1.5 Divide Whole Numbers |
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Learning Objectives By the end of this section, you will be able to: Use division notation Model division of whole numbers Divide whole numbers Translate word phrases to math notation Divide whole numbers in applications Be Prepared 1.7Before you get started, take this readiness quiz. Multiply: 27·3.27·3. If you missed this problem, review Example 1.44. Be Prepared 1.8Subtract: 43−26.43−26. If you missed this problem, review Example 1.32 Be Prepared 1.9Multiply: 62(87).62(87). If you missed this problem, review Example 1.45. Use Division NotationSo far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the 1212 cookies in Figure 1.13 and want to package them in bags with 44 cookies in each bag. How many bags would we need? Figure 1.13You might put 44 cookies in first bag, 44 in the second bag, and so on until you run out of cookies. Doing it this way, you would fill 33 bags. In other words, starting with the 1212 cookies, you would take away, or subtract, 44 cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition. Instead of subtracting 44 repeatedly, we can write 12÷412÷4We read this as twelve divided by four and the result is the quotient of 1212 and 4.4. The quotient is 33 because we can subtract 44 from 1212 exactly 33 times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is 1212 and the divisor is 4.4. In the past you may have used the notation 412412, but this division also can be written as 12÷4,12/4,124.12÷4,12/4,124. In each case the 1212 is the dividend and the 44 is the divisor. Operation Symbols for DivisionTo represent and describe division, we can use symbols and words. Operation Notation Expression Read as Result DivisionDivision ÷÷ abab babaa/ba/b 12÷412÷4 124124 41241212/412/4 Twelve divided by fourTwelve divided by four the quotient of 12 and 4the quotient of 12 and 4Division is performed on two numbers at a time. When translating from math notation to English words, or English words to math notation, look for the words of and and to identify the numbers. Example 1.56Translate from math notation to words. ⓐ 64÷864÷8 ⓑ 427427 ⓒ 428428 Solution ⓐ We read this as sixty-four divided by eight and the result is the quotient of sixty-four and eight. ⓑ We read this as forty-two divided by seven and the result is the quotient of forty-two and seven. ⓒ We read this as twenty-eight divided by four and the result is the quotient of twenty-eight and four. Try It 1.111Translate from math notation to words: ⓐ 84÷784÷7 ⓑ 186186 ⓒ 824824 Try It 1.112Translate from math notation to words: ⓐ 72÷972÷9 ⓑ 213213 ⓒ 654654 Model Division of Whole NumbersAs we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly. Manipulative Mathematics Doing the Manipulative Math Worksheets activity "Model Division of Whole Numbers" will help you develop a better understanding of dividing whole numbers. Example 1.57Model the division: 24÷8.24÷8. SolutionTo find the quotient 24÷8,24÷8, we want to know how many groups of 88 are in 24.24. Model the dividend. Start with 2424 counters. The divisor tell us the number of counters we want in each group. Form groups of 88 counters. Count the number of groups. There are 33 groups. 24 ÷ 8 = 3 24 ÷ 8 = 3 Try It 1.113Model: 24÷6.24÷6. Try It 1.114Model: 42÷7.42÷7. Divide Whole NumbersWe said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know 12÷4=312÷4=3 because 3·4=12.3·4=12. Knowing all the multiplication number facts is very important when doing division. We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In Example 1.57, we know 24÷8=324÷8=3 is correct because 3·8=24.3·8=24. Example 1.58Divide. Then check by multiplying. ⓐ 42÷642÷6 ⓑ 729729 ⓒ 763763 Solution ⓐ 42÷642÷6 Divide 42 by 6. 77 Check by multiplying. 7·67·6 42✓42✓ ⓑ 729729 Divide 72 by 9. 88 Check by multiplying.8·98·9 72✓72✓ ⓒ 763763 Divide 63 by 7. 99 Check by multiplying. 9·79·7 63✓63✓ Try It 1.115Divide. Then check by multiplying: ⓐ 54÷654÷6 ⓑ 279279 Try It 1.116Divide. Then check by multiplying: ⓐ 369369 ⓑ 840840 What is the quotient when you divide a number by itself? 