1. Pre-Algebra Lesson 5-1 Comparing and Ordering Rational Numbers

2. 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? 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 Additional Examples

3. Pre-Algebra Lesson 5-1 Find the LCM of 16 and 36. = 144Multiply. 16 = 2 4 36 = 2 2 3 2 Write the prime factorizations. The LCM of 16 and 36 is 144. LCM = 2 4 3 2 Use the greatest power of each factor. Comparing and Ordering Rational Numbers Additional Examples

4. Pre-Algebra Lesson 5-1 Find the LCM of 5a 4 and 15a. 5a 4 = 5 a 4 15a = 3 5 aWrite the prime factorizations. = 15a 4 Multiply. The LCM of 5a 4 and 15a is 15a 4. LCM = 3 5 a 4 Use the greatest power of each factor. Comparing and Ordering Rational Numbers Additional Examples

5. Pre-Algebra Lesson 5-1 Graph and compare the fractions in each pair. is on the left, so <. 3838 3838 7878 b. – 1313, – 1616 is on the right, so >.– 1616 – 1616 – 1313 a. 7878 3838, 3838 7878 – 1313 – 1616 Comparing and Ordering Rational Numbers Additional Examples

6. 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? 6767 7979 Step 1: Find the LCM of 7 and 9. 7 = 7 and 9 = 3 2 LCM = 7 3 2 = 63 Step 2: Write equivalent fractions with a denominator of 63. 6 9 7 9 7 9 7 = 54 63 49 63 = Step 3: Compare the fractions. 54 63 49 63 6767 7979 > >, so The softball team won the greater fraction of its games. Comparing and Ordering Rational Numbers Additional Examples

7. Pre-Algebra Lesson 5-1 Order,, and from least to greatest. 3737 1414 2323 3737 1414 2323 3 12 7 12 1 21 4 21 2 28 3 28 36 84 21 84 56 84 = = = = = = The LCM of 7, 4, and 3 is 84. Use 84 as the common denominator. 21 84 36 84 56 84 1414 3737 2323 <<, so<<. Comparing and Ordering Rational Numbers Additional Examples

8. Pre-Algebra Fractions and Decimals Lesson 5-2

9. Pre-Algebra Fractions and Decimals 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? 1212 = 1 ÷ 2 = 0.5 Since = 0.5 and 0.5 > 0.4, Scott did not fill the tank. 1212 1212 Additional Examples

10. Pre-Algebra Fractions and Decimals Lesson 5-2 Write each fraction as a decimal. State the block of digits that repeats. a. b. 5656 5 ÷ 6 = 0.83333 … Divide. Place a bar over the digit that repeats. = 0.83 5656 = 0.83; the digit that repeats is 3. 7 11 7 ÷ 11 = 0.636363 …Divide. = 0.63Place a bar over the block of digits that repeats. = 0.63; the block of digits that repeats is 63. 7 11 Additional Examples

11. Pre-Algebra Fractions and Decimals Lesson 5-2 Write the numbers in order, from least to greatest. –0.8,,, 0.125 3 12 5454 – –1.25 < –0.8 < 0.125 < 0.25 Compare the decimals. 3 ÷ 12 = 0.25 Change the fractions to decimals. –5 ÷ 4 = –1.25 From least to greatest, the numbers are, –0.8, 0.125, and. 5454 – 3 12 Additional Examples

12. Pre-Algebra Fractions and Decimals Lesson 5-2 Write 1.72 as a mixed number in simplest form. Keep the whole number 1. Write seventy-two hundredths as a fraction. 1.72 = 1 72 100 Divide the numerator and denominator of the fraction by the GCF, 4. 72 ÷ 4 100 ÷ 4 = 1 Simplify. 18 25 1.72 = 1 Additional Examples

13. Pre-Algebra Fractions and Decimals Lesson 5-2 Write 0.18 as a fraction in simplest form. n Let the variable n equal the decimal. = 0.18 As a fraction in simplest form, 0.18 =. 2 11 100n= 18.18Because 2 digits repeat, multiply each side by 10 2, or 100. 18 99 99n 99 = Divide each side by 99. n 18 ÷ 9 99 ÷ 9 = Divide the numerator and denominator by the GCF, 9. 2 11 = Simplify. 100n= 18.18 n= 0.18 – 99n= 18 The Subtraction Property of Equality lets you subtract the same value from each side of the equation. So, subtract to eliminate 0.18. Additional Examples

14. Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Find each sum or difference. Simplify if possible. a. b. 4949 2929 + 4949 2929 + = 4 + 2 9 Add the numerators. 6969 =Simplify. 2323 = 5b5b 12 b – – = 5b5b 12 b 12 – 5 b Subtract the numerators. 7b7b = Simplify. Additional Examples

15. Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Simplify each difference. a. b. 1616 3434 – 2y2y – 5 16 4 – 18 24 = Use the Order of Operations to simplify. –14 24 = Simplify. = –7 24 Simplify. 2y2y – = 5 16 2 16 – 5 y y 16 Rewrite using a common denominator. = 32 – 5y 16y Simplify. 1616 3434 – = 1 4 – 3 6 6 4 Use a common denominator. Additional Examples

16. Pre-Algebra Adding and Subtracting Fractions Lesson 5-3 Suppose one day you rode a bicycle for 3 hours, and jogged for 1 hours. How many hours did you exercise? 1414 1212 You exercised for 4 hours. 3434 1212 3 + = + 1 1414 7272 5454 Write mixed numbers as improper fractions.= 7 4 + 5 2 2 4 Rewrite using a common denominator. = 28 + 10 8 Use the Order of Operations to simplify. = =4 6868 38 8 Write as a mixed number. = 3434 4Simplify. Additional Examples

17. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Find. 2323 5757 2323 5757 = 2 5 Multiply the numerators. = 10 21 Simplify. Multiply the denominators. 3 7 Additional Examples

18. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 3434 2323 a.Find. 2323 3434 3434 = 2323 2323 11 12 Divide the common factors. = 1212 Multiply. b. 5w5w 3w 17 Find. 5w5w 3w 17 = 5w5w 3w 17 1 1 Divide the common factors. 15 17 =Multiply. Additional Examples

19. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Keesha’s desktop is a rectangle 3 ft long and 1 ft wide. What is the area of her desktop? 1212 1212 A Area of a rectangle = length width.= 3 1 1212 1212 = 7272 3232 Write 3 and 1 as improper fractions, and. 1212 1212 7272 3232 Multiply.= 21 4 Write as a mixed number. 1414 = 5 The area of Keesha’s desk is 5 ft 2. 1414 Additional Examples

20. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 a. Find ÷. 3535 7 10 3535 3535 7 10 ÷ = 10 7 Multiply by the reciprocal of the divisor. = 3535 1 10 7 2 Divide the common factors. = 6767 Multiply. Additional Examples

21. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 (continued) b. Find ÷. 94q94q = 3232 Simplify. 27 8q 94q94q ÷ = 4q94q9 Multiply by the reciprocal of the divisor. 27 8q 27 8q = 1 4q94q9 1 Divide the common factors. 27 8q 31 21 = 1 1212 Write as a mixed number. Additional Examples

22. Pre-Algebra Multiplying and Dividing Fractions Lesson 5-4 Find 4 ÷ (–3 ). 1212 3838 4 ÷ (–3 ) = ÷ (– ) Change to improper fractions. 27 8 1212 3838 9292 = (– )Multiply by –, the reciprocal of –. 8 27 8 27 9292 1 1 = – Divide the common factors. 8 27 9292 4 3 = –, or –1Simplify. 4343 1313 Additional Examples

23. Pre-Algebra Using Customary Units of Measurement Lesson 5-5 Choose an appropriate unit of measure. Explain your choice. a. weight of a hummingbird b. length of a soccer field Measure its weight in ounces because a hummingbird is very light. Measure its length in yards because it is too long to measure in feet or inches and too short to measure in miles. Additional Examples

24. Pre-Algebra Using Customary Units of Measurement Lesson 5-5 Use dimensional analysis to convert 68 fluid ounces to cups. 68 fl oz = Use a conversion factor that changes fluid ounces to cups. 68 fl oz 1 1 c 8 fl oz 17 = Divide the common factors and units. 68 fl oz 1 c 8 fl oz 2 17 2 = c Simplify. = 8 c Write as a mixed number. 1212 There are 8 c in 68 fl oz. 1212 Additional Examples

