107.DECK SUPPORTS After a winter storm, a homeowner has an engineering firm inspect his damaged deck. Their report concludes that the original foundation poles were not sunk deep enough, by a factor of 3. What signed number represents the depth to which the poles should have been sunk?

Ground level

Existing poles 6 feet deep

Poles should be this deep

108.DIETING After giving a patient a physical exam, a physician felt that the patient should begin a diet. The two options that were discussed are shown in the following table.

a.Find the expected weight loss from Plan 1. Express the answer as a signed number.

b.Find the expected weight loss from Plan 2. Express the answer as a signed number.

c.With which plan should the patient expect to lose more weight? Explain why the patient might not choose it.

109.ADVERTISING The paid attendance for the last night of the 2008 Rodeo Houston was 71,906. Suppose a local country music radio station gave a sports bag, worth $3, to everyone that attended. Find the signed number that expresses the radio station’s financial loss from this giveaway.

110.HEALTH CARE A health care provider for a company estimates that 75 hours per week are lost by employees suffering from stress-related or preventable illness. In a 52-week year, how many hours are lost? Use a signed number to answer.

WRITING

111.Explain why the product of a positive number and a negative number is negative, using 5( 3) as an example.

112.Explain the multiplication rule for integers that is shown in the pattern of signs below.

( )( )

( )( )( )

( )( )( )( )

( )( )( )( )( )

113.When a number is multiplied by 1, the result is the opposite of the original number. Explain why.

114.A student claimed, “A positive and a negative is negative.” What is wrong with this statement?

REVIEW

115.List the first ten prime numbers.

116.ENROLLMENT The number of students attending a college went from 10,250 to 12,300 in one year. What was the increase in enrollment?

117.Divide: 175 4

118.What does the symbol mean?

Dividing Integers

1 Divide two integers.

In this section, we will develop rules for division of integers, just as we did earlier for multiplication of integers.

1 Divide two integers.

Recall from Section 1.5 that every division has a related multiplication statement. For example,

2Identify division of 0 and division by 0.

3Solve application problems by dividing integers.

Strategy We will use the rule for dividing two integers that have different (unlike) signs.

WHY Each division involves a positive and a negative integer.

Solution

a. Find the absolute values: 0 14 0 14 and 07 0 7.

14 2 Divide the absolute values, 14 by 7, to get 2.

7

Then make the final answer negative.

To check, we multiply the quotient, 2, and the divisor, 7. We should get the dividend, 14.

d.Recall from Section 1.5, that if a divisor has ending zeros, we can simplify the division by removing the same number of ending zeros in the divisor and dividend.

There are two zeros in the divisor.

Strategy We will use the rule for dividing two integers that have the same (like) signs.

WHY In each case, we are asked to find the quotient of two negative integers.

Solution

a. Find the absolute values: 0 12 0 12 and 0 3 0 3.

12

3 4 Divide the absolute values, 12 by 3, to get 4.

The final answer is positive.

Check: 4( 3) 12 The result checks.

Self Check 1

Divide and check the result:

45

a.

5

b. 28 ( 4)

336

c.14

d.18,000 300

Now Try Problems 13,15,21,and 27

Self Check 2

Divide and check the result:

27

a.3

b.24 ( 4)

301

c.7

d.400 ( 20)

Now Try Problems 33,37,41,and 43

Strategy In each case, we need to determine if we have division of 0 or division by 0.

WHY Division of 0 by a nonzero integer is defined, and the answer is 0. However, division of a nonzero integer by 0 is undefined; there is no answer.

Solution

3 Solve application problems by dividing integers.

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

Form We can express the drop in the price of the house for the year as $11,400. The phrase reduced by an equal amount each month indicates division.

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

Solve To find the quotient, we use the rule for dividing two integers that have different signs. First, we find the absolute values: 0 11,400 0 11,400 and 012 0 12.

State The negative result indicates that the price of the house was reduced by $950 each month.

Check We can use estimation to check the result. A reduction of $1,000 each month would cause the price to drop $12,000 in 1 year. It seems reasonable that a reduction of $950 each month would cause the price to drop $11,400 in a year.

Now Try Problems 45 and 47

Self Check 4

SELLING BOATS The owner of a sail boat reduced the price of the boat by an equal amount each month, because there were no interested buyers.After 8 months, and a $960 reduction in price, the boat sold. By how much was the price of the boat reduced each month?

Now Try Problem 81

S E C T I O N 2.5 STUDY SET

VOCABULARY

Fill in the blanks.

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

12 ( 4) 3

CONCEPTS

5.Write the related multiplication statement for each division.

