68Chapter 1 Whole Numbers

106.POPULATION To find the population density of a state, divide its population by its land area (in square miles). The result is the number of people per square mile. Use the data in the table to approximate the population density for each state.

WRITING

107.Explain how 24 6 can be calculated by repeated subtraction.

108.Explain why division of 0 is possible, but division by 0 is impossible.

109.DIVISIBILTY TEST FOR 7 Use the following rule to show that 308 is divisible by 7. Show each of the steps of your solution in writing.

Subtract twice the units digit from the number formed by the remaining digits. If that result is divisible by 7, then the original number is divisible by 7.

110.DIVISIBILTY TEST FOR 11 Use the following rule to show that 1,848 is divisible by 11. Show each of the steps of your solution in writing.

Start with the digit in the one’s place. From it, subtract the digit in the ten’s place.To that result, add the digit in the hundred’s place. From that result, subtract the digit in the thousands place, and so on. If the final result is a number divisible by 11, the original number is divisible by 11.

REVIEW

111.Add: 2,903 378

112.Subtract: 2,903 378

113.Multiply: 2,903 378

114.DISCOUNTS A car, originally priced at $17,550, is being sold for $13,970. By how many dollars has the price been decreased?

1Apply the steps of a problemsolving strategy.

Problem Solving

2Solve problems requiring more than one operation.

3Recognize unimportant information in application problems.

The operations of addition, subtraction, multiplication, and division are powerful tools that can be used to solve a wide variety of real-world problems.

1 Apply the steps of a problem-solving strategy.

To become a good problem solver, you need a plan to follow, such as the following five-step strategy.

Strategy for Problem Solving

1.Analyze the problem by reading it carefully. What information is given? What are you asked to find? What vocabulary is given? Often, a diagram or table will help you visualize the facts of the problem.

2.Form a plan by translating the words of the problem to numbers and symbols.

3.Solve the problem by performing the calculations.

4.State the conclusion clearly. Be sure to include the units (such as feet, seconds, or pounds) in your answer.

5.Check the result. An estimate is often helpful to see whether an answer is reasonable.

One place setting like that shown on the right costs $94. What is the total cost to purchase these place settings for a restaurant that seats 115 people?

Analyze At this stage, it is helpful to list the given facts and what you are to find.

•One place setting costs $94. Given

•115 place settings will be

purchased. Given

•What is the total cost to

purchase 115 place settings? Find

Form The key word total suggests addition. In this case, the total cost to purchase the place settings is the sum of one hundred fifteen 94’s. This repeated addition can be calculated more simply by multiplication.

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

Self Check 1

BEDDING One set of bed linens costs $134. What is the total cost to purchase linens for an 85-bed hotel?

Now Try Problem 17

Self Check 2

LOWFAT MILK A glass of lowfat milk has 56 fewer calories than a glass of whole milk. If a glass of whole milk has 146 calories, how many calories are there in a glass of lowfat milk?

Now Try Problem 19

Solve Use vertical form to perform the multiplication:

11594 460 10 350 10,810

State It will cost $10,810 to purchase 115 place settings.

Check We can estimate to check the result. If we use $100 to approximate the cost of one place setting, then the cost of 115 place settings is about 115 $100 or $11,500. Since the estimate, $11,500, and the result, $10,810, are close, the result seems reasonable.

A glass of nonfat milk has 63 fewer calories than a glass of whole milk. If a glass of whole milk has 146 calories, how many calories are there in a glass of nonfat milk?

Analyze

Form The word fewer indicates subtraction.

Caution! We must be careful when translating subtraction because order is important. Since the 146 calories in a glass of whole milk is to be made 63 calories fewer, we reverse those numbers as we translate from English words to math symbols.

nonfat milk

Solve Use vertical form to perform the subtraction:

146

63

83

State A glass of nonfat milk has 83 calories.

Check We can use addition to check.

8363

A diagram is often helpful when analyzing the problem.

Tunneling A tunnel boring machine can drill through solid rock at a rate of 33 feet per day. How many days will it take the machine to tunnel through 7,920 feet of solid rock?

AP Image

Analyze

In the diagram below, we see that the daily tunneling separates a distance of 7,920 feet into equal-sized lengths of 33 feet. That indicates division.

Self Check 3

OIL WELLS An offshore oil drilling rig can drill through the ocean floor at a rate of 17 feet per hour. How many hours will it take the machine to drill 578 feet to reach a pocket of crude oil?

