UNIT TWO: Prealgebra in a Technical World 2.5 Dividing Integers & Order of Operations SWBAT 1. Divide integers. 2. Write exact quotients using mixed numbers. 3. Use order of operations to simplify integer expressions. 4. Solve applied problems using integers. Dividing Integers Once people had started multiplying, they developed an operation for finding a missing factor. We call that operation division. Another name for the missing factor is the quotient. The quotient is the result when we divide one number by another. 3 For instance, we know that 2) 6 , because 2 ∙ 3 = 6. What about −2) − 6 ? Since (−2) + (−2) + (−2) = −6, the answer is 3. So, 3 is the missing factor in the multiplication (−2) ∙ ? = −6. The good news is that when the signs are the same, just as in multiplication, the answer in division is positive. RULE: To divide two numbers with the same sign, divide their absolute values. The quotient is positive. What about division problems with different signs? To divide 2) − 6 we check for the number that makes this multiplication true: 2 ∙ ? = −6. In this equation, the missing factor must be −3. Similarly, when we divide −2) 6 , we are looking for the missing factor that makes (−2) ∙ ? = 6 true and that number is −3. The rule of signs for division matches the rule for multiplication: RULE: To divide two numbers with different signs, divide their absolute values. The quotient is negative. 137 138 SECTION 2.5 Dividing Integers & Order of Operations If we use (+) to represent any positive number and (−) to represent any negative number, then: Multiplication Division (+) ∙ (+) = (+) (+) ÷ (+) = (+) (−) ∙ (−) = (+) (−) ÷ (−) = (+) (+) ∙ (−) = (−) (+) ÷ (−) = (−) (−) ∙ (+) = (−) (−) ÷ (+) = (−) Check Point 1 Tell whether the quotient is positive (+) or negative(−), then divide. Division Sign Quotient Division a. 𝟏𝟒 ÷ (−𝟐) d. (−𝟓𝟒) ÷ (−𝟔) b. (−𝟒𝟎) ÷ 𝟖 e. c. (−𝟑𝟐) ÷ (−𝟖) Sign Quotient 𝟐𝟕 ÷ (−𝟗) f. (−𝟖𝟏) ÷ (−𝟗) RULE: 𝟎 ÷ 𝒂 = 𝟎, and 𝒂 ÷ 𝟎 is undefined. Dividing into 0 is different from dividing by 0. We can divide into 0, 0 ÷ 4 = 0, because 4 ∙ 0 = 0. (If I have no cookies and four people, each person can have 0 cookies.) However, we cannot divide by 0, 4 ÷ 0 = ____ does not have an answer. The missing factor in 0 · ? = 4 , does not exist. We say that dividing by zero is “undefined.” Order of Operations The order of operations is used for all real numbers. In this section we practice both using the correct order of operations and simplifying integer expressions. UNIT TWO: Prealgebra in a Technical World Example 1: 5 + (−4)2 + 21 ÷ (−15 + 8) Think it through: 5 + (−4)2 + 21 ÷ (−15 + 8) Parentheses first 5 + (−4)2 + 21 ÷ (−7) Exponents 5 + 16 + 21 ÷ (−7) Multiplication and Division left to right 5 + 16 + −3 Addition and Subtraction left to right 21 + −3 Addition (and we subtract to find the sum!) ANSWER: 𝟓 + (−𝟒)𝟐 + 𝟐𝟏 ÷ (−𝟏𝟓 + 𝟖) = 𝟏𝟖 Example 2: Simplify 3 + (−6)2 ÷ 22 ∙ 7. Think it through: 3 + (−6)2 ÷ 22 ∙ 7 3 + 36 ÷ 4 ∙ 7 Exponents left to right (when possible, simplify all) Multiplication and Division left to right 3+9∙7 Multiplication and Division left to right 3 + 63 Addition and Subtraction left to right ANSWER: 𝟑 + (−𝟔)𝟐 ÷ 𝟐𝟐 ∙ 𝟕 = 𝟔𝟔 139 140 SECTION 2.5 Dividing Integers & Order of Operations Example 3: Simplify −2+(5−1)(−3+7) 9−2 . Think it through: Hint: The fraction bar is treated like parentheses. Simplify the numerator and denominator first, and then divide. −2+(𝟓−𝟏)(−𝟑+𝟕) Groupings first (when possible, simplify all) 9−2 −2+(4)(4) Multiply and Divide left to right 9−2 −2+𝟏𝟔 Add and Subtract (when possible, simplify all) 9−2 14 Divide 7 ANSWER: −2+(5−1)(−3+7) 9−2 Check Point 2 Simplify −6 − (−14 ÷ 7) ∙ 32 = 𝟐 UNIT TWO: Prealgebra in a Technical World Check Point 3 Jasmine and Joanie both simplified the expression 3 − 36 ÷ (−9 + 13) ∙ (−3)2 + 1. Jasmine’s work is shown below. Joanie’s answer is −77, and she checked her work. Jasmine has made a mistake. Tell what the error is and on which row Jasmine made the error. 