Copyright © Cengage Learning. All rights reserved. 1 Whole Numbers Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. S E C T I O N 1.5 Dividing Whole Numbers Copyright © Cengage Learning. All rights reserved.

Objectives Write the related multiplication statement for a division. Use properties of division to divide whole numbers. Perform long division (no remainder). Perform long division (with a remainder).

Objectives 5. Use tests for divisibility. Divide whole numbers that end with zeros. Estimate quotients of whole numbers. Solve application problems by dividing whole numbers.

Dividing Whole Numbers Division of whole numbers is used by everyone. For example, to find how many 6-ounce servings a chef can get from a 48-ounce roast, he divides 48 by 6. To split a $36,000 inheritance equally, a brother and sister divide the amount by 2. A professor divides the 35 students in her class into groups of 5 for discussion.

Write the related multiplication statement for a division 1

Write the related multiplication statement for a division To divide whole numbers, think of separating a quantity into equal-sized groups. For example, if we start with a set of 12 stars and divide them into groups of 4 stars, we will obtain 3 groups. A set of 12 stars. There are 3 groups of 4 stars.

Write the related multiplication statement for a division We can write this division problem using a division symbol , a long division symbol or a fraction bar We call the number being divided the dividend and the number that we are dividing by is called the divisor. The answer is called the quotient. Division symbol Long division symbol Fraction bar We read each form as “12 divided by 4 equals (or is) 3.”

Write the related multiplication statement for a division Recall that multiplication is repeated addition. Likewise, division is repeated subtraction. To divide 12 by 4, we ask, “How many 4’s can be subtracted from 12?” Since exactly three 4’s can be subtracted from 12 to get 0, we know that 12  4 = 3. Subtract 4 one time. Subtract 4 a second time. Subtract 4 a third time.

Write the related multiplication statement for a division Another way to answer a division problem is to think in terms of multiplication. For example, the division 12  4 asks the question, “What must I multiply 4 by to get 12?” Since the answer is 3, we know that 12  4 = 3 because 3  4 = 12 We call 3  4 = 12 the related multiplication statement for the division 12  4 = 3.

Write the related multiplication statement for a division In general, to write the related multiplication statement for a division, we use: Quotient  divisor = dividend

Example 1 Write the related multiplication statement for each division. a. b. c. Strategy: We will identify the quotient, the divisor, and the dividend in each division statement. Solution: a. 10  5 = 2 because 2  5 = 10. Dividend Quotient Divisor

Example 1 – Solution b. because 4  6 = 24. c. because 7  3 = 21. cont’d b. because 4  6 = 24. c. because 7  3 = 21. 4 is the quotient, 6 is the divisor, and 24 is the dividend. 7 is the quotient, 3 is the divisor, and 21 is the dividend.

Use properties of division to divide whole numbers 2

Use properties of division to divide whole numbers Recall that the product of any whole number and 1 is that whole number. We can use that fact to establish two important properties of division. Consider the following examples where a whole number is divided by 1: 8  1 = 8 because 8  1 = 8. because 4  1 = 4. because 20  1 = 20.

Use properties of division to divide whole numbers These examples illustrate that any whole number divided by 1 is equal to the number itself. Consider the following examples where a whole number is divided by itself: 6  6 = 1 because 1  6 = 6. because 1  9 = 9. because 1  35 = 35. These examples illustrate that any nonzero whole number divided by itself is equal to 1.

Use properties of division to divide whole numbers Recall that the product of any whole number and 0 is 0. We can use that fact to establish another property of division. Consider the following examples where 0 is divided by a whole number: 0  2 = 0 because 0  2 = 0.

Use properties of division to divide whole numbers because 0  7 = 0. because 0  42 = 0. These examples illustrate that 0 divided by any nonzero whole number is equal to 0.

Use properties of division to divide whole numbers We cannot divide a whole number by 0. To illustrate why, we will attempt to find the quotient when 2 is divided by 0 using the related multiplication statement shown below. Division statement Related multiplication statement ?  0 = 0. There is no number that gives 2 when multiplied by 0.

Use properties of division to divide whole numbers Since does not have a quotient, we say that division of 2 by 0 is undefined. Our observations about division of 0 and division by 0 are listed below.

Perform long division (no remainder) 3

Perform long division (no remainder) A process called long division can be used to divide larger whole numbers.

Example 2 Divide using long division: 2,514  6. Check the result. Strategy: We will write the problem in long-division form and follow a four-step process: estimate, multiply, subtract, and bring down. Solution: To help you understand the process, each step of this division is explained separately. Your solution need only look like the last step.

Example 2 – Solution cont’d We write the problem in the form The quotient will appear above the long division symbol. Since 6 will not divide 2, we divide 25 by 6. Next, we multiply 4 and 6, and subtract their product, 24, from 25, to get 1.

Example 2 – Solution cont’d Now we bring down the next digit in the dividend, the 1, and again estimate, multiply, and subtract.

Example 2 – Solution cont’d To complete the process, we bring down the last digit in the dividend, the 4, and estimate, multiply, and subtract one final time.

Example 2 – Solution cont’d To check the result, we see if the product of the quotient and the divisor equals the dividend. The check confirms that 2,514  6 = 419.

Perform long division (no remainder) We can see how the long division process works if we write the names of the place value columns above the quotient. The solution for Example 2 is shown below in more detail.