Now Try Problem 21

Self Check 4

ANATOMY A human skeleton consists of 29 bones in the skull; 26 bones in the spine; 25 bones in the ribs and breastbone; 64 bones in the shoulders, arms, and hands; and 62 bones in the pelvis, legs and feet. In all, how many bones make up the human skeleton?

Now Try Problem 23

240

33 7,9206 6

132132

0000 0

State It will take the tunneling machine 240 days to drill 7,920 feet through solid rock.

Check We can check using multiplication.

24033 720 7200

7920 Quotient divisor dividend. The result checks.

Sometimes it is helpful to organize the given facts of a problem in a table.

Orchestras An orchestra consists of a 19-piece woodwind section, a 23-piece brass section, a 54-piece string section, and a two-person percussion section. In all, how many musicians make up the orchestra?

Analyze We can use a table to organize the facts of the problem.

2 Solve problems requiring more than one operation.

Sometimes more than one operation is needed to solve a problem.

EXAMPLE 5 How many 6-ounce servings are there in a 5-gallon bottle of water? (Hint: There are 128 fluid ounces in 1 gallon.)

Analyze The diagram on the next page is helpful in understanding the problem.

•Since each of the 5 gallons of water is 128 ounces, the total number of ounces is the sum of five 128’s. This repeated addition can be calculated using multiplication.

•Since equal-sized servings of water come from the bottle, this suggests division.

•Therefore, to solve this problem, we need to perform two operations: multiplication and division.

Self Check 5

BOTTLED WATER How many 8-ounce servings are there in a 3-gallon bottle of water? (Hint: There are 128 fluid ounces in

1 gallon.)

Now Try Problem 25

Caution! As you read a problem, it is easy to miss numbers that are written in words. It is helpful to circle those words and write the corresponding number above.

47

Forty-seven people were riding on a bus on Route 66. It arrived at the 7th Street stop at 5:30 PM, where 11 people paid the $1.50 fare to board after 16 riders had exited. As the driver pulled away from that stop at 5:32 PM, how many riders were on the bus?

If we carefully reread the problem, we see that the phrase to board indicates addition and the word exited indicates subtraction.

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

Solve We will solve the problem in horizontal form. Recall from Section 1.3 that the operations of addition and subtraction must be performed as they occur, from left to right.

State There were 42 riders on the bus after the 7th Street stop.

Check The addition can be checked with estimation. To check the subtraction, use: Difference subtrahend minuend.

ANSWERS TO SELF CHECKS

1. It will cost $11,390 to purchase 85 sets of bed linens. 2. There are 90 calories in a glass of lowfat milk. 3. It will take the drilling rig 34 hours to drill 578 feet. 4. There are 206 bones in the human skeleton. 5. In a 3-gallon bottle of water, there are 48 8-ounce servings. 6. There were 46 riders when the bus left the 103rd Street stop.

Self Check 6

BUS SERVICE Thirty-four people were riding on bus number 481. At 11:45 AM, it arrived at the 103rd Street stop where 6 people got off and 18 people paid the 75¢ fare to board. As the driver pulled away from the stop at 11:47 AM, how many riders were on the bus?

Now Try Problem 27

S E C T I O N 1.6 STUDY SET

VOCABULARY

Fill in the blanks.

2. To solve application problems, which are usually given in words, we those words into numbers and mathematical symbols.

Tell whether addition, subtraction, multiplication, or division is indicated by each of the following words and phrases.

CONCEPTS

13.Write the following steps of the problem-solving strategy in the correct order:

State, Check, Analyze, Form, Solve

14.A 12-ounce Mountain Dew has 55 milligrams

of caffeine. Fill in the blanks to translate the following statement to numbers and symbols.

The number of milligrams

of caffeine in a 12-ounce

Dr Pepper

15.Multiply 15 and 8. Then divide that result by 3.

16.Subtract 27 from 100. Then multiply that result by 6.

GUIDED PRACTICE

Solve the following problems. See Example 1.

17.TRUCKING An automobile transport is loaded with 9 new Chevrolet Malibu sedans, each valued at $21,605. What is the total value of the cars carried by the transport?

18. GOLD MEDALS Michael Phelps won 8 gold medals at the 2008 Summer Olympic Games in China. At that time, the actual value of a gold medal was estimated to be about $144. What was the total value of Phelps’ gold medals?

Solve the following problems. See Example 2.

