Objectives

1Write the related multiplication statement for a division.

2Use properties of division to divide whole numbers.

3Perform long division (no remainder).

4Perform long division (with a remainder).

5Use tests for divisibility.

6Divide whole numbers that end with zeros.

7Estimate quotients of whole numbers.

8Solve application problems by dividing whole numbers.

S E C T I O N 1.5

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.

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.

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.

We read each form as “12 divided by 4 equals (or is) 3.”

Recall from Section 1.4 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?”

12

4

8

4

4

4

0

Since exactly three 4’s can be subtracted from 12 to get 0, we know that 12 4 3.

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. In general, to write the related multiplication statement for a division, we use:

Quotient divisor dividend

Strategy We will identify the quotient, the divisor, and the dividend in each division statement.

WHY A related multiplication statement has the following form: Quotient divisor dividend.

Solution

4

b. 6 24 because 4 6 24. 4 is the quotient, 6 is the divisor, and 24 is the dividend.

c. 213 7 because 7 3 21. 7 is the quotient, 3 is the divisor, and 21 is the dividend.

The Language of Mathematics To describe the special relationship between multiplication and division, we say that they are inverse operations.

2 Use properties of division to divide whole numbers.

Recall from Section 1.4 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.

4

1 4 because 4 1 4.

201 20 because 20 1 20.

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.

1

9 9 because 1 9 9.

35

35 1 because 1 35 35.

These examples illustrate that any nonzero whole number divided by itself is equal to 1.

Properties of Division

Self Check 1

Write the related multiplication statement for each division.

a. 8 2 4

8

b.7 56

36

c.4 9

Now Try Problems 19 and 23

Division statement

20 ?

Related multiplication statement

? 0 2

There is no number that gives 2 when multiplied by 0.

Self Check 2

Divide using long division: 2,968 4. Check the result.

Now Try Problem 31

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.

WHY The repeated subtraction process would take too long to perform and the related multiplication statement (? 6 = 2,514) is too difficult to solve.

Solution

To help you understand the process, each step of this division is explained separately. Your solution need only look like the last step.

We write the problem in the form 6 2514. The quotient will appear above the long division symbol. Since 6 will not divide 2,

6 2514

we divide 25 by 6.

4

419

6 2514

24 11

6

54

54 0

Ask: “How many times will 6 divide 54?” We

estimate that 54 6 is 9, and we write the

9 in the ones column above the long division symbol. Multiply 9 and 6, and subtract their product, 54, from 54, to get 0.

5454 0

Self Check 3

Divide using long division:

57 45,885

Now Try Problem 35

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.

Strategy We will follow a four-step process: estimate, multiply, subtract, and bring down.

WHY This is how long division is performed.

Solution

Since 23 will not divide 8, we divide 83 by 23.

4

23 832 Ask: “How many times will 23 divide 83?” Since 23 is about 20, we can estimate the answer to that question by thinking 8 2 is 4, and we write the 4 in the tens column of the quotient.

4

23 832 Multiply 4 and 23, and write their product, 92, under the 83. Because 9292 is greater than 83, the estimate of 4 for the tens column of the quotient is too large. We will erase the 4 and decrease the estimate of the quotient

by 1 and try again.

Self Check 4

Divide: 34 792. Check the result.

Now Try Problem 39

Self Check 5

28,992

Divide:

629

Now Try Problem 43

25

518 13011

1036

2651

2590 61

Bring down the 1 from the ones column of the dividend. Ask: “How many times will 518 divide 2,651?” We can estimate the answer to

that question by thinking 26 5 is about 5, and we write the 5 in the ones column of the quotient. Multiply 5 and 518, and subtract their product, 2,590, from 2,651, to get a remainder of 61.

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.

Tests for Divisibility

A number is divisible by

•2 if its last digit is divisible by 2.

•3 if the sum of its digits is divisible by 3.

•4 if the number formed by its last two digits is divisible by 4.

•5 if its last digit is 0 or 5.

•6 if it is divisible by 2 and 3.

•9 if the sum of its digits is divisible by 9.

•10 if its last digit is 0.

