1Multiply decimals.

2Multiply decimals by powers of 10.

3Multiply signed decimals.

4Evaluate exponential expressions that have decimal bases.

5Use the order of operations rule.

6Evaluate formulas.

7Estimate products of decimals.

8Solve application problems by multiplying decimals.

Multiplying Decimals

Since decimal numbers are base-ten numbers, multiplication of decimals is similar to multiplication of whole numbers. However, when multiplying decimals, there is one additional step—we must determine where to write the decimal point in the product.

1 Multiply decimals.

To develop a rule for multiplying decimals, we will consider the multiplication 0.3 0.17 and find the product in a roundabout way. First, we write 0.3 and 0.17 as fractions and multiply them in that form. Then we express the resulting fraction as a decimal.

From this example, we can make observations about multiplying decimals.

• The digits in the answer are found by multiplying 3 and 17.

Self Check 1

Multiply: 2.7 4.3

Now Try Problem 9

Multiplying Decimals

To multiply two decimals:

1.Multiply the decimals as if they were whole numbers.

2.Find the total number of decimal places in both factors.

3.Insert a decimal point in the result from step 1 so that the answer has the same number of decimal places as the total found in step 2.

EXAMPLE 1 Multiply: 5.9 3.4

Strategy We will ignore the decimal points and multiply 5.9 and 3.4 as if they were whole numbers. Then we will write a decimal point in that result so that the final answer has two decimal places.

Self Check 2

Multiply: (0.0002)7.2

Now Try Problem 13

Self Check 3

Multiply: 178(4.7)

Now Try Problem 17

2345.1

23 4

1170 0

1193.4

Move 1 place from the right to the left and insert a decimal point in the answer.

One zero in 10 Two zeros in 100

These observations illustrate the following rule.

Three zeros in 1,000

8.675 1,000 8675

It moves 3 places to the right.

EXAMPLE 4 Multiply: a. 2.81 10 b. 0.076(10,000)

Strategy For each multiplication, we will identify the factor that is a power of 10, and count the number of zeros that it has.

WHY To find the product of a decimal and a power of 10 that is greater than 1, we move the decimal point to the right the same number of places as there are zeros in the power of 10.

Solution

a. 2.81 10 28.1 Since 10 has 1 zero, move the decimal point 1 place to the right.

Numbers such as 10, 100, and 1,000 are powers of 10 that are greater than 1. There are also powers of 10 that are less than 1, such as 0.1, 0.01, and 0.001.To develop a rule to find the product when multiplying a decimal by one tenth, one hundredth, one thousandth, and so on, we will consider three examples:

Self Check 4

Multiply:

a.0.721 100

b.6.08(1,000)

Now Try Problems 21 and 23

Self Check 5

Multiply:

a.0.1(129.9)

b.0.002 0.00001

Now Try Problems 25 and 27

Self Check 6

Write each number in standard notation:

a.567.1 million

b.50.82 billion

c.4.133 trillion

Now Try Now Try Problems 29, 31, and 33

Multiplying a Decimal by 0.1, 0.01, 0.001, and So On

To find the product of a decimal and 0.1, 0.01, 0.001, and so on, move the decimal point to the left the same number of decimal places as there are in the power of 10.

EXAMPLE 5 Multiply: a. 145.8 0.01 b. 9.76(0.0001)

Strategy For each multiplication, we will identify the factor of the form 0.1, 0.01, and 0.001, and count the number of decimal places that it has.

WHY To find the product of a decimal and a power of 10 that is less than 1, we move the decimal point to the left the same number of decimal places as there are in the power of 10.

Solution

b. 9.76(0.0001) 0.000976 Since 0.0001 has four decimal places, move the decimal point in 9.76 four places to the left. This requires that

three placeholder zeros (shown in blue) be inserted in front of the 9.