19.TV HISTORY There were 95 fewer episodes of

I Love Lucy made than episodes of The Beverly Hillbillies. If there are 274 episodes of The Beverly Hillbillies, how many episodes of I Love Lucy are there?

20.PETS In 2007, the number of American households owning a cat was estimated to be 5,561,000 fewer than the number of households owning a dog. If 43,021,000 households owned a dog, how many owned a cat? (Source: U.S. Pet Ownership & Demographics Sourcebook, 2007 Edition)

Solve the following problems. See Example 3.

21.CHOCOLATE A study found that 7 grams of dark chocolate per day is the ideal amount to protect against the risk of a heart attack. How many daily servings are there in a bar of dark chocolate weighing 98 grams? (Source: ScienceDaily.com)

22.TRAVELING A tourism website claims travelers can see Europe for $95 a day. If a tourist saved $2,185 for a vacation, how many days can he spend in Europe?

Solve the following problems. Use a table to organize the facts of the problem. See Example 4.

23.THEATER The play Romeo and Juliet by William Shakespeare has five acts. The first act has 5 scenes. The second act has 6 scenes. The third and fourth acts each have 5 scenes, and the last act has 3 scenes. In all, how many scenes are there in the play?

24.STATEHOOD From 1800 to 1850, 15 states joined the Union. From 1851 to 1900, an additional 14 states entered. Three states joined from 1901 to 1950. Since then, Alaska and Hawaii are the only others to enter the Union. In all, how many states have joined the Union since 1800?

Solve the following problems. Use a diagram to show the facts of the problem. See Example 5.

25.BAKING A baker uses 3-ounce pieces of bread dough to make dinner rolls. How many dinner rolls can he make from 5 pounds of dough? (Hint: There are 16 ounces in one pound.)

26.DOOR MATS There are 7 square yards of carpeting left on a roll. How many 4-square-foot door mats can be made from the roll? (Hint: There are 9 square feet in one square yard.)

Solve the following problems. See Example 6.

27.LAPTOPS A file folder named “Finances” on a student’s Thinkpad T60 contained 81 documents. To free up 3 megabytes of storage space, he deleted

26 documents from that folder. Then, 48 hours later, he inserted 13 new documents (2 megabytes) into it. How many documents are now in the student’s “Finances” folder?

28. iPHONES A student had

135 text messages saved on her 16-gigabyte iPhone. She deleted 27 text messages (600 kilobytes) to free up some storage space. Over

TRY IT YOURSELF

29.FORESTS Canada has 2,342,949 fewer square miles of forest than Russia. The United States has

71,730 fewer square miles of forest than Canada. If Russia has 3,287,243 square miles of forest (the most of any country in the world), how many square miles does the United States have? (Source: Maps of World.com)

30.VACATION DAYS Workers in France average 5 fewer days of vacation a year than Italians.

Americans average 24 fewer vacation days than the French. If the Italians average 42 vacation days each year (the most in the world), how many does the average American worker have a year? (Source: infoplease.com)

31.BATMAN As of 2008, the worldwide box office revenue for the following Batman films are The Dark Knight (2008): $998 million, Batman (1989):

$411 million, Batman Forever (1995): $337 million,

Batman Begins (2005): $372 million, Batman Returns

(1992): $267 million, and Batman & Robin (1997): $238 million. What is the total box office revenue for the films? (Source: Wikipedia)

32.SOAP OPERAS The total number of viewers of the top 4 TV soap operas for the week of December 1, 2008, were: The Young and the Restless (5,016,000),

The Bold and the Beautiful (3,587,000), General Hospital (2,853,000), and As the World Turns

(2,694,000). What is the total number of viewers of these programs for that week? (Source: soapoperanetwork.com)

33.MED SCHOOL There were 375 fewer applications to U.S. medical schools submitted by women in 2007 compared to 2008. If 20,735 applications were submitted by women in 2008, how many were submitted in 2007? (Source: AAMC: Data Warehouse)

34.AEROBICS A 30-minute high-impact aerobic workout burns 302 calories. A 30-minute low-impact workout burns 64 fewer calories. How many calories are burned during the 30-minute low-impact workout?

35.TRAVEL How much money will a family of six save on airfare if they take advantage of the offer shown in the advertisement?

Discount Airfare

Roundtrip per person

Los Angeles/Orlando

WAS: $593

NOW! $516

36.DISCOUNT LODGING A hotel is offering rooms that normally go for $129 per night for only $99 a night. How many dollars would a traveler save if he stays in such a room for 5 nights?

