1.6 Division of Whole Numbers


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1 1.6 Division of Whole Numbers 1.6 OBJECTIVES 1. Use repeated subtraction to divide whole numbers 2. Check the results of a division problem 3. Divide whole numbers using long division 4. Estimate a quotient Overcoming Math Anxiety Learn to Take Useful Notes Although some students find it easier to be organized than do other students, every student can become a better note taker. Below are some hints that can help you learn to take more useful notes. Good note taking begins with your preparation for class. Note that the first several items refer to your preparation. 1. Read the assigned material before the lecture. This helps you become familiar with both the vocabulary and the concepts. 2. Review your notes from the previous class meeting. If you already have a concept in your lecture notes and the idea is referred to again, you then simply jot a note to refer back to previous material in your notes. 3. Get to class a few minutes early. Have your materials out and ready to go when your instructor walks in the door. 4. Be ready to listen as soon as the instructor walks in the door. 5. Have more than one pencil sharpened and ready to use. 6. Have a highlighter or colored pen available to mark particularly important segments of your notes. 7. Know how to spell. This includes both plain old English words and technical words that have already been presented in class. If you can spell them, you won t have to waste time trying to figure out How to spell them. 8. Be aggressive in notetaking. Don t wait for an idea to strike you it s better to have too much material than too little. 9. If the professor repeats something, write it down. 10. Take notes, not dictation. That means being able to develop your own form of shorthand. 11. Develop abbreviations for words that are used frequently in the course. Real numbers R Natural numbers N 12. Use symbols when you can. & and E there exists B but I such that 5 for each H is an element of therefore 13. Skip lines. Leave visual breaks between definitions, lists, or explanations. 14. If you miss something, leave a blank in your notes. You can fill it in later. If you try to copy it from your neighbor during the lecture, both of you will lose more material. 15. Get together with your classmates. Do so after lecture and pool your notes. That way, you can be sure you have everything down. It will also help make sure you understand what you have written down. 71
2 72 CHAPTER 1 WHOLE NUMBERS We now examine a fourth arithmetic operation, division. Just as multiplication was repeated addition, division is repeated subtraction. Division asks how many times one number is contained in another. OBJECTIVE 1 Example 1 Dividing by Using Subtraction Joel needs to set up 48 chairs in the student union for a concert. If there is room for 8 chairs per row, how many rows will it take to set up all 48 chairs? This problem can be solved by subtraction. Each row subtracts another 8 chairs Because 8 can be subtracted from 48 six times, there will be 6 rows. This can also be seen as a division problem NOTE Each of these notations represent the same division problem, 48 divided by 8 is or 8B48 or 6 8 No matter which notation we use, we call the 48 the dividend, the 8 the divisor, and the 6 the quotient. CHECK YOURSELF 1 Carlotta is creating a garden path made of bricks. She has 72 bricks. Each row will have 6 bricks in it. How many rows can she make? Units Analysis When dividing a denominate number by an abstract number, the result has the units of the denominate number. Here are a couple of examples. 76 trombones 4 19 trombones $55 11 $5 When one denominate number is divided by another, the result has the units of the dividend over the units of the divisor. 144 mi 6 gal 24 mi/gal (which we read as miles per gallon ) $120 8 h 15 dollars/h ( dollars per hour )