1515=1because1·15=151515=1because1·15=15Dividing any number (except 0)(except 0) by itself produces a quotient of 1.1. Also, any number divided by 11 produces a quotient of the number. These two ideas are stated in the Division Properties of One. Division Properties of One Any number (except 0) divided by itself is one. a÷a=1a÷a=1 Any number divided by one is the same number. a÷1=aa÷1=a Table 1.6 Example 1.59Divide. Then check by multiplying: ⓐ 11÷1111÷11 ⓑ 191191 ⓒ 1717 Solution ⓐ 11÷1111÷11 A number divided by itself is 1. 11 Check by multiplying.1·111·11 11✓11✓ ⓑ 191191 A number divided by 1 equals itself. 1919 Check by multiplying.19·119·1 19✓19✓ ⓒ 1717 A number divided by 1 equals itself. 77 Check by multiplying.7·17·1 7✓7✓ Try It 1.117Divide. Then check by multiplying: ⓐ 14÷1414÷14 ⓑ 271271 Try It 1.118Divide. Then check by multiplying: ⓐ 161161 ⓑ 1414 Suppose we have $0,$0, and want to divide it among 33 people. How much would each person get? Each person would get $0.$0. Zero divided by any number is 0.0. Now suppose that we want to divide $10$10 by 0.0. That means we would want to find a number that we multiply by 00 to get 10.10. This cannot happen because 00 times any number is 0.0. Division by zero is said to be undefined. These two ideas make up the Division Properties of Zero. Division Properties of Zero Zero divided by any number is 0. 0÷a=00÷a=0 Dividing a number by zero is undefined. a÷0a÷0 undefined Table 1.7Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away 00 from 10?10? Because subtracting 00 will never change the total, we will never get an answer. So we cannot divide a number by 0.0. Example 1.60Divide. Check by multiplying: ⓐ 0÷30÷3 ⓑ 10/0.10/0. Solution ⓐ 0÷30÷3 Zero divided by any number is zero. 00 Check by multiplying.0·30·3 0✓0✓ ⓑ 10/010/0 Division by zero is undefined. undefined Try It 1.119Divide. Then check by multiplying: ⓐ 0÷20÷2 ⓑ 17/017/0 Try It 1.120Divide. Then check by multiplying: ⓐ 0÷60÷6 ⓑ 13/013/0 When the divisor or the dividend has more than one digit, it is usually easier to use the 412412 notation. This process is called long division. Let’s work through the process by dividing 7878 by 3.3. Divide the first digit of the dividend, 7, by the divisor, 3. The divisor 3 can go into 7 two times since 2×3=62×3=6. Write the 2 above the 7 in the quotient. Multiply the 2 in the quotient by 3 and write the product, 6, under the 7. Subtract that product from the first digit in the dividend. Subtract 7−67−6. Write the difference, 1, under the first digit in the dividend. Bring down the next digit of the dividend. Bring down the 8. Divide 18 by the divisor, 3. The divisor 3 goes into 18 six times. Write 6 in the quotient above the 8. Multiply the 6 in the quotient by the divisor and write the product, 18, under the dividend. Subtract 18 from 18.We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished. So78÷3=26.So78÷3=26.Check by multiplying the quotient times the divisor to get the dividend. Multiply 26×326×3 to make sure that product equals the dividend, 78.78. 216×3___78✓216×3___78✓It does, so our answer is correct. How To Divide whole numbers. Step 1. Divide the first digit of the dividend by the divisor. If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on. Step 2. Write the quotient above the dividend. Step 3. Multiply the quotient by the divisor and write the product under the dividend. Step 4. Subtract that product from the dividend. Step 5. Bring down the next digit of the dividend. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down. Step 7. Check by multiplying the quotient times the divisor. Example 1.61Divide 2,596÷4.2,596÷4. Check by multiplying: Solution Let's rewrite the problem to set it up for long division. Divide the first digit of the dividend, 2, by the divisor, 4. Since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. The divisor 4 goes into 25 six times. We write the 6 in the quotient above the 5. Multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. Subtract that product from the first two digits in the dividend. Subtract 25−2425−24. Write the difference, 1, under the second digit in the dividend. Now bring down the 9 and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. Bring down the 6 and repeat these steps. There are 9 fours in 36. Write the 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. So 2,596÷4=6492,596÷4=649. Check