3 − 36 ÷ (−9 + 13) ∙ (−3)2 + 1 (This is the original problem.) (1) 3 − 36 ÷ (4) ∙ (−3)2 + 1 (2) 3 − 36 ÷ (4) ∙ 9 + 1 (3) 3 − 36 ÷ 36 + 1 (4) (5) 3−1 +1 3 (This is Jasmine’s answer.) a. The error occurs in Row __________. b. Jasmine's error is __________________________________________________________ ____________________________________________________________________________ ____________________________________________________________________________ Writing Exact Quotients Using Mixed Numbers The rules for signs are the same when the exact quotient in our division problem is written as a mixed number (as an integer and a fraction). When we divide exactly, we leave no remainders. Instead we write a fraction to complete the division. STUDY SKILLS: If you have forgotten how to do long division, now is the time to review. Ask your instructor for extra problems, with solutions. If you need instructions or tutoring, use the Tutoring Center and the Web! 141 142 SECTION 2.5 Dividing Integers & Order of Operations Example 4 Calculate −51 ÷ 8 exactly. Think it through: If you have forgotten how to write exact quotients using fractions, or you have forgotten how to use the long division algorithm, practice many of these problems until you can divide automatically. You can find many more practice problems, with answers so that you can check your work, at the class Web site. Order of Operations & Applied Problems In the last section we estimated the Fahrenheit temperature when given temperature in degrees Celsius. That formula is written 𝐹 ≈ 2𝐶 + 30 or 2𝐶 + 30 ≈ 𝐹 . In the table at right we have used the formula Celsius 0° 20° 40° to convert several Celsius readings to estimated Fahrenheit degrees. We can put an order to the operations in this formula: 2𝐶 + 30 ≈ 𝐹 degrees Celsius → times 2 → plus 30 ≈ degrees Fahrenheit 𝟐𝑪 + 𝟑𝟎 : 2(0) + 30 2(20) + 30 2(40) + 30 Fahrenheit ≈ 30° ≈ 70° ≈ 110° UNIT TWO: Prealgebra in a Technical World If we reverse this order, and perform the opposite operation, we create a formula that reverses the process and estimates the Celsius temperature when given the temperature in degrees Fahrenheit. degrees Fahrenheit → minus 30 →divided by 2 ≈ degrees Celsius (𝐹 − 30) ÷ 2 ≈ 𝐶 In the new table, we use this new formula to estimate Celsius temperature when given temperature in degrees Fahrenheit. In algebra you will study this process of finding different formulas, Fahrenheit 30° 70° 110° (𝑭 − 𝟑𝟎) ÷ 𝟐 ≈ 𝑪 (30 − 30) ÷ 2 (70 − 30) ÷ 2 (110 − 30) ÷ 2 Celsius ≈ 0° ≈ 20° ≈ 40° but here we will simply use our new formula: 𝐶 ≈ (𝐹 − 30) ÷ 2. Example 5: Today a child dies every 15 seconds because of a water-related disease1, yet water can easily be pasteurized. All of the viruses, bacteria, protozoa and parasites can be killed by bringing water to just 150°F. Most of the developing world uses Celsius temperature readings. What is the approximate temperature needed to pasteurize water in degrees Celsius? Understand: Find the degrees Celsius for 150℉. Plan: Use the formula 𝐶 ≈ (𝐹 − 30) ÷ 2 with 𝐹 = 150 Solve: Substitute 150 for 𝐹 and simplify result: 𝐶 ≈ (150 − 30) ÷ 2 = 120 ÷ 2 = 60. Check: Approximate temperature needed to pasteurize water is 𝟔𝟎°𝑪. Compared with result in the table on the last page, 60°𝐶 is reasonable. The calculations are most probably accurate. 1 Water. (2009) Web site http://water.org/facts Accessed 1 August 2009. 143 144 SECTION 2.5 Dividing Integers & Order of Operations Check Point 4 A greenhouse is to be maintained between 74℉ and 78℉ to attain the best germination rate for tomato seeds. The greenhouse venting system was made in Canada and must be set using degrees Celsius. What is a good approximation for a Celsius temperature setting for tomato seed germination that will make sure the temperature is within the given Fahrenheit range? ______________________________________________________________________________ ______________________________________________________________________________ Sentence:______________________________________________________________________ ______________________________________________________________________________ Check Point 5 Chloe is in the National Guard and is stationed in Bagdad, Iraq. While Armed Forces Radio gives the daily temperature in degrees Fahrenheit, Chloe reads the thermometer in her barracks. The lowest reading she has ever seen is -2°C and the highest reading she has seen is 46°C. What are these two readings approximately equal to in degrees Fahrenheit? ________________ ______________________________________________________________________________ ______________________________________________________________________________ Sentence:______________________________________________________________________ ______________________________________________________________________________ UNIT TWO: Prealgebra in a Technical World 2.5 Exercise Set Name _______________________________ Skills Determine the sign of each quotient first, then divide. Remember, division by 0 is undefined. 