There are tests for divisibility by a number other than 2, 3, 4, 5, 6, 9, or 10, but they are more complicated. See problems 109 and 110 of Study Set 1.5 for some examples.

Strategy We will look at the last digit, the last two digits, and the sum of the digits of each number.

WHY The divisibility rules call for these types of examination.

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

Self Check 6

Is 73,311,435 divisible by:

Now Try Problems 49 and 53

Self Check 7

Divide: a. 50 10

b.62,000 100

c.12,000 1,500

Now Try Problems 55 and 57

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.)

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.

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.

Strategy We will look for ending zeros in each divisor.

WHY If a divisor has ending zeros, we can simplify the division by removing the same number of ending zeros in the divisor and dividend.

Solution

There is one zero in the divisor.

a.80 10 8 1 8

Remove one zero from the dividend and the divisor, and divide.

There are two zeros in the divisor.

b.47,000 100 470 1 470

Remove two zeros from the dividend and the divisor, and divide.

c. To find

350 9,800

we can drop one zero from the divisor and the dividend and perform the division 35 980.

28

35 98070 280280 0

Thus, 9,800 350 is 28.

7 Estimate quotients of whole numbers.

EXAMPLE 8 Estimate the quotient: 170,715 57

Strategy We will round the dividend and the divisor up and find 180,000 60.

WHY The division can be made easier if the dividend and the divisor end with zeros. Also, 6 divides 18 exactly.

Solution

The estimate is 3,000.

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.

Self Check 8

Estimate the quotient: 33,642 42

Now Try Problem 59

8 Solve application problems by dividing whole numbers.

Application problems that involve forming equal-sized groups can be solved by division.

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.

WHY Separating 1,990 people into equal-sized groups of 144 indicates division.

Solution

We translate the words of the problem to numbers and symbols.

Use long division to find 1,990 144.

13

144 1,990144 550432 118

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.

Self Check 9

MOVIE TICKETS On a Saturday, 3,924 movie tickets were purchased at an IMAX theater. Each showing of the movie was sold out, except for the last. If the theater seats 346 people, how many times was the movie shown on Saturday? How many people were at the last showing?

Now Try Problem 91

Self Check 10

TOURING A rock band will take a 275-day world tour and spend the same number of days in each of 25 cities. How long will they stay in each city?

Now Try Problem 97

Timeshares Every year, the 73 part-owners of a timeshare resort condominium get use of it for an equal number of days. How many days does each part-owner get to stay at the condo? (Use a 365-day year.)

Strategy We will divide 365 by 73.

WHY Since the part-owners get use of the condo for an equal number of days, the phrase “How many days does each” indicates division.

Solution

We translate the words of the problem to numbers and symbols.

Use long division to find 365 73.

5

73 365365 0

Each part-owner gets to stay at the condo for 5 days during the year.

Using Your CALCULATOR

The Division Key

Bottled water

A beverage company production run of 604,800 bottles of mountain spring water will be shipped to stores on pallets that hold

1,728 bottles each. We can find the number of full pallets to be shipped using the division key on a calculator.

On some calculator models, the ENTER key is pressed instead of for the result to be displayed.

The beverage company will ship 350 full pallets of bottled water.

S E C T I O N 1.5 STUDY SET

VOCABULARY

Fill in the blanks.

1.In the three division problems shown below, label the dividend, divisor, and the quotient.

12 4 3

CONCEPTS

7.a. Divide the objects below into groups of 3. How many groups of 3 are there?

•• • • • • • • • • • • • • • • • • • • •

b.Divide the objects below into groups of 4. How many groups of 4 are there? How many objects are left over?

** * * * * * * * * * * * * * * * * * * * *

8.Tell whether each statement is true or false.

a.Any whole number divided by 1 is equal to that number.

b.Any nonzero whole number divided by itself is equal to 1.

c.Zero divided by any nonzero number is undefined.

d.Division of a number by 0 is equal to 0.

11. Find the first digit of each quotient.

13.To check whether the division 9 333 is correct, we use multiplication:

9

14.a. A number is divisible by if its last digit is divisible by 2.

b.A number is divisible by 6 if it is divisible by and .

d. A number is divisible by if its last digit is 0.