Quite often, newspapers, websites, and television programs present large numbers in a shorthand notation that involves a decimal in combination with a place-value column name. For example,

•As of December 31, 2008, Sony had sold 21.3 million Playstation 3 units worldwide. (Source: Sony Computer Entertainment)

•Boston’s Big Dig was the most expensive single highway project in U.S. history. It cost about $14.63 billion. (Source: Roadtraffic-technology.com)

•The distance that light travels in one year is about 5.878 trillion miles. (Source: Encyclopaedia Britannica)

We can use the rule for multiplying a decimal by a power of ten to write these large numbers in standard form.

Strategy We will express each of the large numbers as the product of a decimal and a power of 10.

WHY Then we can use the rule for multiplying a decimal by a power of 10 to find their product. The result will be in the required standard form.

Solution

a.21.3 million 21.3 1 million

21.3 1,000,000 Write 1 million in standard form.

b.14.63 billion 14.63 1 billion

14.63 1,000,000,000 Write 1 billion in standard form.

c.5.9 trillion 5.9 1 trillion

5.9 1,000,000,000,000 Write 1 trillion in standard form.

3 Multiply signed decimals.

The rules for multiplying integers also hold for multiplying signed decimals. The product of two decimals with like signs is positive, and the product of two decimals with unlike signs is negative.

EXAMPLE 7 Multiply: a. 1.8(4.5) b. ( 1,000)( 59.08)

Strategy In part a, we will use the rule for multiplying signed decimals that have different (unlike) signs. In part b, we will use the rule for multiplying signed decimals that have the same (like) signs.

WHY In part a, one factor is negative and one is positive. In part b, both factors are negative.

Solution

a. Find the absolute values: 0 1.8 0 1.8 and 04.5 0 4.5. Since the decimals have unlike signs, their product is negative.

b.Find the absolute values: 0 1,000 0 1,000 and 0 59.08 0 59.08. Since the decimals have like signs, their product is positive.

( 1,000)( 59.08) 1,000(59.08) Multiply the absolute values, 1,000 and

4 Evaluate exponential expressions that have decimal bases.

We have evaluated exponential expressions that have whole number bases, integer bases, and fractional bases. The base of an exponential expression can also be a positive or a negative decimal.

EXAMPLE 8 Evaluate: a. (2.4)2 b. ( 0.05)2

Strategy We will write each exponential expression as a product of repeated factors, and then perform the multiplication.This requires that we identify the base and the exponent.

WHY The exponent tells the number of times the base is to be written as a factor.

Self Check 7

Multiply:

a.6.6( 5.5)

b.44.968( 100)

Now Try Problems 37 and 41

Self Check 8

Evaluate:

a.( 1.3)2

b.(0.09)2

Now Try Problems 45 and 47

Self Check 9

Evaluate:

2 0 4.4 5.6 0 ( 0.8)2

Now Try Problem 49

Self Check 10

Evaluate V 1.3pr3 for p 3.14 and r 3.

Now Try Problem 53

Solution

decimals with like signs is positive.

5 Use the order of operations rule.

Recall that the order of operations rule is used to evaluate expressions that involve more than one operation.

Strategy The absolute value bars are grouping symbols. We will perform the addition within them first.

WHY By the order of operations rule, we must perform all calculations within parentheses and other grouping symbols (such as absolute value bars) first.

6 Evaluate formulas.

Recall that to evaluate a formula, we replace the letters (called variables) with specific numbers and then use the order of operations rule.

EXAMPLE 10 Evaluate the formula S 6.28r(h r) for h 3.1 and r 6.

Strategy In the given formula, we will replace the letter r with 6 and h with 3.1.

WHY Then we can use the order of operations rule to find the value of the expression on the right side of the symbol.

Solution

7 Estimate products of decimals.

Estimation can be used to check the reasonableness of an answer to a decimal multiplication.There are several ways to estimate,but the objective is the same:Simplify the numbers in the problem so that the calculations can be made easily and quickly.

EXAMPLE 11

a.Estimate using front-end rounding: 27 6.41

b.Estimate by rounding each factor to the nearest tenth: 13.91 5.27

c.Estimate by rounding: 0.1245(101.4)

Strategy We will use rounding to approximate the factors. Then we will find the product of the approximations.