37.PAINTING One gallon of latex paint covers

350 square feet. How many gallons are needed if the total area of walls and ceilings to be painted is 9,800 square feet, and if two coats must be

applied?

38.ASPHALT One bucket of asphalt sealcoat covers 420 square feet. How many buckets are needed if a 5,040-square-foot playground is to be sealed with two coats?

39.iPODS The iPod shown has 80 gigabytes (GB) of storage space. From the information in the bar graph, determine how many gigabytes of storage space are used and how many are free to use.

40.MULTIPLE BIRTHS Refer to the table on the next page.

a.Find the total number of children born

in a twin, triplet, or quadruplet birth for the year 2006.

b.Find the total number of children born

in a twin, triplet, or quadruplet birth for the year 2005.

c.In which year were more children born in these ways? How many more?

Source: National Vital Statistics Report, 2009

41.TREES The height of the tallest known tree

(a California Coastal Redwood) is 379 feet. Some scientists believe the tallest a tree can grow is 47 feet more than this because it is difficult for water to be raised from the ground any more than that to support further growth. What do the scientists believe to be the maximum height that a tree can reach? (Source: BBC News)

© Zack Frank, 2009. Used under license from Shutterstock.com

42.CAFFEINE A 12-ounce can of regular Pepsi-Cola contains 38 milligrams of caffeine. The same size can of Pepsi One has 18 more milligrams of caffeine. How many milligrams of caffeine are there in a can of Pepsi One? (Source: wilstar.com)

43.TIME There are 60 minutes in an hour, 24 hours in a day, and 7 days in a week. How many minutes are there in a week?

44.LENGTH There are 12 inches in a foot, 3 feet in a yard, and 1,760 yards in a mile. How many inches are in a mile?

45.FIREPLACES A contractor ordered twelve pallets of fireplace brick. Each pallet holds 516 bricks. If it takes 430 bricks to build a fireplace, how many fireplaces can be built from this order? How many bricks will be left over?

46.ROOFING A roofer ordered 108 squares of shingles. (A square covers 100 square feet of roof.) In a new development, the houses have 2,800 square-feet roofs. How many can be completely roofed with this

order?

47.CROSSWORD PUZZLES A crossword puzzle is made up of 15 rows and 15 columns of small squares. Forty-six of the squares are blacked out. When completed, how many squares in the crossword puzzle will contain letters?

48.CHESS A chessboard consists of 8 rows, with

8 squares in each row. Each of the two players has 16 chess pieces to place on the board, one per square. At the start of the game, how many squares on the board do not have chess pieces on them?

49.CREDIT CARDS The balance on 10/23/10 on Visa account number 623415 was $1,989. If purchases of $125 and $296 were charged to the card on 10/24/10, a payment of $1,680 was credited on 10/31/10, and no other charges or payments were made, what is the new balance on 11/1/10?

50.ARIZONA The average high temperature in Phoenix in January is 65°F. By May, it rises by 29°F, by July it rises another 11°, and by December it falls 39°. What is the average temperature in Phoenix in December? (Source: countrystudies.us)

51.RUNNING Rod Bellears, age 59, has run the

1212 miles from his Upper Skene Street home to his business at Moolap Concrete Products and back every day for more than 20 years. That distance is equal to three times around the Earth. If one trip around the Earth is 7,926 miles, how far has

Mr. Bellears run over the years? (Source: Greelong News)

52.DIAPERS Each year in the United States, 18 billion disposable diapers are used. Laid end-to-end, that’s enough to reach to the moon and back 9 times. If the distance from the Earth to the moon is about 238,855 miles, how far do the disposable diapers extend? (Source: diapersandwipers.com)

53.DVDs A shopper purchased four Blu-ray DVDs: Planet Earth ($59), Wall-E ($26), Elf ($23), and Blade Runner ($37). There was $11 sales tax. If he paid for the DVDs with $20 bills, how many bills were needed? How much did he receive back in change?

61.Write an application problem that would have the following solution. Use the phrase how much does each in the problem.

410,000

6 2,460,000

62.Write an application problem that would have the following solution. Use the word twice in the problem.

55

2 110

REVIEW

63.Check the following addition by adding upward. Is the sum correct?

3,714

2,489

781

5,500303 12,987

64.Check the following subtraction using addition. Is the difference correct?

42,4031,675 40,728

65.Check the following multiplication using estimation. Does the product seem reasonable?

7359 6,407

66.Check the following division using multiplication. Is the quotient correct?

407

27 10,989