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Comparing and Ordering Rational Numbers

Comparing and Ordering Rational Numbers. PRE-ALGEBRA LESSON 5-1. Write the prime factorization of 364. 2 2 x 7 x 13. 5-1. Comparing and Ordering Rational Numbers. PRE-ALGEBRA LESSON 5-1. (For help, go to Lesson 4-3.). Write the prime factorization of each number.

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Comparing and Ordering Rational Numbers

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  1. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Write the prime factorization of 364. 22 x 7 x 13 5-1

  2. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 (For help, go to Lesson 4-3.) Write the prime factorization of each number. 1. 20 2. 125 3. 45 4. 186 5. 621 6. 1,575 Check Skills You’ll Need 5-1

  3. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Solutions 1. 20 2. 125 3. 45 2 • 10 5 • 25 3 • 15 2 • 2 • 5 5 • 5 • 5 3 • 3 • 5 20 = 22 • 5 125 = 53 45 = 32 • 5 4. 186 5. 621 6. 1,575 2 • 93 3 • 207 3 • 525 2 • 3 • 31 3 • 3 • 69 3 • 3 • 175 186 = 2 • 3 • 31 3 • 3 • 3 • 23 3 • 3 • 5 • 35 621 = 33 • 23 3 • 3 • 5 • 5 • 7 1,575 = 32 • 52 • 7 5-1

  4. 7, 14, 21, 28, 35, 42, . . . List the multiples of 7. 3, 6, 9, 12, 15, 18, 21, . . . List the multiples of 3. The LCM is 21. In 21 days both teams will have games again. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Today, the school’s baseball and soccer teams had games. The baseball team plays every 7 days. The soccer team plays every 3 days. When will the teams have games on the same day again? Quick Check 5-1

  5. 16 = 24 Write the prime factorizations. 36 = 22 • 32 LCM = 24 • 32 Use the greatest power of each factor. = 144 Multiply. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Find the LCM of 16 and 36. The LCM of 16 and 36 is 144. Quick Check 5-1

  6. 5a4 = 5 • a4 15a = 3 • 5 • a Write the prime factorizations. LCM = 3 • 5 • a4 Use the greatest power of each factor. = 15a4 Multiply. Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Find the LCM of 5a4 and 15a. The LCM of 5a4 and 15a is 15a4. Quick Check 5-1

  7. 7 8 3 8 , a. 3 8 7 8 3 8 3 8 7 8 is on the left, so < . 1 3 1 6 , – – b. 1 3 1 6 – – 1 6 1 6 – – is on the right, so > . 1 3 – Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Graph and compare the fractions in each pair. Quick Check 5-1

  8. Step 1  Find the LCM of 7 and 9. 7 = 7 and 9 = 32 LCM = 7 • 32 = 63 Step 2  Write equivalent fractions with a denominator of 63. = 6 • 9 7 • 9 = 7 • 7 9 • 7 Step 3 Compare the fractions. > , so > 7 9 6 7 7 9 6 7 54 63 49 63 54 63 49 63 Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 The softball team won of its games and the hockey team won of its games. Which team won the greater fraction of its games? Quick Check The softball team won the greater fraction of its games. 5-1

  9. 3 • 12 7 • 12 = = The LCM of 7, 4, and 3 is 84. Use 84 as the common denominator. 1 • 21 4 • 21 = = 2 • 28 3 • 28 = = < < , so < < . 1 4 3 7 3 7 2 3 2 3 2 3 3 7 1 4 1 4 56 84 21 84 36 84 21 84 36 84 56 84 Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Order , , and from least to greatest. Quick Check 5-1

  10. 8 10 3 16 3 16 – – < < 3 16 8 10 3 16 Comparing and Ordering Rational Numbers PRE-ALGEBRA LESSON 5-1 Find the LCM of each pair of numbers. 1. 8, 6 2. 12, 16 3. Compare and order , – , and – from least to greatest. 24 48 5-1

  11. Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Without actually dividing, which expression is the largest? 6.1 ÷2.8 12.6 ÷ 1.1 9.305 ÷ 2.99 12.6 ÷ 1.1 5-2

  12. Fractions and Decimals PRE-ALGEBRA LESSON 5-2 (For help, go to Skills Handbook page 728.) Write the decimals in order from least to greatest. 1. 2.41, 0.241, 24.1, 12.4 2. 1.030, 13.03, 1.300, 1.003 3. 0.1, 0.01, –0.1, –0.01 Check Skills You’ll Need 5-2