WHY Rounding produces factors that contain fewer digits. Such numbers are easier to multiply.

Solution

a.To estimate 27 6.41 by front-end rounding, we begin by rounding both factors to their largest place value.

The estimate is 180. If we calculate 27 6.41, the product is exactly 173.07. The estimate is close: It’s about 7 more than 173.07.

b. To estimate 13.91 5.27, we will round both decimals to the nearest tenth.

13.9113.9 Round to the nearest tenth.

417

6950

73.67

The estimate is 73.67. If we calculate 13.91 5.27, the product is exactly 73.3057. The estimate is close: It’s just slightly more than 73.3057.

c.Since 101.4 is approximately 100, we can estimate 0.1245(101.4) using 0.1245(100).

0.1245(100) 12.45 Since 100 has two zeros, move the decimal point in

0.1245 two places to the right.

The estimate is 12.45. If we calculate 0.1245(101.4), the product is exactly 12.6243. Note that the estimate is close: It’s slightly less than 12.6243.

8 Solve application problems by multiplying decimals.

Application problems that involve repeated addition are often more easily solved using multiplication.

Coins Banks wrap pennies in rolls of 50 coins. If a penny is 1.55 millimeters thick, how tall is a stack of 50 pennies?

Self Check 11

a.Estimate using front-end rounding: 4.337 65

b.Estimate by rounding the factors to the nearest tenth: 3.092 11.642

c.Estimate by rounding: 0.7899(985.34)

Now Try Problems 61 and 63

Self Check 12

COINS Banks wrap nickels in rolls of 40 coins. If a nickel is 1.95 millimeters thick, how tall is a stack of 40 nickels?

Now Try Problem 97

Self Check 13

WEEKLY EARNINGS A pharmacy assistant’s basic workweek is 40 hours. After her daily shift is over, she can work overtime

at a rate of 1.5 times her regular rate of $15.90 per hour. How much money will she earn in a week if she works 4 hours of overtime?

Now Try Problem 113

Solve Use vertical form to perform the multiplication:

1.5550

000

7750

77.50

State A stack of 50 pennies is 77.5 millimeters tall.

Check We can estimate to check the result. If we use 2 millimeters to approximate the thickness of one penny, then the height of a stack of 50 pennies is about 2 50 millimeters 100 millimeters. The result, 77.5 mm, seems reasonable.

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

A cashier’s basic workweek is 40 hours. After his daily shift is over, he can work overtime at a rate 1.5 times his regular rate of $13.10 per hour. How much money will he earn in a week if he works 6 hours of overtime?

Analyze

• A cashier’s basic workweek is 40 hours. Given

•His overtime pay rate is 1.5 times his regular rate of $13.10 per hour. Given

•How much money will he earn in a week if he works his regular shift

Form To find the cashier’s overtime pay rate, we multiply 1.5 times his regular pay rate, $13.10.

13.101.5

6550

13100

19.650

The cashier’s overtime pay rate is $19.65 per hour.

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

State The cashier will earn a total of $641.90 for the week.

13.1040

0000

5240

524.00

5 3 3

19.65

6

117.90

1

524.00

117.90

641.90

S E C T I O N 4.3 STUDY SET

VOCABULARY

Fill in the blanks.

1.In the multiplication problem shown below, label each factor, the partial products, and the product.

3.4

2. Numbers such as 10, 100, and 1,000 are called of 10.

CONCEPTS

Fill in the blanks.

3.Insert a decimal point in the correct place for each product shown below. Write placeholder zeros, if necessary.

4.Fill in the blanks.

a.To find the product of a decimal and 10, 100, 1,000, and so on, move the decimal point to the

the same number of places as there are zeros in the power of 10.

b.To find the product of a decimal and 0.1, 0.01, 0.001, and so on, move the decimal point to the

the same number of places as there are in the power of 10.