25. Pre-Algebra Using Customary Units of Measurement Lesson 5-5 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? 1212 1212 Since 7 pints > 6 pints, you get more lemonade for your money at Jill’s stand. 1212 3 qt = qt Use a conversion factor that changes quarts to pints 1212 7272 2 pt 1 qt = Divide the common factors and units. 1 2 pt 1 qt 7 qt 2 1 = 7 ptMultiply. Additional Examples

26. Pre-Algebra Problem Solving Strategy: Work Backward 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. Additional Examples

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

28. Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 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? 1313 3434 Words plus is Let n = the increase. Equation + n = fraction school recycles the increase student goal 3434 1313 Additional Examples

29. Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 (continued) + n = 1313 3434 1313 – + n = – Subtract from each side. 1313 3434 1313 1313 n = Use 3 4 as the common denominator. 3 3 – 1 4 3 4 n = Use the Order of Operations. 9 – 4 12 n = Simplify. 5 12 To meet the student council goal, the school needs to recycle more of its waste paper. 5 12 Additional Examples

30. Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 (continued) 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. 9 12 1313 3434 5 12 4 12 5 12 Additional Examples

31. Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Solve x – =. 2323 1919 x – + = +Add to each side. 2323 1919 2323 2323 2323 x – = 2323 1919 x = Use 9 3 as the common denominator. 1 3 + 9 2 9 3 x = Use the Order of Operations. 3 + 18 27 x = Divide the common factors. 21 27 7 9 x = Simplify. 7979 Additional Examples

32. Pre-Algebra Solving Equations by Adding or Subtracting Fractions Lesson 5-7 Solve q – 6 = –1. 1212 3535 q – 6 = – 1 1212 3535 q = Use the Order of Operations. –16 + 65 10 q = Simplify. 49 10 q = –8 2 + 5 13 5 2 Use 5 2 as the common denominator. q = – + Write mixed numbers as improper fractions. 8585 13 2 q = 4 Write as a mixed number. 9 10 1212 q – 6 + 6 = – 1 + 6Add 6 to each side. 3535 1212 1212 1212 Additional Examples

33. Pre-Algebra Lesson 5-8 Solve 7y =. 1313 (7y) = Multiply each side by, the reciprocal of 7. 1313 1717 1717 1717 7y = 1313 y = Simplify. 1 21 Solving Equations by Multiplying Fractions Additional Examples

34. Pre-Algebra Lesson 5-8 5252 w = Divide the common factors. 13 15 1 3 w = 2 Write as a mixed number. 1616 Solve w =. 2525 13 15 w = Multiply each side by, the reciprocal of. 2525 5252 5252 5252 2525 13 15 w = 2525 13 15 w = Simplify. 13 6 Solving Equations by Multiplying Fractions Additional Examples

35. Pre-Algebra Lesson 5-8 Solve – c =. 4949 20 27 – c = 4949 20 27 c = Divide common factors. 27 4 20 9 1 3 1 5 = – Simplify. 3535 – – c = – Multiply each side by –, the reciprocal of –. 4949 27 20 27 20 27 20 27 Solving Equations by Multiplying Fractions Additional Examples

36. Pre-Algebra Lesson 5-8 How many 2 -t trucks can you place on a rail car that has a carrying capacity of 15 t? 1212 Words times is Let n = the number of trucks. Equation 2 n = 15 weight of each truck the number of trucks carrying capacity 1212 Solving Equations by Multiplying Fractions Additional Examples

37. Pre-Algebra Lesson 5-8 (continued) 1212 2 n = 15 n = Divide common factors. 2 15 5 1 3 1 n = 15Write 2 as. 5252 5252 1212 n = 15Multiply each side by, the reciprocal of. 5252 5252 2525 2525 2525 = 6 Simplify. You can place 6 trucks on the rail car. Solving Equations by Multiplying Fractions Additional Examples

38. Pre-Algebra Powers of Products and Quotients Lesson 5-9 Simplify (3z 5 ) 4. (3z 5 ) 4 = 3 4 (z 5 ) 4 Raise each factor to the fourth power. = 3 4 z 5 4 Use the Rule for Raising a Power to a Power. = 3 4 z 20 Multiply exponents. = 81z 20 Simplify. Additional Examples

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

40. Pre-Algebra Powers of Products and Quotients Lesson 5-9 Find the area of a square with side length. x4x4 A = s 2 s = length of a side x242x242 = x 2 16 = The area of the square is square units. x 2 16 = x4x4 2 Additional Examples