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Comparing and Ordering Rational Numbers

Comparing and Ordering Rational Numbers. Lesson 5-1. 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. Lesson 5-1. Additional Examples.

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Comparing and Ordering Rational Numbers

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  1. Comparing and Ordering Rational Numbers Lesson 5-1

  2. 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 Lesson 5-1 Additional Examples 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?

  3. 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 Lesson 5-1 Additional Examples Find the LCM of 16 and 36. The LCM of 16 and 36 is 144.

  4. 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 Lesson 5-1 Additional Examples Find the LCM of 5a4 and 15a. The LCM of 5a4 and 15a is 15a4.

  5. 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 Lesson 5-1 Additional Examples Graph and compare the fractions in each pair.

  6. 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 Lesson 5-1 Additional Examples The softball team won of its games and the hockey team won of its games. Which team won the greater fraction of its games? The softball team won the greater fraction of its games.

  7. 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 3 7 2 3 1 4 1 4 56 84 21 84 36 84 21 84 36 84 56 84 Comparing and Ordering Rational Numbers Lesson 5-1 Additional Examples Order , , and from least to greatest.

  8. Fractions and Decimals Lesson 5-2

  9. = 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 Lesson 5-2 Additional Examples 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?

  10. 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 Lesson 5-2 Additional Examples Write each fraction as a decimal. State the block of digits that repeats. a. b.

  11. –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 Lesson 5-2 Additional Examples Write the numbers in order, from least to greatest.

  12. 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 Lesson 5-2 Additional Examples Write 1.72 as a mixed number in simplest form.

  13. 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 Lesson 5-2 Additional Examples Write 0.18 as a fraction in simplest form.

  14. 4 + 2 9 + = Add the numerators. = Simplify. = Simplify. 12 – 5 b Subtract the numerators. – = = Simplify. 4 9 7 b 2 9 4 9 5 b 6 9 2 3 5 b 2 9 12 b 12 b Adding and Subtracting Fractions Lesson 5-3 Additional Examples Find each sum or difference. Simplify if possible. a. + – b.

  15. 1 • 4 – 3 • 6 6• 4 – = Use 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 3 4 1 6 3 4 1 6 2 y 5 16 5 16 Adding and Subtracting Fractions Lesson 5-3 Additional Examples Simplify each difference. – a. – b.

  16. 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. 7 2 1 4 1 2 3 4 1 2 1 4 3 4 6 8 5 4 Adding and Subtracting Fractions Lesson 5-3 Additional Examples Suppose one day you rode a bicycle for 3 hours, and jogged for 1 hours. How many hours did you exercise?

  17. Multiply the numerators. 2 • 5 Multiply the denominators. 3 • 7 • = = Simplify. 2 3 5 7 2 3 5 7 10 21 Multiplying and Dividing Fractions Lesson 5-4 Additional Examples Find • .

  18. 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 3 4 2 3 5 w 3 4 5 w 5 w 2 3 1 2 2 3 15 17 Multiplying and Dividing Fractions Lesson 5-4 Additional Examples a. Find • . b. Find • .

  19. 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 Lesson 5-4 Additional Examples Keesha’s desktop is a rectangle 3 ft long and 1 ft wide. What is the area of her desktop?

  20. 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 10 10 7 Multiplying and Dividing Fractions Lesson 5-4 Additional Examples a. Find ÷ .

  21. Multiply by the reciprocal of the divisor. ÷ = • 4q 9 3 1 1 = • Divide the common factors. 9 4q 1 2 1 = Simplify. = 1 Write as a mixed number. 3 2 1 2 27 8q 9 4q 27 8q 27 8q 4q 9 27 8q Multiplying and Dividing Fractions Lesson 5-4 Additional Examples (continued) b. Find ÷ .

  22. 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. 9 2 1 3 1 2 9 2 3 8 1 2 4 3 9 2 3 8 Multiplying and Dividing Fractions Lesson 5-4 Additional Examples Find 4 ÷ (–3 ).

  23. Using Customary Units of Measurement Lesson 5-5 Additional Examples 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.

  24. 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 Lesson 5-5 Additional Examples Use dimensional analysis to convert 68 fluid ounces to cups.

  25. 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 Lesson 5-5 Additional Examples 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

  26. Problem Solving Strategy: Work Backward Lesson 5-6 Additional Examples 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.

  27. Problem Solving Strategy: Work Backward Lesson 5-6 Additional Examples (continued) You should leave home at 5:45 A.M.