5.Determine whether the sign of each result is positive or negative. You do not have to find the product.

a.7.6( 1.8)

b.4.09 2.274

6.a. When we move its decimal point to the right, does a decimal number get larger or smaller?

b.When we move its decimal point to the left, does a decimal number get larger or smaller?

NOTATION

7.a. List the first five powers of 10 that are greater than 1.

b.List the first five powers of 10 that are less than 1.

8.Write each number in standard notation.

a.one million

b.one billion

c.one trillion

GUIDED PRACTICE

Write each number in standard notation. See Example 6.

Evaluate each expression. See Example 9.

49.( 0.2)2 4 0 2.3 1.5 0

50.( 0.3)2 6 0 6.4 1.7 0

51.( 0.8)2 7 0 5.1 4.8 0

52.( 0.4)2 6 0 6.2 3.5 0

Evaluate each formula. See Example 10.

53. A P Prt for P 85.50, r 0.08, and t 5

54. A P Prt for P 99.95, r 0.05, and t 10

55.A lw for l 5.3 and w 7.2

56.A 0.5bh for b 7.5 and h 6.8

57.P 2l 2w for l 3.7 and w 3.6

58. P a b c for a 12.91, b 19, and c 23.6

59.C 2pr for p 3.14 and r 2.5

60.A pr2 for p 3.14 and r 4.2

Estimate each product using front-end rounding.

See Example 11.

Estimate each product by rounding the factors to the nearest tenth. See Example 11.

63. 17.11 3.85 64. 18.33 6.46

TRY IT YOURSELF

69.( 0.7 0.5)(2.4 3.1)

70.( 8.1 7.8)(0.3 0.7)

85.7(8.1781)

86.5(4.7199)

87.1,000(0.02239)

88.100(0.0897)

89.(0.5 0.6)2( 3.2)

90.( 5.1)(4.9 3.4)2

91.0.2(306)( 0.4)

92.0.3(417)( 0.5)

93.0.01( 0 2.6 6.7 0)2

94.0.01( 0 8.16 9.9 0)2

APPLICATIONS

97.REAMS OF PAPER Find the thickness of a 500-sheet ream of copier paper if each sheet is 0.0038 inch thick.

98.MILEAGE CLAIMS Each month, a salesman is reimbursed by his company for any work-related travel that he does in his own car at the rate of $0.445 per mile. How much will the salesman receive if he traveled a total of 120 miles in his car on business in the month of June?

99.SALARIES Use the following formula to determine the annual salary of a recording engineer who works 38 hours per week at a rate of $37.35 per hour. Round the result to the nearest hundred dollars.

100.PAYCHECKS If you are paid every other week, your monthly gross income is your gross income from one paycheck times 2.17. Find the monthly gross income of a supermarket clerk who earns $1,095.70 every two weeks. Round the result to the nearest cent.

101.BAKERY SUPPLIES A bakery buys various types of nuts as ingredients for cookies. Complete the table by filling in the cost of each purchase.

102.NEW HOMES Find the cost to build the home shown below if construction costs are $92.55 per square foot.

103.BIOLOGY Cells contain DNA. In humans, it determines such traits as eye color, hair color, and height. A model of DNA appears below. If 1 Å (angstrom) 0.000000004 inch, find the

dimensions of 34 Å, 3.4 Å, and 10 Å, shown in the illustration.

34 Å

3.4 Å

10 Å

104.TACHOMETERS

a.Estimate the decimal number to which the tachometer needle points in the illustration below.

b.What engine speed (in rpm) does the tachometer indicate?

RPM x 1000

105.CITY PLANNING The streets shown in blue on the city map below are 0.35 mile apart. Find the distance of each trip between the two given locations.

a.The airport to the Convention Center

b.City Hall to the Convention Center

c.The airport to City Hall

Airport

106.RETROFITS The illustration below shows the current widths of the three columns of a freeway overpass. A computer analysis indicated that the width of each column should actually be 1.4 times what it currently is to withstand the stresses of an earthquake. According to the analysis, how wide should each of the columns be?