by multiplying.It equals the dividend, so our answer is correct. Try It 1.121Divide. Then check by multiplying: 2,636÷42,636÷4 Try It 1.122Divide. Then check by multiplying: 2,716÷42,716÷4 Example 1.62Divide 4,506÷6.4,506÷6. Check by multiplying: Solution Let's rewrite the problem to set it up for long division. First we try to divide 6 into 4. Since that won't work, we try 6 into 45.There are 7 sixes in 45. We write the 7 over the 5. Multiply the 7 by 6 and subtract this product from 45. Now bring down the 0 and repeat these steps. There are 5 sixes in 30.Write the 5 over the 0. Multiply the 5 by 6 and subtract this product from 30. Now bring down the 6 and repeat these steps. There is 1 six in 6. Write the 1 over the 6. Multiply 1 by 6 and subtract this product from 6. Check by multiplying.It equals the dividend, so our answer is correct. Try It 1.123Divide. Then check by multiplying: 4,305÷5.4,305÷5. Try It 1.124Divide. Then check by multiplying: 3,906÷6.3,906÷6. Example 1.63Divide 7,263÷9.7,263÷9. Check by multiplying. Solution Let's rewrite the problem to set it up for long division. First we try to divide 9 into 7. Since that won't work, we try 9 into 72. There are 8 nines in 72. We write the 8 over the 2. Multiply the 8 by 9 and subtract this product from 72. Now bring down the 6 and repeat these steps. There are 0 nines in 6. Write the 0 over the 6. Multiply the 0 by 9 and subtract this product from 6. Now bring down the 3 and repeat these steps. There are 7 nines in 63. Write the 7 over the 3. Multiply the 7 by 9 and subtract this product from 63. Check by multiplying.It equals the dividend, so our answer is correct. Try It 1.125Divide. Then check by multiplying: 4,928÷7.4,928÷7. Try It 1.126Divide. Then check by multiplying: 5,663÷7.5,663÷7. So far all the division problems have worked out evenly. For example, if we had 2424 cookies and wanted to make bags of 88 cookies, we would have 33 bags. But what if there were 2828 cookies and we wanted to make bags of 8?8? Start with the 2828 cookies as shown in Figure 1.14. Figure 1.14Try to put the cookies in groups of eight as in Figure 1.15. Figure 1.15There are 33 groups of eight cookies, and 44 cookies left over. We call the 44 cookies that are left over the remainder and show it by writing R4 next to the 3.3. (The R stands for remainder.) To check this division we multiply 33 times 88 to get 24,24, and then add the remainder of 4.4. 3×8___24+4___283×8___24+4___28 Example 1.64Divide 1,439÷4.1,439÷4. Check by multiplying. Solution Let's rewrite the problem to set it up for long division. First we try to divide 4 into 1. Since that won't work, we try 4 into 14. There are 3 fours in 14. We write the 3 over the 4. Multiply the 3 by 4 and subtract this product from 14. Now bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract this product from 23. Now bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract this product from 39. There are no more numbers to bring down, so we are done. The remainder is 3. Check by multiplying.So 1,439÷41,439÷4 is 359359 with a remainder of 3.3. Our answer is correct. Try It 1.127Divide. Then check by multiplying: 3,812÷8.3,812÷8. Try It 1.128Divide. Then check by multiplying: 4,319÷8.4,319÷8. Example 1.65Divide and then check by multiplying: 1,461÷13.1,461÷13. Solution Let's rewrite the problem to set it up for long division. 131,461131,461 First we try to divide 13 into 1. Since that won't work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4. Multiply the 1 by 13 and subtract this product from 14. Now bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16. Now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. There are no more numbers to bring down, so we are done. The remainder is 5. 1,462÷131,462÷13 is 112 with a remainder of 5. Check by multiplying.Our answer is correct. Try It 1.129Divide. Then check by multiplying: 1,493÷13.1,493÷13. Try It 1.130Divide. Then check by multiplying: 1,461÷12.1,461÷12. Example 1.66Divide and check by multiplying: 74,521÷241.74,521÷241. Solution Let's rewrite the problem to set it up for long division. 