Section 1.6 Dividing Whole Numbers

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1 Section 1.6 Dividing Whole Numbers We begin this section by looking at an example that involves division of whole numbers. Dale works at a Farmer s Market. There are 245 apples that he needs to put in boxes of 8 apples each. How many full boxes can he make, and how many apples will be left over? To answer this question, we begin by imagining that we have already placed 8 apples in the first box. This means we now have = 237 apples. If we place 8 apples in the second box, we will now have only = 229 apples. By subtracting this way we will be able to determine the number of full boxes that can be made, and the number that will be left over. However, this process would take too long. Instead, we will use division to answer the question because division represents repeated subtraction of the same number. What we need to do is to divide the 245 apples by groups of 8. This is written as divided by divided by divided by 8 Note the location in which the 245 and the 8 were placed. If you are given the expression 245 8, the 245 must be placed inside the division box and the 8 outside. This placement has nothing to do with the fact that 245 is greater than 8. To perform the division, we look at the leftmost digit of the number 245 (it s a 2) and notice that the maximum number of full boxes of 8 apples that can be made with 2 apples is zero (8 doesn t fit into a 2). We indicate this as follows: We then consider the two leftmost digits of 245 (think of them as a 24). We then determine that the number of full boxes containing 8 apples each, that can be made with 24 apples, is 3 since 8 3=24. We express this as Copyright 2014 Luis Soto-Ortiz 73

2 We bring down the last digit of 245 (it s a 5) and notice that the number of full boxes of 8 apples that can be made with 5 apples is zero. We express this as follows divisor quotient dividend remainder Notice that 030 = 30. In the division above, we call 30 the quotient, 8 is the divisor, 245 is the dividend and 5 is the remainder. Answer: Dale can make 30 full boxes of 8 apples each, and there will be 5 apples left over. Example An employer will distribute $7,344 evenly among his 6 workers. How much will each worker receive? To distribute evenly means to divide $7,344 by 6. Copyright 2014 Luis Soto-Ortiz 74

3 We notice that if there were only $7 and 6 workers, each worker would get $1 and there would be $1 left over. In other words, 6 fits a maximum of 1 time into 7, since 6x1=6 but 6x2=12 is bigger than 7. We bring down the next digit (3). We notice that 6 fits a maximum of 2 times into 13, since 6x2=12 but 6x3=18 is bigger than 13. We bring down the next digit (4). We notice that 6 fits a maximum of 2 times into 14, since 6x2=12 but 6x3=18 is bigger than 14. We bring down the last digit (4). We notice that 6 fits a maximum of 4 times into 24, since 6x4=24 but 6x5=30 is bigger than 24. The zero remainder means that the money was distributed evenly among the 6 workers and there was no money left over. Answer: By distributing $7,344 evenly among the 6 workers, each worker will receive $1,224. The zero remainder means that $7,344 can be divided equally among 6 people and everyone will get an amount in full dollars and no cents. Copyright 2014 Luis Soto-Ortiz 75

4 Example Perform the following division: $ We notice that 15 does not fit into 4 at all, so we consider the first two digits (40) instead. We know that 15 fits a maximum of 2 times into 40, since 15x2=30 but 15x3=45 is bigger than Next, we bring down the 6. Counting by 15 s or from a multiplication table you should notice that 15 fits a maximum of 7 times into 106, since 15x7=105 but 15x8=120 which is bigger than Finally, bring down the last digit (5). We notice that 15 fits a maximum of 1 time into 15, since 15x1=15 but 15x2=30. After subtracting, we get a zero remainder. This means that 15 fits exactly 271 times into 4,065 because 15x271 = 4,065. Remember that 0271 = 271. Answer: $ = $271 Copyright 2014 Luis Soto-Ortiz 76

5 Example Perform the following division: This expression means 734,904 8 as well as We notice that 8 does not fit into 7 at all, so we consider the first two digits (73) instead. We know that 8 fits a maximum of 9 times into 73 because 8x9=72. After subtracting, we bring down the 4 and notice that 8 fits a maximum of 1 time into 14, since 8x1=8 but 8x2=16 which is greater than We bring down the 9 and notice that 8 fits a maximum of 8 times into 69, since 8x8=64 but 8x9=72 which is greater than 69. We then subtract 64 from We bring down the 0 and notice that 8 fits a maximum of 6 times into 50, since 8x6=48 but 8x7=56 which is greater than 50. We then subtract 48 from 50. Copyright 2014 Luis Soto-Ortiz 77

6 Finally, we bring down the last digit (4) and notice that 8 fits a maximum of 3 times into 24, since 8x3=24 but 8x4=32 which is greater than 24. We then subtract. We are done. The dividend is 734,904, the divisor is 8 the quotient is 91,863 and the remainder is 0. Answer: = 91,863 Example At a fundraiser event, a total of $3,360 were raised. If 32 people attended the event and contributed the same amount, how much did each person contribute? Since 32 does not fit into 3, we consider the first two digits (33) instead. We know that 32 fits a maximum of 1 time into 33 because 32x1=32 but 32x2=64 which is bigger than 33. We then subtract 32 from 33. Copyright 2014 Luis Soto-Ortiz 78

7 Next, we bring down the 6 but notice that 32 does not fit into 16 at all. We must indicate this by writing a 0 in the quotient. We bring down the next digit (0) and consider 160. Counting by 32 s or by multiplication, we determine that 32 fits a maximum of 5 times into 160. We are done. The dividend is 3,360, the divisor is 32 the quotient is 105 and the remainder is 0. Answer: Each person who attended the fundraiser contributed $105. Classwork 1.6 Perform each division. Identify the dividend, divisor, quotient and remainder Divide: 378, Divide: 71, Divide: 8. Divide:, Copyright 2014 Luis Soto-Ortiz 79

8 9. Divide: 10. Divide:, Divide: 4,662, Divide: 635, Divide: 18. Divide: 19. Divide: 20. Divide:,, 21. Divide: 657,314, Divide: 3,120, Divide: 24. Divide: 25. Divide:, Copyright 2014 Luis Soto-Ortiz 80

9 CW 1.6 Solutions: 1. 48,218 R R R9 4. 3,361 R ,897 R ,590 R ,644 R ,629 R ,977 R ,043 R ,063 R R R ,984 R ,256 R ,654 R ,605 R ,697 R ,575 R ,957 R ,328, ,380 R ,682,308 R ,531 R ,397 R3 Copyright 2014 Luis Soto-Ortiz 81

12 HW 1.6 Solutions: 1. 15,721 R R R ,111 R ,818 R ,937 R R0 8. 3,596 R0 9. 8,640 R ,806 R R ,065 R R ,052 R ,627 R ,673 R ,634 R ,475 R ,320 R ,614 R ,938,749 R ,995 R ,023 R ,69 R ,674 R178 Copyright 2014 Luis Soto-Ortiz 84

Section 2.1 Factors and Multiples

Section 2.1 Factors and Multiples When you want to prepare a salad, you select certain ingredients (lettuce, tomatoes, broccoli, celery, olives, etc.) to give the salad a specific taste. You can think