  13. Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Solutions 1. 0.241, 2.41, 12.4, 24.1 2. 1.003, 1.030, 1.300, 13.03 3. –0.1, –0.01, 0.01, 0.1 5-2

  14. = 1 ÷ 2 = 0.5 Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank. 1 2 1 2 1 2 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 The fuel tank of Scott’s new lawn mower holds gal of gasoline. Scott poured 0.4 gal into the tank. Did Scott fill the tank? Quick Check 5-2

  15. 5 ÷ 6 = 0.83333 … Divide. = 0.83 = 0.83; the digit that repeats is 3. Place a bar over the digit that repeats. 7 ÷ 11 = 0.636363 … Divide. = 0.63 Place a bar over the block of digits that repeats. = 0.63; the block of digits that repeats is 63. 5 6 5 6 7 11 7 11 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Write each fraction as a decimal. State the block of digits that repeats. a. b. Quick Check 5-2

  16. –0.8, , , 0.125 3 ÷ 12 = 0.25 Change the fractions to decimals. –5 ÷ 4 = –1.25 – – –1.25 < –0.8 < 0.125 < 0.25  Compare the decimals. From least to greatest, the numbers are , –0.8, 0.125, and . 5 4 5 4 3 12 3 12 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Write the numbers in order, from least to greatest. Quick Check 5-2

  17. 1.72 = 1 Keep the whole number 1. Write seventy-two hundredths as a fraction. 72 ÷ 4 100 ÷ 4 = 1 Divide the numerator and denominator of the fraction by the GCF, 4. 1.72 = 1 Simplify. 18 25 72 100 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Write 1.72 as a mixed number in simplest form. Quick Check 5-2

  18. Let the variable n equal the decimal. n = 0.18 100n = 18.18 Because 2 digits repeat, multiply each side by 102, or 100. 100n = 18.18 The Subtraction Property of Equality lets you subtract the same value from each side of the equation. So, subtract to eliminate 0.18. – n = 0.18 99n = 18 Divide each side by 99. = Divide the numerator and denominator by the GCF, 9. 18 ÷ 9 99 ÷ 9 n = = Simplify. As a fraction in simplest form, 0.18 = . 2 11 18 99 2 11 99n 99 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 Write 0.18 as a fraction in simplest form. Quick Check 5-2

  19. –0.35, – , 0625, 6 5 1 3 1 25 7 33 6 6 5 1 3 17 20 Fractions and Decimals PRE-ALGEBRA LESSON 5-2 1. Order , – , 0.625, –0.35 from least to greatest. 2. Write as a decimal. Write each decimal as a fraction or mixed number in simplest form. 3. 6.04 4. 0.21 0.85 5-2

  20. Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Five consecutive multiples of 6 have a sum of 270. What are they? 42, 48, 54, 60, 66 5-3

  21. Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 (For help, go to Lesson 5-1.) Find the LCM of each group of numbers or expressions. 1. 4, 8 2. 9, 18 3. 2n, 10 4. 3, 6, 9 5. 8, 5, 4 6. 10, n Check Skills You’ll Need 5-3

  22. Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Solutions 1. 4 = 222. 9 = 323. 2n = 2 • n 8 = 23 18 = 2 • 32 5 = 5 LCM = 23 LCM = 2 • 32 LCM = 2 • 5 • n = 8 = 18 = 10n 4. 3 = 3 5. 8 = 236. 10 = 2 • 5 6 = 2 • 3 5 = 5 n = n 9 = 32 4 = 22 LCM = 2 • 5 • n LCM = 2 • 32 LCM = 23 • 5 = 10n = 18 = 40 5-3

  23. 4 + 2 9 + = Add the numerators. = Simplify. = Simplify. 12 – 5 b Subtract the numerators. – = = Simplify. 4 9 5 b 2 9 4 9 7 b 6 9 2 3 5 b 2 9 12 b 12 b Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Find each sum or difference. Simplify if possible. a. + – b. Quick Check 5-3

  24. 1 • 4 – 3 • 6 6• 4 – = Rewrite using a common denominator. 4 – 18 24 = Use the Order of Operations to simplify. = Simplify. –14 24 = Simplify. –7 24 2 • 16 – 5 • y y •16 – = Rewrite using a common denominator. = Simplify. 32 – 5y 16y 2 y 1 6 3 4 1 6 3 4 2 y 5 16 5 16 Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Simplify each difference. – a. – b. Quick Check 5-3