24174,52124174,521 First we try to divide 241 into 7. Since that won’t work, we try 241 into 74. That still won’t work, so we try 241 into 745. Since 2 divides into 7 three times, we try 3. Since 3×241=7233×241=723, we write the 3 over the 5 in 745. Note that 4 would be too large because 4×241=9644×241=964, which is greater than 745. Multiply the 3 by 241 and subtract this product from 745. Now bring down the 2 and repeat these steps. 241 does not divide into 222. We write a 0 over the 2 as a placeholder and then continue. Now bring down the 1 and repeat these steps. Try 9. Since 9×241=2,1699×241=2,169, we write the 9 over the 1. Multiply the 9 by 241 and subtract this product from 2,221. There are no more numbers to bring down, so we are finished. The remainder is 52. So 74,521÷24174,521÷241is 309 with a remainder of 52. Check by multiplying.Sometimes it might not be obvious how many times the divisor goes into digits of the dividend. We will have to guess and check numbers to find the greatest number that goes into the digits without exceeding them. Try It 1.131Divide. Then check by multiplying: 78,641÷256.78,641÷256. Try It 1.132Divide. Then check by multiplying: 76,461÷248.76,461÷248. Translate Word Phrases to Math NotationEarlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in Table 1.8. Operation Word Phrase Example Expression Division divided byquotient ofdivided into 1212 divided by 44the quotient of 1212 and 4444 divided into 1212 12÷412÷412412412/412/4412412 Table 1.8 Example 1.67Translate and simplify: the quotient of 5151 and 17.17. SolutionThe word quotient tells us to divide. the quotient of 51 and 17 Translate. 51 ÷ 17 Divide. 3 the quotient of 51 and 17 Translate. 51 ÷ 17 Divide. 3 We could just as correctly have translated the quotient of 5151 and 1717 using the notation 17 51 or 51 17 . 17 51 or 51 17 . Try It 1.133Translate and simplify: the quotient of 9191 and 13.13. Try It 1.134Translate and simplify: the quotient of 5252 and 13.13. Divide Whole Numbers in ApplicationsWe will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question. Example 1.68Cecelia bought a 160-ounce160-ounce box of oatmeal at the big box store. She wants to divide the 160160 ounces of oatmeal into 8-ounce8-ounce servings. She will put each serving into a plastic bag so she can take one bag to work each day. How many servings will she get from the big box? SolutionWe are asked to find the how many servings she will get from the big box. Write a phrase. 160 ounces divided by 8 ounces Translate to math notation. 160÷8160÷8 Simplify by dividing. 2020 Write a sentence to answer the question.Cecelia will get 20 servings from the big box. Try It 1.135Marcus is setting out animal crackers for snacks at the preschool. He wants to put 99 crackers in each cup. One box of animal crackers contains 135135 crackers. How many cups can he fill from one box of crackers? Try It 1.136Andrea is making bows for the girls in her dance class to wear at the recital. Each bow takes 44 feet of ribbon, and 3636 feet of ribbon are on one spool. How many bows can Andrea make from one spool of ribbon? Media ACCESS ADDITIONAL ONLINE RESOURCES Dividing Whole Numbers Dividing Whole Numbers No Remainder Dividing Whole Numbers With Remainder Section 1.5 Exercises Practice Makes PerfectUse Division Notation In the following exercises, translate from math notation to words. 343.54 ÷ 9 54 ÷ 9 344.56 7 56 7 345.32 8 32 8 346.6 42 6 42 347.48 ÷ 6 48 ÷ 6 348.63 9 63 9 349.7 63 7 63 350.72 ÷ 8 72 ÷ 8 Model Division of Whole Numbers In the following exercises, model the division. 351.15 ÷ 5 15 ÷ 5 352.10 ÷ 5 10 ÷ 5 353.14 7 14 7 354.18 6 18 6 355.4 20 4 20 356.3 15 3 15 357.24 ÷ 6 24 ÷ 6 358.16 ÷ 4 16 ÷ 4 Divide Whole Numbers In the following exercises, divide. Then check by multiplying. 359.18 ÷ 2 18 ÷ 2 360.14 ÷ 2 14 ÷ 2 361.27 3 27 3 362.30 3 30 3 363.4 28 4 28 364.4 36 4 36 365.45 5 45 5 366.35 5 35 5 367.72 / 8 72 / 8 368.8 64 8 64 369.35 7 35 7 370.42 ÷ 7 42 ÷ 7 371.15 15 15 15 372.12 12 12 12 373.43 ÷ 43 43 ÷ 43 374.37 ÷ 37 37 ÷ 37 375.23 1 23 1 376.29 1 29 1 377.19 ÷ 1 19 ÷ 1 378.17 ÷ 1 17 ÷ 1 379.0 ÷ 4 0 ÷ 4 380.0 ÷ 8 0 ÷ 8 381.5 0 5 0 382.9 0 9 0 383.26 0 26 0 384.32 0 32 0 385.12 0 12 0 386.16 0 16 0 387.72 ÷ 3 72 ÷ 3 388.57 ÷ 3 57 ÷ 3 389.96 8 96 8 390.78 6 78 6 391.5 465 5 465 392.4 528 4 528 393.924 ÷ 7 924 ÷ 7 394.861 ÷ 7 861 ÷ 7 395.5,226 6 5,226 6 396.3,776 8 3,776 8 397.4 31,324 4 31,324 398.5 46,855 5 46,855 399.7,209 ÷ 3 7,209 ÷ 3 400.4,806 ÷ 3 4,806 ÷ 3 401.5,406 ÷ 6 5,406 ÷ 6 402.3,208 ÷ 4 3,208 ÷ 4 403.4 2,816 4 2,816 404.6 3,624 6 3,624 405.91,881 9 91,881 9 406.83,256 8 83,256 8 407.2,470 ÷ 7 2,470 ÷ 7 408.3,741 ÷ 7 3,741 ÷ 7 409.8 55,305 8 55,305 410.9 51,492 9 51,492 411.431,174 5 431,174 5 412. 