1. 24 ÷ (−6) 2. −32 ÷ (−16) 3. −99 ÷ 11 4. −5 ÷ (−5) 5. −39 ÷ (−13) 6. −144 ÷ 12 7. −1 ÷ 0 8. 63 ÷ 0 9. 0 −22 10. 14 −7 11. 15 −1 12. 13. 21 −7 14. −60 12 15. 99 −11 16. − 42 −3 18. −56 −7 19. 63 −3 20. −19 −1 17. 256 0 1 −1 21. 5 −5 22. 0 −34 23. 72 −9 24. −45 −9 25. −169 −13 26. 242 −121 27. −18 0 28. 0 −412 Simplify using order of operation: 29. 2 − 3 ∙ 4 30. 7 − 5 ∙ (2) 31. −5 − (−2)(4) 32. −4 − (4)(3) 33. 2 − 3 ∙ 4 + 32 34. −5[(−2 + (−4)] 35. 3 − 5 ∙ 7 + 42 36. −4[(−1) + (−5)] 37. (5 − 8)(3 − 9) 145 146 SECTION 2.5 Dividing Integers & Order of Operations 38. (6 − 8)(2 − 3)) 39. 32 (7) − 4 40. 23 (−5) − 6 41. −(4 − 6)(2) 42. −4(6) + 24 ÷ 3 43. −(4 − 6)(−5) 44. 6(−7) − 66 ÷ (−6) 45. 12 − 62 ÷ 3(22 ) 46. 2 − 102 ÷ 5(−4) 47. −3 − 42 (4 − 8) ÷ 2 48. 4 − |4 − 8| 49. 20 ÷ 22 − 4(−5) 51. |23 − 42 | − 92 52. 9 − |4 − 7|3 50. 3 − |8 − 9| Fraction bars are grouping symbols. Work these carefully, step-by-step, using correct order of operations. 53. 2 ∙ 43 − 8 ∙ 16 = 43 − 24 54. 3 − 4(5) ÷ 2 + 1 = 7 − 42 UNIT TWO: Prealgebra in a Technical World Applications UPS 55. What would the approximate Celsius temperature be if the Fahrenheit temperature is: a. 10℉ ≈ b. −6℉ ≈ c. 2℉ ≈ d. 30℉ ≈ 56. What would the approximate Fahrenheit temperature be if the Celsius temperature is: a. −5℃ ≈ b. −28℃ ≈ c. 30℃ ≈ d. 45℃ ≈ 57. Olaf, who lived in Florida, went to visit his cousin in Denmark. He had a hard time adjusting to the cool weather there. When he checked the temperature gauge, it read 16℃ . If his gauge in Florida was measuring the temperature, what would the approximate temperature be in ℉? Sentence: _______________________________________________________________ 58. Sarah has credit card slips for items she purchased this week: $15.95, $22.49, $49.24, and 5 lunch receipts each for $5.99. If her weekly paycheck was $234.79 and she had to pay her brother back $150 that she borrowed from him, will she be able to pay off her credit card? Sentence: _______________________________________________________________ 59. Dirk's monthly profits at his men's store for the quarter were: What was Dirk's average profit (or loss for the quarter)? January: −$3,935 February: −$1,412 March: −$5,212 Sentence: _______________________________________________________________________ 147 148 SECTION 2.5 Dividing Integers & Order of Operations 60. Susan is working in Excel, and enters $124 in Cell A1, but she doesn't have a dollar amount for Cell B1, so she enters 0. Then she writes a formula in Cell C1 that says to divide the number in A1 by the number in B1. Excel does not show a number in Cell C1. Instead, Excel reads,"#DIV?0." Why did Excel give Susan this answer? Sentence:_________________________________________________________________ 61. Bubba enters the following problem into his calculator: 200 ÷ 50. His calculator gives him the answer "Error." Can you tell Bubba what he probably entered incorrectly? Sentence:_________________________________________________________________ Review and Extend In order to master order of operations, practice problems that have many steps. You will need to copy these carefully to your own paper and work the following step by step. 62. 8 − 2 ∙ 53 − 6(37 − 102 ) −2(−3 + 5)2 ∙ (25 − 3 ∙ 5) 63. 9 − |32 − 52 | + (32 − 5)2 −|32 − 42 | + (−2)2 64. (4 − 6)2 − (9 − 15) 10 − |2 ∙ 7 − 42 | − 4( ) (500 − 30 ÷ 6)(−5) + 3 ∙ (−7) 5 − 32 The goal of this chapter is to learn the laws of signs. Before calculating, you should always know the sign of your answer. Without simplifying, write P for positive, N for negative to show the results for the following problems. 65. −12 − 4 66. −14/7 67. −2(−4)(−7) 68. 54 + (−56) 69. −15 + (−12) 70. 5 − (−2) 71. 19 − 30 72. (3)(−3) ÷ 3 73. (−12) ÷ (−2) 74. 4(−12) 75. −3(−4)3 76. 9 + (−3) 77. −5 + 4 78. (−3)(−20) 79. 4(−3) −12 80. −14 − (−15)