  25. 3 1 + = + Write mixed numbers as improper fractions. 7 • 4 + 5 • 2 2 • 4 = Rewrite using a common denominator. = Use the Order of Operations to simplify. 28 + 10 8 = 38 8 Write as a mixed number. = 4 = 4 Simplify. You exercised for 4 hours. 3 4 1 4 1 2 3 4 1 2 1 4 6 8 7 2 5 4 Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Suppose one day you rode a bicycle for 3 hours, and jogged for 1 hours. How many hours did you exercise? Quick Check 5-3

  26. 3 5 8 1 2 8c + 21 14c 14 15 – 4 7 1 8 3 5 1 3 3 4 2 24 10 24 3 2c Adding and Subtracting Fractions PRE-ALGEBRA LESSON 5-3 Find each sum or difference. 1.2. 5 – 2 3. – – 4. + + 5-3

  27. 1 5 7 8 21 5 31 8 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Write each number as an improper fraction. a. 4 b. 3 5-4

  28. 1 3 4 9 4 5 1 7 7 8 3 10 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 (For help, go to Skills Handbook page 738.) Write each mixed number as an improper fraction. 1. 2 2. 3 3. 1 4. 4 5. 7 6. 5 Check Skills You’ll Need 5-4

  29. Solutions 1.2.3. 3 = 3 + = + 30 + 3 10 = = 13 9 3 = 1 = 2 = 2 + 1 = 1 + = + = + 6 + 1 3 9 + 4 9 = 2 = 7 3 4 9 1 3 1 3 6 3 1 3 4 9 1 3 4 9 9 9 4 9 3 10 3 10 30 10 3 10 33 10 3 10 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 5-4

  30. 56 + 7 8 35 + 1 7 20 + 4 5 5 = 5 + 4 = 4 + 7 = 7 + 1 7 1 7 4 5 4 5 4 5 1 7 7 8 7 8 7 8 1 7 7 8 4 5 63 8 56 8 35 7 20 5 24 5 36 7 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Solutions (continued) 4.5.6. = + = + = + = = = 4 = 7 = 5 = 5-4

  31. Multiply the numerators. 2 • 5 Multiply the denominators. 3 • 7 • = = Simplify. 5 7 2 3 5 7 2 3 10 21 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Find • . Quick Check 5-4

  32. 1 1 • = • Divide the common factors. 2 1 = Multiply. 3w 17 3w 17 3w 17 1 • = • Divide the common factors. 1 = Multiply. 2 3 3 4 5 w 2 3 5 w 2 3 5 w 1 2 3 4 3 4 2 3 15 17 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 a. Find • . b. Find • . Quick Check 5-4

  33. 7 2 1 2 1 2 3 2 A = 3 • 1 Area of a rectangle = length • width. 1 2 1 2 7 2 3 2 = Write 3 and 1 as improper fractions, and . • = Multiply. = 5 Write as a mixed number. The area of Keesha’s desk is 5 ft2. 1 2 1 2 1 4 1 4 21 4 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Keesha’s desktop is a rectangle 3 ft long and 1 ft wide. What is the area of her desktop? Quick Check 5-4

  34. Multiply by the reciprocal of the divisor. ÷ = • 10 7 2 = • Divide the common factors. 7 10 1 = Multiply. 3 5 6 7 3 5 3 5 3 5 7 20 10 7 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 a. Find ÷ . 5-4

  35. Multiply by the reciprocal of the divisor. ÷ = • 4q 9 3 1 1 = • Divide the common factors. 9 4q 1 2 1 = Multiply. = 1 Write as a mixed number. 3 2 1 2 9 4q 27 8q 27 8q 27 8q 4q 9 27 8q Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 (continued) b. Find ÷ . Quick Check 5-4

  36. 27 8 4 ÷ (–3 ) = ÷ (– ) Change to improper fractions. 27 8 = • (– ) Multiply by – , the reciprocal of – . 8 27 8 27 4 1 8 27 = • – Divide the common factors. 1 3 = – , or –1 Simplify. 3 8 9 2 9 2 1 3 1 2 9 2 3 8 1 2 4 3 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Find 4 ÷ (–3 ). Quick Check 5-4

  37. – ÷ • 12g 25 ÷ 1 5 , or 3 1 3 6 5 1 5 7 32 10 3 1 20 21 20 , or 1 , or 1 – 3 8 3 8 11 16 5 2g 7 12 3z 4 5z 7 Multiplying and Dividing Fractions PRE-ALGEBRA LESSON 5-4 Simplify each expression. 1.2. 3. 4. • 5-4

  38. 1 2 Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 In 2002, a disc jockey broadcast her show for 27 h each week. How many hours did she broadcast that year? 1,430 h 5-5