3 DIVISION OF WHOLE NUMBERS SECTION To solve a problem that requires division, you first set up the problem as a division statement. Example 2 illustrates this. Example 2 Writing a Division Statement Write a division statement that corresponds to the following situation. You need not do the division. The staff at the Wok Inn Restaurant splits all tips at the end of each shift. Yesterday s evening shift collected a total of $224. How much should each of the seven employees get in tips? $224 7 employees Note that the units for the answer will be dollars per employee. CHECK YOURSELF 2 Write a division statement that corresponds to the following situation. You need not do the division. All nine sections of basic math skills at SCC (Sum Community College) are full. There are a total of 315 students in the classes. How many students are in each class? What are the units for the answer? In Section 1.5, we used a rectangular array of stars to represent multiplication. These same arrays can represent division. Just as and , so is it true that and OBJECTIVE 2 NOTE For a division problem to check, the product of the divisor and the quotient must equal the dividend. or or This relationship allows us to check our division results by doing multiplication. Example 3 Checking Division by Using Multiplication 3 (a) 7B21 Check: (b) Check:
4 74 CHAPTER 1 WHOLE NUMBERS CHECK YOURSELF 3 Complete the division statements and check your results. (a) 9B45 (b) 28 7 NOTE Because , we say that 36 is exactly divisible by 9. In our examples so far, the product of the divisor and the quotient has been equal to the dividend. This means that the dividend is exactly divisible by the divisor. That is not always the case. In Example 4, we are again using repeated subtraction. Example 4 Dividing by Using Subtraction, Leaving a Remainder NOTE The remainder must be smaller than the divisor or we could subtract again. How many times is 5 contained in 23? We see that 5 is contained 4 times in 23, but 3 is left over. 23 is not exactly divisible by 5. The leftover 3 is called the remainder in the division. CHECK YOURSELF 4 How many times is 7 contained in 38? Property: Remainder Dividend divisor quotient remainder We can check our work in a division problem with a remainder as follows. Example 5 Checking Division by a SingleDigit Number NOTE Another way to write the result is 4r3 5B23 The r stands for remainder. NOTE The multiplication is done before the 3 is added. Using the work of Example 4, we can write 4 5B23 with remainder 3 To apply the remainder property, we have Divisor Quotient Dividend Remainder Evaluate The division checks. CHECK YOURSELF 5 7B38. Check your answer. We must be careful when 0 is involved in a division problem.there are two special cases.
5 DIVISION OF WHOLE NUMBERS SECTION Property: Division and Zero 1. 0 divided by any whole number (except 0) is Division by 0 is undefined. The first case involving zero occurs when we are dividing into zero. Example 6 Dividing into Zero because CHECK YOURSELF 6 (a) 0 7 (b) 0 12 Our second case illustrates what happens when 0 is the divisor. Here we have a special problem. Example 7 Dividing by Zero 8 0? This means that 8 0? Can 0 times some number ever be 8? From our multiplication facts, the answer is no! There is no answer to this problem, so we say that 8 0 is undefined. CHECK YOURSELF 7 Decide whether each problem results in 0 or is undefined. (a) 9 0 (b) 0 9 (c) 0 15 (d) 15 0 It is easy to divide when small whole numbers are involved, because much of the work can be done mentally. In working with larger numbers, we turn to a process called long division. This is a method for performing the steps of repeated subtraction. To start, we can look at an example in which we subtract multiples of the divisor. OBJECTIVE 3 NOTE With larger numbers, repeated subtraction is just too timeconsuming to be practical. Example 8 Divide 176 by 8. Because 20 eights are 160, we know that there are at least 20 eights in 176. Step 1 Write 20 8B eights Dividing by a SingleDigit Number Subtracting 160 is just a shortcut for subtracting eight 20 times.
