar? \$9.90 3. A sweater regularly sells for \$49. It is on sale for 20% off. What is the sale price? \$39.20 4. You buy a baseball cap for \$13. This price is 35% off the regular price. Find the regular price. \$20.00 5-6

50. Simple Interest LESSON 5-7 Problem of the Day Luis’s mother has two older sisters who are twins and 6 yr older than she is. She has a brother who is half her age. The sum of all their combined ages is 145 yr. How old is each? mother 38, older twins 44, brother 19 5-7