16.We can simplify the division 43,800 200 by

NOTATION

17.Write three symbols that can be used for division.

18.In a division, 35 R 4 means “a quotient of 35 and a

of 4.”

GUIDED PRACTICE

Fill in the blanks. See Example 1.

21.44 11 4 because .

22.120 12 10 because .

Write the related multiplication statement for each division.

See Example 1.

23. 21 3 7

72

25. 12 6

Divide using long division. Check the result. See Example 2.

28. 72 4

Divide using long division. Check the result. See Example 3.

Divide using long division. Check the result. See Example 4.

Divide using long division. Check the result. See Example 5.

If the given number is divisible by 2, 3, 4, 5, 6, 9, or 10, enter a checkmark in the box. See Example 6.

47.2,940

48.5,850

49.43,785

50.72,954

51.181,223

52.379,157

53.9,499,200

54.6,653,100

Use a division shortcut to find each quotient. See Example 7.

APPLICATIONS

88.RUNNING Brian runs 7 miles each day. In how many days will Brian run 371 miles?

89.DUMP TRUCKS A 15-cubic-yard dump truck must haul 405 cubic yards of dirt to a construction site. How many trips must the truck make?

90.STOCKING SHELVES After receiving a delivery of 288 bags of potato chips, a store clerk stocked each shelf of an empty display with 36 bags. How many shelves of the display did he stock with potato chips?

91.LUNCH TIME A fifth grade teacher received 50 half-pint cartons of milk to distribute evenly to his class of 23 students. How many cartons did each child get? How many cartons were left over?

92.BUBBLE WRAP A furniture manufacturer uses an 11-foot-long strip of bubble wrap to protect a lamp when it is boxed and shipped to a customer. How many lamps can be packaged in this way from a 200-foot-long roll of bubble wrap? How many feet will be left on the roll?

93.GARDENING A metal can holds 640 fluid ounces of gasoline. How many times can the 68-ounce tank of a lawnmower be filled from the

can? How many ounces of gasoline will be left in the can?

94.BEVERAGES A plastic container holds 896 ounces of punch. How many 6-ounce cups of punch can be served from the container? How many ounces will be left over?

95.LIFT SYSTEMS If the bus weighs 58,000 pounds, how much weight is on each jack?

96.LOTTERY WINNERS In 2008, a group of 22 postal workers, who had been buying Pennsylvania Lotto tickets for years, won a $10,282,800 jackpot. If they split the prize evenly, how much money did each person win?

97.TEXTBOOK SALES A store received $25,200 on the sale of 240 algebra textbooks. What was the cost of each book?

98.DRAINING POOLS A 950,000-gallon pool is emptied in 20 hours. How many gallons of water are drained each hour?

99.MILEAGE A tour bus has a range of 700 miles on one tank (140 gallons) of gasoline. How far does the bus travel on one gallon of gas?

100.WATER MANAGEMENT The Susquehanna River discharges 1,719,000 cubic feet of water into Chesapeake Bay in 45 seconds. How many cubic feet of water is discharged in one second?

101.ORDERING SNACKS How many dozen doughnuts must be ordered for a meeting if

156 people are expected to attend, and each person will be served one doughnut?

102.TIME A millennium is a period of time equal to one thousand years. How many decades are in a millennium?

103.VOLLEYBALL A total of 216 girls are going to play in a city volleyball league. How many girls should be put on each team if the following requirements must be met?

•All the teams are to have the same number of players.

•A reasonable number of players on a team is 7 to 10.

•For scheduling purposes, there must be an even number of teams (2, 4, 6, 8, and so on).

104.WINDSCREENS A farmer intends to plant pine trees 12 feet apart to form a windscreen for her crops. How many trees should she buy if the length of the field is 744 feet?

105.ENTRY-LEVEL JOBS The typical starting salaries for 2008 college graduates majoring in nursing, marketing, and history are shown below. Complete the last column of the table.

College major Yearly salary Monthly salary

Nursing $52,128

Marketing $43,464

History $35,952

Source: CNN.com/living