6 76 CHAPTER 1 WHOLE NUMBERS After subtracting the 20 eights, or 160, we are left with 16. There are 2 eights in 16, and so we continue. Step B eights 16 0 Adding 20 and 2 gives us the quotient, 22. Subtracting the 2 eights, we have a 0 remainder. So CHECK YOURSELF 8 Verify the results of Example 8, using multiplication. The next step is to simplify this repeatedsubtraction process one step further. The result is the longdivision method. Example 9 Dividing by a SingleDigit Number NOTE Because the 4 is smaller than the divisor, we have a remainder of 4. NOTE Verify that this is true and that the division checks. Divide 358 by 6. The dividend is 358. We look at the first digit, 3. We cannot divide 6 into 3, so we look at the first two digits, 35. There are 5 sixes in 35, and so we write 5 above the tens digit of the dividend. 5 6B358 Now multiply 5 6, place the product below 35, and subtract. 5 6B Because the remainder, 5, is smaller than the divisor, 6, we bring down 8, the ones digit of the dividend. 5 6B Now divide 6 into 58. There are 9 sixes in 58, and so 9 is the ones digit of the quotient. Multiply 9 6 and subtract to complete the process. 59 6B When we place 5 as the tens digit, we really mean 5 tens, or 50. We have actually subtracted 50 sixes (300) from 358. We now have: r4 To check:
7 DIVISION OF WHOLE NUMBERS SECTION CHECK YOURSELF 9 Divide 7B453 Long division becomes a bit more complicated when we have a twodigit divisor. It becomes, in part, a matter of trial and error. We round the divisor and dividend to form a trial divisor and a trial dividend. We then estimate the proper quotient and must determine whether our estimate is correct. Example 10 Dividing by a TwoDigit Number Divide 38B293 7 NOTE Think: 4B29. Round the divisor and dividend to the nearest ten. So 38 is rounded to 40, and 293 is rounded to 290. The trial divisor is then 40, and the trial dividend is 290. Now look at the nonzero digits in the trial divisor and dividend. They are 4 and 29. We know that there are 7 fours in 29, and so 7 is our first estimate of the quotient. Now we will see if 7 works. 7 38B Your estimate Multiply The product, 266, is less than 293, and so we can subtract. The remainder, 27, is less than the divisor, 38, and so the process is complete r27 Check: You should verify that this statement is true. CHECK YOURSELF 10 Divide. 57B482 Because this process is based on estimation, our first guess will sometimes be wrong. Example 11 Dividing by a TwoDigit Number 8 NOTE Think: 5B43. Divide 54B428 Rounding to the nearest ten, we have a trial divisor of 50 and a trial dividend of 430. Looking at the nonzero digits, how many fives are in 43? There are 8. This is our first estimate. 8 54B Too large We multiply Do you see what s wrong? The product, 432, is too large. We cannot subtract. Our estimate of the quotient must be adjusted downward.
8 78 CHAPTER 1 WHOLE NUMBERS NOTE If we tried 6 as the quotient 6 54B We have 104, which is too large to be a remainder. We adjust the quotient downward to 7. We can now complete the division. 7 54B We have r50 Check: CHECK YOURSELF 11 Divide. 63B557 We have to be careful when a 0 appears as a digit in the quotient. Next, we look at an example in which this happens with a twodigit divisor. Example 12 Dividing with Large Dividends Divide 32B9871 NOTE Our divisor, 32, divides into 98, the first two digits of the dividend. Rounding to the nearest ten, we have a trial divisor of 30 and a trial dividend of 100. Think, How many threes are in 10? There are 3, and this is our first estimate of the quotient. 3 32B Everything seems fine so far! Bring down 7, the next digit of the dividend B We continue by multiplying by 0. After subtraction, we bring down 1, the last digit of the dividend B Now do you see the difficulty? We cannot divide 32 into 27, and so we place 0 in the tens place of the quotient to indicate this fact.
9 DIVISION OF WHOLE NUMBERS SECTION Another problem develops here. We round 32 to 30 for our trial divisor, and we round 271 to 270, which is the trial dividend at this point. Our estimate of the last digit of the quotient must be B Too large We cannot subtract. The trial quotient must be adjusted downward to 8. We can now complete the division B , r15 Check: 9, CHECK YOURSELF 12 Divide. 43B8857 Because of the availability of the handheld calculator, it is rarely necessary that people find the exact answer when performing long division. On the other hand, it is frequently important that one be able to either estimate the result of long division or confirm that a given answer (particularly from a calculator) is reasonable. As a result, the emphasis in this section will be to improve your estimation skills in division. In Example 13, we divide a fourdigit number by a twodigit number. Generally, we round the divisor to the nearest ten and the dividend to the nearest hundred. OBJECTIVE 4 Example 13 The Ramirez family took a trip of 2,394 mi in their new car, using 77 gal of gas. Estimate their gas mileage (mi/gal). Our estimate will be based on dividing 2,400 by B2400 They got approximately 30 mi/gal. Estimating the Result of a Division Application
10 80 CHAPTER 1 WHOLE NUMBERS CHECK YOURSELF 13 Troy flew a light plane on a trip of 2,844 mi that took 21 h. What was his approximate speed in miles per hour? As before, we may have to combine operations to solve an application of the mathematics you have learned. Example 14 Estimating the Result of a Division Application Charles purchases a used car for $8,574. He agrees to make payments for 4 years. Interest charges will be $978. Approximately what should his monthly payments be? First, we find the amount that Charles owes: $8,574 $978 $9,552 Now, to find the monthly payment, we divide that amount by 48 (months). To estimate the payment, we divide $9,600 by 50 months B9600 The payments will be approximately $192 per month. CHECK YOURSELF 14 One $10 bag of fertilizer will cover 310 ft 2. Approximately what would it cost to cover 2,200 ft 2? Using a Scientific Calculator to Divide Of course, division is easily done using your calculator. However, as we will see, some special things come up when we use a calculator to divide. First we outline the steps of division as it is done on a calculator. Divide 35B2380. RECALL A graphing calculator uses the Enter key rather than. Step 1 Enter the dividend Step 2 Press the divide key. Step 3 Enter the divisor. 35 Step 4 Press the equals key. The desired quotient is now in your display. The display shows 68. We have already mentioned some of the difficulties related to division with 0. We will experiment on the calculator.