297,277 4 297,277 4 413.130,016 ÷ 3 130,016 ÷ 3 414.105,609 ÷ 2 105,609 ÷ 2 415.15 5,735 15 5,735 416.4,933 21 4,933 21 417.56,883 ÷ 67 56,883 ÷ 67 418.43,725 / 75 43,725 / 75 419.30,144 314 30,144 314 420.26,145 ÷ 415 26,145 ÷ 415 421.273 542,195 273 542,195 422.816,243 ÷ 462 816,243 ÷ 462 Mixed Practice In the following exercises, simplify. 423.15 ( 204 ) 15 ( 204 ) 424.74 · 391 74 · 391 425.256 − 184 256 − 184 426.305 − 262 305 − 262 427.719 + 341 719 + 341 428.647 + 528 647 + 528 429.25 875 25 875 430.1104 ÷ 23 1104 ÷ 23 Translate Word Phrases to Algebraic Expressions In the following exercises, translate and simplify. 431.the quotient of 4545 and 1515 432.the quotient of 6464 and 1616 433.the quotient of 288288 and 2424 434.the quotient of 256256 and 3232 Divide Whole Numbers in Applications In the following exercises, solve. 435.Trail mix Ric bought 6464 ounces of trail mix. He wants to divide it into small bags, with 22 ounces of trail mix in each bag. How many bags can Ric fill? 436.Crackers Evie bought a 4242 ounce box of crackers. She wants to divide it into bags with 33 ounces of crackers in each bag. How many bags can Evie fill? 437.Astronomy class There are 125125 students in an astronomy class. The professor assigns them into groups of 5.5. How many groups of students are there? 438.Flower shop Melissa’s flower shop got a shipment of 152152 roses. She wants to make bouquets of 88 roses each. How many bouquets can Melissa make? 439.Baking One roll of plastic wrap is 4848 feet long. Marta uses 33 feet of plastic wrap to wrap each cake she bakes. How many cakes can she wrap from one roll? 440.Dental floss One package of dental floss is 5454 feet long. Brian uses 22 feet of dental floss every day. How many days will one package of dental floss last Brian? Mixed Practice In the following exercises, solve. 441.Miles per gallon Susana’s hybrid car gets 4545 miles per gallon. Her son’s truck gets 1717 miles per gallon. What is the difference in miles per gallon between Susana’s car and her son’s truck? 442.Distance Mayra lives 5353 miles from her mother’s house and 7171 miles from her mother-in-law’s house. How much farther is Mayra from her mother-in-law’s house than from her mother’s house? 443.Field trip The 4545 students in a Geology class will go on a field trip, using the college’s vans. Each van can hold 99 students. How many vans will they need for the field trip? 444.Potting soil Aki bought a 128128 ounce bag of potting soil. How many 44 ounce pots can he fill from the bag? 445.Hiking Bill hiked 88 miles on the first day of his backpacking trip, 1414 miles the second day, 1111 miles the third day, and 1717 miles the fourth day. What is the total number of miles Bill hiked? 446.Reading Last night Emily read 66 pages in her Business textbook, 2626 pages in her History text, 1515 pages in her Psychology text, and 99 pages in her math text. What is the total number of pages Emily read? 447.Patients LaVonne treats 1212 patients each day in her dental office. Last week she worked 44 days. How many patients did she treat last week? 448.Scouts There are 1414 boys in Dave’s scout troop. At summer camp, each boy earned 55 merit badges. What was the total number of merit badges earned by Dave’s scout troop at summer camp? Writing Exercises 449.Contact lenses Jenna puts in a new pair of contact lenses every 1414 days. How many pairs of contact lenses does she need for 365365 days? 450.Cat food One bag of cat food feeds Lara’s cat for 2525 days. How many bags of cat food does Lara need for 365365 days? Everyday Math 451.Explain how you use the multiplication facts to help with division. 452.Oswaldo divided 300300 by 88 and said his answer was 3737 with a remainder of 4.4. How can you check to make sure he is correct? Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not? |
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