  39. 1 4 8 1 1 4 1 2 1 3 1 12 16 1 Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 (For help, go to Lesson 5-4.) Find each product. 1. 4 • 2. 3 • 3. 1 • 4. 4 • Check Skills You’ll Need 5-5

  40. 4 • = • 3 • = • = = = 4 • = • 1 • = • = = = = 1 4 1 1 4 9 6 1 2 3 2 1 4 5 4 8 1 1 2 1 3 9 2 1 3 1 3 8 1 1 12 1 12 16 1 16 1 4 12 40 4 13 4 208 4 Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 Solutions 1. 2. = 52 3.4. = 10 5-5

  41. Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 Choose an appropriate unit of measure. Explain your choice. a. weight of a hummingbird • Measure its weight in ounces because a hummingbird is very light. b. length of a soccer field • Measure its length in yards because it is too long to measure in feet or inches and too short to measure in miles. Quick Check 5-5

  42. Use a conversion factor that changes fluid ounces to cups. 1 c 8 fl oz 68 fl oz = • 17 68 fl oz • 1 c 8 fl oz = Divide the common factors and units. 2 17 2 = c Simplify. 68 fl oz 1 = 8 c Write as a mixed number. There are 8 c in 68 fl oz. 1 2 1 2 Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 Use dimensional analysis to convert 68 fluid ounces to cups. Quick Check 5-5

  43. Use a conversion factor that changes quarts to pints 3 qt = qt • 1 = • Divide the common factors and units. 1 = 7 pt Multiply. Since 7 pints > 6 pints, you get more lemonade for your money at Jill’s stand. 1 2 1 2 7 2 7 qt 2 2 pt 1 qt 2 pt 1 qt Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 1 2 Fred’s fruit stand sells homemade lemonade in 6 -pint bottles for $1.99. Jill’s fruit stand stand sells homemade lemonade in 3 -qt containers for the same price. At which stand do you get more lemonade for your money? 1 2 Quick Check 5-5

  44. 1 2 Using Customary Units of Measurement PRE-ALGEBRA LESSON 5-5 Choose an appropriate unit of measure. Explain your choice. 1. weight of a dog 2. length of a football field 3. Use dimensional analysis to convert 1 mi to ft. Pounds; dogs are too heavy to measure in ounces and too light to measure in tons. Yards; it is too long to measure in feet or inches and too short to measure in miles. 7,920 ft 5-5

  45. equal: and , and , and 6 7 4 5 36 42 16 20 6 15 18 20 Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 Use mental math to decide whether or not each pair of fractions is yes, yes, no 5-6

  46. 7 3 1 2 1 4 1 3 1 5 3 5 3 7 3 3 3 5 3 7 3 5 3 3 5 3 13 3 3 13 1 10 3 13 Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 (For help, go to Lesson 5-2.) Order from least to greatest. 1.2. 3.4. , , , , , , , – – – – , , , , , , Check Skills You’ll Need 5-6

  47. , , , , 7 3 1 5 1 3 5 3 3 3 1 2 3 7 1 4 5 3 3 5 3 3 1 3 3 3 1 5 3 5 3 3 1 4 3 5 3 5 7 3 3 7 7 3 3 5 3 7 1 2 3 3 13 3 13 3 1 10 1 10 3 13 3 13 3 13 3 5 Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 Solutions 1.2. , , , 0.50, 0.3, 0.25, 0.20, 0.10 2.3, 1.6, 4.3, 1 , , , , , , , 3.4. – – – – , , , , , , 0.43, 0.6, 0.23, 1 –0.43, –0.6, –0.23, 1 – – – – , , , , , , 5-6

  48. Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 Your flight leaves the airport at 10:00 A.M. You must arrive 2 hours early to check your luggage. The drive to the airport takes about 90 minutes. A stop for breakfast takes about 30 minutes. It will take about 15 minutes to park and get to the terminal. At what time should you leave home? Move the hands of the clock to find the time you should leave home. Write the starting time for each event. 5-6

  49. Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 (continued) You should leave home at 5:45 A.M. Quick Check 5-6

  50. Problem Solving Strategy: Work Backward PRE-ALGEBRA LESSON 5-6 Solve by working backward. 1. Jason has a total of 212 sheep. Last spring, 52 lambs were born, but 9 died. How many sheep did he have before the lambs were born? 2. The bus picks up Tom every morning at 7:20 A.M. Tom walks his dog every morning before school for 36 minutes. It takes him 25 minutes to get ready for school and 7 minutes to walk to the bus stop. At what time does Tom need to wake up in order to get to the bus stop on time? 169 sheep 6:12 A.M. 5-6

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