11 DIVISION OF WHOLE NUMBERS SECTION Example 15 Using a Scientific Calculator to Divide To find 0 5, we use this sequence: 0 5 Display 0 There is no problem with this. Zero divided by any whole number other than 0 is just 0. CHECK YOURSELF 15 What is the result when you use your calculator to perform the given operation? 0 17 We see what happens when dividing zero by another number, but what happens when we try to divide by zero? More importantly to this section, how does the calculator handle division by zero? Example 16 illustrates this concept. Example 16 Using a Scientific Calculator to Divide To find 5 0, we use this sequence: 5 0 Display Error NOTE You may find that you must clear your calculator after trying this. If we try this sequence, the calculator gives us an error! Do you see why? Division by 0 is not allowed. Try this on your calculator to see how this error is indicated. CHECK YOURSELF 16 What is the result when you use your calculator to perform the given operation? 17 0 Another special problem comes up when a remainder is involved in a division problem. NOTE Be aware that the calculator will not give you a remainder in the form we have been using in this chapter. Example 17 Dividing 293 by 38 gives 7 with remainder Quotient Using a Scientific Calculator to Divide Remainder 7 is the wholenumber part of the quotient as before is the decimal form of the remainder, 27, as a fraction of 38.
12 82 CHAPTER 1 WHOLE NUMBERS CHECK YOURSELF 17 What is the result when you use your calculator to perform the given operation? The calculator can also help you combine division with other operations. Example 18 Using a Scientific Calculator to Divide To find , use this sequence: Display 12 Do you see that the calculator has done the division as the first step? CHECK YOURSELF 18 Use your calculator to compute Example 19 Using a Scientific Calculator to Divide To find 6 3 2, use this sequence: Display 4 CHECK YOURSELF 19 Use your calculator to compute
13 DIVISION OF WHOLE NUMBERS SECTION READING YOUR TEXT The following fillintheblank exercises are designed to assure that you understand the key vocabulary used in this section. Each sentence comes directly from the section. You will find the correct answers in Appendix C. Section 1.6 (a) If you read the assigned material before the lecture, you will become familiar with both the and the concepts. (b) Just as multiplying was repeated addition, division is repeated. (c) You can check division by using. (d) Division by is undefined. CHECK YOURSELF ANSWERS students 9 classes; students per class 3. (a) 5; ; (b) 4; with remainder 3 6. (a) 0; (b) 0 7. (a) undefined; (b) 0; (c) 0; (d) undefined with remainder with remainder with remainder with remainder mi/h 14. $ Error message
14 Boost your GRADE at ALEKS.com! 1.6 Exercises Practice Problems SelfTests NetTutor Name Section eprofessors Videos Date 1. Given , 8 is the, 48 is the, and 6 is the. 2. In the statement 5B45 9, 9 is the, 5 is the, and 45 is the. 3. Find 36 9 by repeated subtraction. ANSWERS Find 40 8 by repeated subtraction. 5. Problem Solving Stefanie is planting rows of tomato plants. She wants to plant 63 plants with 9 plants per row. How many rows will she have? 6. Construction Nick is designing a parking lot for a small office building. He must make room for 42 cars with 7 cars per row. How many rows should he plan for? Divide and identify the correct units for the quotient pages 4 8. $ ,900 km gal mi 4 h ft 3 s 13. 3,720 h 5 months cal 7 g Divide using long division and check your work B B B Divide B B B B B B B 22, B 43, B B B B B B 4321 Solve the applications. 37. Statistics Ramon bought 56 bags of candy. There were 8 bags in each box. How many boxes were there? 38. Statistics There are 32 students who are taking a field trip. If each car can hold 4 students, how many cars are needed for the field trip? 84 SECTION 1.6