Perform long division (no remainder) The extra zeros (shown in the steps highlighted in red and blue) are often omitted. We can use long division to perform divisions when the divisor has more than one digit. The estimation step is often made easier if we approximate the divisor.

Perform long division (with a remainder) 4

Perform long division (with a remainder) Sometimes, it is not possible to separate a group of objects into a whole number of equal-sized groups. For example, if we start with a set of 14 stars and divide them into groups of 4 stars, we will have 3 groups of 4 stars and 2 stars left over. We call the left over part the remainder. A set of 14 stars. There are 3 groups of 4 stars. There are 2 stars left over.

Perform long division (with a remainder) In the next long division example, there is a remainder. To check such a problem, we add the remainder to the product of the quotient and divisor. The result should equal the dividend. (Quotient  divisor) + remainder = dividend Recall that the operation within the parentheses must be performed first.

Example 4 Divide: Check the result. Strategy: We will follow a four-step process: estimate, multiply, subtract, and bring down. Solution: Since 23 will not divide 8, we divide 83 by 23.

Example 4 – Solution cont’d

Example 4 – Solution cont’d

Example 4 – Solution cont’d The quotient is 36, and the remainder is 4. We can write this result as 36 R 4. To check the result, we multiply the divisor by the quotient and then add the remainder. The result should be the dividend.

Example 4 – Solution Check: (36  23) + 4 = 828 + 4 = 832 cont’d Check: (36  23) + 4 = 828 + 4 = 832 Since 832 is the dividend, the answer 36 R 4 is correct. Quotient Divisor Remainder Dividend

Use tests for divisibility 5

Use tests for divisibility We have seen that some divisions end with a 0 remainder and others do not. The word divisible is used to describe such situations. Since 27  3 = 9, with a 0 remainder, we say that 27 is divisible by 3. Since 27  5 = 5 R 2, we say that 27 is not divisible by 5. There are tests to help us decide whether one number is divisible by another.

Use tests for divisibility There are tests for divisibility by a number other than 2, 3, 4, 5, 6, 9, or 10, but they are more complicated.

Example 6 Is 534,840 divisible by: a. 2 b. 3 c. 4 d. 5 e. 6 f. 9 g. 10 Strategy: We will look at the last digit, the last two digits, and the sum of the digits of each number. Solution: a. 534,840 is divisible by 2, because its last digit 0 is divisible by 2. b. 534,840 is divisible by 3, because the sum of its digits is divisible by 3. 5 + 3 + 4 + 8 + 4 + 0 = 24 and 24  3 = 8

Example 6 – Solution cont’d c. 534,840 is divisible by 4, because the number formed by its last two digits is divisible by 4. 40  4 = 10 d. 534,840 divisible by 5, because its last digit is 0 or 5. e. 534,840 is divisible by 6, because it is divisible by 2 and 3. (See parts a and b.)

Example 6 – Solution cont’d f. 534,840 is not divisible by 9, because the sum of its digits is not divisible by 9. There is a remainder. 24  9 = 2 R 6 g. 534,840 is divisible by 10, because its last digit is 0.

Divide whole numbers that end with zeros 6

Divide whole numbers that end with zeros There is a shortcut for dividing a dividend by a divisor when both end with zeros. We simply remove the ending zeros in the divisor and remove the same number of ending zeros in the dividend.

Example 7 Divide: a. 80  10 b. 47,000  100 c. Strategy: We will look for ending zeros in each divisor. Solution: a. 80  10 = 8  1 There is one zero in the divisor. Remove one zero from the dividend and the divisor, and divide. = 8

Example 7 – Solution b. 47,000  100 = 470  1 c. To find cont’d b. 47,000  100 = 470  1 c. To find we can drop one zero from the divisor and the dividend and perform the division There are two zeros in the divisor. Remove two zeros from the dividend and the divisor, and divide. = 470

Example 7 – Solution cont’d Thus, 9,800  350 is 28.

Estimate quotients of whole numbers 7

Estimate quotients of whole numbers To estimate quotients, we use a method that approximates both the dividend and the divisor so that they divide easily. There is one rule of thumb for this method: If possible, round both numbers up or both numbers down.

Example 8 Estimate the quotient: 170,715  57 Strategy: We will round the dividend and the divisor up and find 180,000  60. Solution: 170,715  57 180,000  60 = 3,000 The estimate is 3,000. The dividend is approximately To divide, drop one zero from 180,000 and from 60 and find 18,000  6. The divisor is approximately

Example 8 – Solution cont’d If we calculate 170,715  57, the quotient is exactly 2,995. Note that the estimate is close: It’s just 5 more than 2,995.

Solve application problems by dividing whole numbers 8

Solve application problems by dividing whole numbers Application problems that involve forming equal-sized groups can be solved by division.

Example 9 – Managing a Soup Kitchen A soup kitchen plans to feed 1,990 people. Because of space limitations, only 144 people can be served at one time. How many group seatings will be necessary to feed everyone? How many will be served at the last seating? Strategy: We will divide 1,990 by 144.

Example 9 – Solution We translate the words of the problem to numbers and symbols.

Example 9 – Solution Use long division to find 1,990  144. cont’d Use long division to find 1,990  144. The quotient is 13, and the remainder is 118. This indicates that fourteen group seatings are needed: 13 full-capacity seatings and one partial seating to serve the remaining 118 people.