6.Using multiplication, check to determine whether720 45 12.

7.Fill in the blanks.

To divide two integers, divide their absolute values.

a. The quotient of two integers that have the same (like) signs is .

b. The quotient of two integers that have different (unlike) signs is .

8.If a divisor has ending zeros, we can simplify the division by removing the same number of ending zeros in the divisor and dividend. Fill in the blank:2,400 60 240

9.Fill in the blanks.

a.If 0 is divided by any nonzero integer, the quotient is .

10.What operation can be used to solve problems that involve forming equal-sized groups?

11.Determine whether each statement is always true, sometimes true, or never true.

a.The product of a positive integer and a negative integer is negative.

b.The sum of a positive integer and a negative integer is negative.

c.The quotient of a positive integer and a negative integer is negative.

12.Determine whether each statement is always true, sometimes true, or never true.

a.The product of two negative integers is positive.

b.The sum of two negative integers is negative.

c.The quotient of two negative integers is negative.

GUIDED PRACTICE

Divide and check the result. See Example 1.

25.9,000 300

26.12,000 600

27.250,000 5,000

28.420,000 7,000

Divide and check the result. See Example 2.

55.Find the quotient of 45 and 9.

56.Find the quotient of 36 and 4.

Use a calculator to perform each division.

APPLICATIONS

Use signed numbers to solve each problem.

81.LOWERING PRICES A furniture store owner reduced the price of an oak table an equal amount each week, because it was not selling. After six weeks, and a $210 reduction in price, the table was purchased. By how much was the price of the table reduced each week?

82.TEMPERATURE DROP During a five-hour period, the temperature steadily dropped 20°F. By how many degrees did the temperature change each hour?

83.SUBMARINES In a series of three equal dives, a submarine is programmed to reach a depth of 3,030 feet below the ocean surface. What signed number describes how deep each of the dives will be?

84.GRAND CANYON A mule train is to travel from a stable on the rim of the Grand Canyon to a camp on the canyon floor, approximately 5,500 feet below the rim. If the guide wants the mules to be rested after every 500 feet of descent, how many stops will be made on the trip?

85.CHEMISTRY During an experiment, a solution was steadily chilled and the times and temperatures were recorded, as shown in the illustration below. By how many degrees did the temperature of the solution change each minute?

86.OCEAN EXPLORATION The Mariana Trench is the deepest part of the world’s oceans. It is located in the North Pacific Ocean near the Philippines and has a maximum depth of 36,201 feet. If a remotecontrolled vessel is sent to the bottom of the trench

in a series of 11 equal descents, how far will the vessel descend on each dive? (Source: marianatrench.com)

87.BASEBALL TRADES At the midway point of the season, a baseball team finds itself 12 games behind the league leader. Team management decides to trade for a talented hitter, in hopes of making up at least half of the deficit in the standings by the end of the year. Where in the league standings does management expect to finish at season’s end?

88.BUDGET DEFICITS A politician proposed a twoyear plan for cutting a county’s $20-million budget deficit, as shown. If this plan is put into effect, how will the deficit change in two years?

89.MARKDOWNS The owner of a clothing store decides to reduce the price on a line of jeans that are not selling. She feels she can afford to lose $300 of projected income on these pants. By how much can she mark down each of the 20 pairs of jeans?

90.WATER STORAGE Over a week’s time, engineers at a city water reservoir released enough water to lower the water level 105 feet. On average, how much did the water level change each day during this period?

91.THE STOCK MARKET On Monday, the value of Maria’s 255 shares of stock was at an all-time high. By Friday, the value had fallen $4,335. What was her per-share loss that week?

92.CUTTING BUDGETS In a cost-cutting effort, a company decides to cut $5,840,000 from its annual budget. To do this, all of the company’s 160 departments will have their budgets reduced by an equal amount. By how much will each department’s budget be reduced?

WRITING

93.Explain why the quotient of two negative integers is positive.

94.How do the rules for multiplying integers compare with the rules for dividing integers?

95.Use a specific example to explain how multiplication can be used as a check for division.

96.Explain what it means when we say that division by 0 is undefined.

97.Explain the division rules for integers that are shown below using symbols.

98.Explain the difference between division of 0 and division by 0.

REVIEW

100.Find the prime factorization of 210.

101.The statement (4 8) 10 4 (8 10) illustrates what property?

102.Is 17 17 a true statement?

103.Does 8 2 2 8?

104.Sharif has scores of 55, 70, 80, and 75 on four mathematics tests. What is his mean (average) score?