107.ELECTRIC BILLS When billing a household, a utility company charges for the number of kilowatthours used. A kilowatt-hour (kwh) is a standard measure of electricity. If the cost of 1 kwh is $0.14277, what is the electric bill for a household that uses 719 kwh in a month? Round the answer to the nearest cent.

108.UTILITY TAXES Some gas companies are required to tax the number of therms used each month by the customer. What are the taxes collected on a monthly usage of 31 therms if the tax rate is $0.00566 per therm? Round the answer to the nearest cent.

109.Write each highlighted number in standard form.

a.CONSERVATION The 19.6-million acre

Arctic National Wildlife Refuge is located in the northeast corner of Alaska. (Source: National Wildlife Federation)

b.POPULATION According to projections by the International Programs Center at the U.S. Census Bureau, at 7:16 P.M. eastern time on Saturday, February 25, 2006, the population of the Earth hit

6.5 billion people.

c.DRIVING The U.S. Department of Transportation estimated that Americans drove a total of 3.026 trillion miles in 2008. (Source: Federal Highway Administration)

110.Write each highlighted number in standard form.

a.MILEAGE Irv Gordon, of Long Island, New York, has driven a record 2.6 million miles in his 1966 Volvo P-1800. (Source: autoblog.com)

b.E-COMMERCE Online spending during the 2008 holiday season (November 1 through December 23) was about $25.5 billion. (Source: pcmag.com)

c.FEDERAL DEBT On March 27, 2009, the U.S. national debt was $11.073 trillion. (Source: National Debt Clock)

111.SOCCER A soccer goal is rectangular and measures 24 feet wide by 8 feet high. Major league soccer officials are proposing to increase its width by 1.5 feet and increase its height by 0.75 foot.

a.What is the area of the goal opening now?

b.What would the area be if the proposal is adopted?

c.How much area would be added?

112.SALT INTAKE Studies done by the Centers for Disease Control and Prevention found that the average American eats 3.436 grams of salt each day. The recommended amount is 1.5 grams per day. How many more grams of salt does the average American eat in one week compared with what the Center recommends?

113.CONCERT SEATING Two types of tickets were sold for a concert. Floor seating costs $12.50 a ticket, and balcony seats cost $15.75.

a.Complete the following table and find the receipts from each type of ticket.

b.Find the total receipts from the sale of both types of tickets.

114.PLUMBING BILLS A corner of the invoice for plumbing work is torn. What is the labor charge for the 4 hours of work? What is the total charge (standard service charge, parts, labor)?

115.WEIGHTLIFTING The barbell is evenly loaded with iron plates. How much plate weight is loaded on the barbell?

45.5 lb

20.5 lb

2.2 lb

116.SWIMMING POOLS Long bricks, called coping, can be used to outline the edge of a swimming pool. How many meters of coping will be needed in the construction of the swimming pool shown?

50 m

30.3 m

117.STORM DAMAGE After a rainstorm, the saturated ground under a hilltop house began to give way. A survey team noted that the house

dropped 0.57 inch initially. In the next three weeks, the house fell 0.09 inch per week. How far did the house fall during this three-week period?

118.WATER USAGE In May, the water level of a reservoir reached its high mark for the year. During the summer months, as water usage increased, the level dropped. In the months of May and June, it fell 4.3 feet each month. In August, and September, because of high temperatures, it fell another 8.7 feet each month. By the beginning of October, how far below the year’s high mark had the water level fallen?

WRITING

119.Explain how to determine where to place the decimal point in the answer when multiplying two decimals.

120.List the similarities and differences between wholenumber multiplication and decimal multiplication.

121.Explain how to multiply a decimal by a power of 10 that is greater than 1, and by a power of ten that is less than 1.

122.Is it easier to multiply the decimals 0.4 and 0.16 or the fractions 104 and 10016 ? Explain why.

123.Why do we have to line up the decimal points when adding, but we do not have to when multiplying?

124.Which vertical form for the following multiplication do you like better? Explain why.

2.7 0.000003

REVIEW

Find the prime factorization of each number. Use exponents in your answer, when helpful.