15 ANSWERS 39. Business and Finance Ticket receipts for a play were $552. If the tickets were $4 each, how many tickets were purchased? Construction Construction of a fence section requires 8 boards. If you have 256 boards, how many sections can you build? 41. Construction The homeowners along a street must share the $2,030 cost of new street lighting. If there are 14 homes, what amount will each owner pay? Business and Finance A bookstore ordered 325 copies of a textbook at a cost of $7,800. What was the cost to the store for an individual textbook? 43. Business and Finance A company distributes $16,488 in yearend bonuses. If each of the 36 employees receives the same amount, what bonus will each receive? 44. Complete the following number cross. Across Down (12 16) , , , , (2 6) (161 7) 16. 1, , , SECTION
16 ANSWERS Estimate the result in the division problems. (Remember to round divisors to the nearest ten and dividends to the nearest hundred.) divided by divided by ,967 divided by ,971 divided by ,812 divided by ,245 divided by 255 In exercises 51 to 54, solve the applications. 51. Technology Jose drove 279 mi on 18 gal of gas. Estimate his mileage. (Hint: Find the number of miles per gallon.) 52. Construction A contractor can build a house in 27 days. Estimate how many houses can be built in 265 days. 53. Construction You are going to recarpet your living room. You have budgeted $1,500 for the carpet and installation. (a) Determine how much carpet you will need to do the job. Draw a sketch to support your measurements. (b) What is the highest price per square yard you can pay and still stay within budget? (c) Go to a local store and determine the total cost of doing the job for three different grades of carpet. Be sure to include padding, labor costs, and any other expenses. (d) What considerations (other than cost) would affect your decision about what type of carpet to install? (e) Write a brief paragraph indicating your final decision and give supporting reasons. 54. Division is the inverse operation of multiplication. Many daily activities have inverses. For each of the following activities, state the inverse activity: (a) Spending money (b) Going to sleep (c) Turning down the volume on your CD player (d) Getting dressed 55. Division is not associative. For example, will produce different results if 8 is divided by 4 and then divided by 2 or if 8 is divided by the result of 4 2. Place parentheses in the proper place so that each expression is true. (a) (b) (c) (d) (e) Is there any situation in which the order of how the operation of division is performed produces the same result? Give an example. 86 SECTION 1.6
17 ANSWERS 56. Division is not commutative. For example, What must be true of the numbers a and b if a b b a? Solve each chapteractivity application. 57. Crafts A set of samesized packages are each 13 in. wide and have a height of 3 in. Find the girth of each package (twice the width plus twice the height). You need a length of wrapping paper equal to one more inch than the girth of the package. What length of wrapping paper do you need for each package? How many packages can you wrap if your wrapping paper is 300 in. (25 ft) long? How long a piece of scrap will you be left with? 58. Business and Finance If a business needs to wrap 50 packages, each as described in exercise 57, how many rolls of 300in.long wrapping paper will they need? What if they needed to wrap 200 such packages? Calculator Exercises Use your calculator to perform the indicated operations , , , , (89 14) 25 Answers 1. Divisor, dividend, quotient pages km mi/h h/month r r r r r ,769 r r r r boxes tickets 41. $ $ mi/gal in.; 33 in.; 9 packages; 3 in SECTION
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