  28. One school recycles about of its waste paper. The student council set a goal of recycling of the school’s waste paper by the end of the year. By how much does the school need to increase its paper recycling to reach the goal? fraction school recycles the increase student goal Words plus is Let n = the increase. Equation + n = 1 3 3 4 3 4 1 3 Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Additional Examples

  29. 1 3 + n = 1 3 1 3 – + n = – Subtract from each side. 3 • 3 – 1 • 4 3 • 4 n = Use 3 • 4 as the common denominator. 9 – 4 12 n = Use the Order of Operations. To meet the student council goal, the school needs to recycle more of its waste paper. n = Simplify. 3 4 3 4 1 3 1 3 5 12 5 12 Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Additional Examples (continued)

  30. Check: Is the answer reasonable? The present fraction of paper waste that is recycled plus the increase must equal the goal. Since + = + = = , the answer is reasonable. 1 3 3 4 9 12 5 12 4 12 5 12 Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Additional Examples (continued)

  31. x – = 2 3 2 3 2 3 x – + = +Add to each side. 1 • 3 + 9 • 2 9 • 3 x = Use 9 • 3 as the common denominator. 3 + 18 27 x = Use the Order of Operations. 7 x = Divide the common factors. 9 x = Simplify. 2 3 7 9 1 9 1 9 2 3 1 9 2 3 21 27 Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Additional Examples Solve x – = .

  32. 1 2 3 5 8 5 Solve q – 6 = –1 . q – 6 = – 1 q – 6 +6 = – 1 + 6Add 6 to each side. 1 2 1 2 1 2 Write mixed numbers as improper fractions. q = – + Use 5 • 2 as the common denominator. –8 • 2 + 5 • 13 5 • 2 q = –16 + 65 10 q = Use the Order of Operations. q = Simplify. q = 4 Write as a mixed number. 3 5 1 2 1 2 3 5 49 10 13 2 9 10 Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Additional Examples

  33. 1 3 7y = 1 7 1 7 1 7 • (7y) = •Multiply each side by , the reciprocal of 7. y = Simplify. 1 3 1 3 1 21 Solving Equations by Multiplying Fractions Lesson 5-8 Additional Examples Solve 7y = .

  34. Solve w = . w = 5 2 5 2 2 5 5 2 • w = • Multiply each side by , the reciprocal of . 1 w = • Divide the common factors. 3 w = Simplify. w = 2 Write as a mixed number. 5 2 1 6 2 5 2 5 2 5 13 15 13 15 13 15 13 15 13 6 Solving Equations by Multiplying Fractions Lesson 5-8 Additional Examples

  35. – c = – – c = –Multiply each side by – , the reciprocal of – . 27 20 20 27 27 20 27 20 1 3 27 • 4 20 • 9 c = Divide common factors. 1 5 = – Simplify. 4 9 3 5 4 9 4 9 20 27 20 27 20 27 Solving Equations by Multiplying Fractions Lesson 5-8 Additional Examples Solve – c = .

  36. 1 2 How many 2 -t trucks can you place on a rail car that has a carrying capacity of 15 t? weight of each truck the number of trucks carrying capacity Words times is Let n = the number of trucks. Equation 2 • n = 15 1 2 Solving Equations by Multiplying Fractions Lesson 5-8 Additional Examples

  37. 2 • n = 15 n = 15 Write 2 as . 1 2 5 2 5 2 • n = • 15 Multiply each side by , the reciprocal of . 2 5 2 5 2 5 3 2 • 15 5 • 1 n = Divide common factors. 1 5 2 1 2 5 2 Solving Equations by Multiplying Fractions Lesson 5-8 Additional Examples (continued) = 6 Simplify. You can place 6 trucks on the rail car.

  38. Powers of Products and Quotients Lesson 5-9 Additional Examples Simplify (3z5)4. (3z5)4 = 34 • (z5)4Raise each factor to the fourth power. = 34 • z5 • 4Use the Rule for Raising a Power to a Power. = 34 • z20 Multiply exponents. = 81z20 Simplify.

  39. Powers of Products and Quotients Lesson 5-9 Additional Examples a. Simplify (–3a)4. (–3a)4 = (–3)4(a)4 = 81a4 b. Simplify –(3a)4. –(3a)4 = (–1)(3a)4 = (–1)(3)4(a)4 = –81a4

  40. Find the area of a square with side length . 2 = = = The area of the square is square units. x 4 x 4 x2 42 x2 16 x2 16 Powers of Products and Quotients Lesson 5-9 Additional Examples A = s2s = length of a side

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