See Exercises 68 and 69 in Section 3.3.

Transcription

1 Many people are familiar with decimal numbers because of sports. Batting averages in baseball and save percentages in hockey are calculated to the nearest thousandth. Quarterback ratings in football and average points per game in basketball are calculated to the nearest tenth. Decimal numbers are also used to record many of the world s sport speed records. For example, the men s swimming record for the 100 meter freestyle is seconds by Cesar Cielo of Brazil and the women s swimming record is 5.07 seconds by Germany s Britta Steffen. Decimal numbers are necessary to solve problems like the one below. In 1996, Lance Armstrong was diagnosed with testicular cancer that had spread to his brain and lungs. Against all odds, in 1999 he won his first Tour de France bicycling race by 7 minutes 37 seconds. In 005 he won the Tour for a record seventh consecutive time. The race covered 4 miles in all kinds of weather conditions and over mountain passes, switchback roads and winding paths. His time was 86 hours 15 minutes seconds. This time was 4 minutes 40 seconds better than the second place finisher, Ivan Basso of Italy. How would you find the average speed of these two cyclists? See Exercises 68 and 69 in Section 3.3.

2 3 Decimal Numbers, Percents, and Square Roots 3.1 Reading, Writing, and Rounding Decimal Numbers 3. Addition and Subtraction with Decimal Numbers 3.3 Multiplication and Division with Decimal Numbers 3.4 Measures of Center 3.5 Decimal Numbers, Fractions, and Scientific Notation 3.6 Basics of Percent 3.7 Applications of Percent 3.8 Simple and Compound Interest 3.9 Square Roots Chapter 3: Index of Key Terms and Ideas Chapter 3: Test Cumulative Review: Chapters 1-3

3 Objectives 3.1 Reading, Writing, and Rounding Decimal Numbers A B Learn to read and write decimal numbers. Learn how to round decimal numbers to indicated places of accuracy. Objective A Reading and Writing Decimal Numbers Technically, what we write to represent numbers are symbols or notations called numerals. Numbers are abstract ideas represented by numerals. There are many notations for numbers. For example, the Romans used V to represent five and the Alexandrian Greek system used capital epsilon, Ε, to represent five. In the Hindu-Arabic system that we use today, the symbol 5 represents five. We will not emphasize the distinction between numbers and numerals (or symbols), but you should remember the fact that numbers are abstract ideas, and we use symbols to represent these ideas so that we can communicate in a meaningful manner. The common decimal notation uses a place value system and a decimal point, with whole numbers written to the left of the decimal point and fractions written to the right of the decimal point. We will say that numbers represented by decimal notation are decimal numbers. The values of several places in this decimal system are shown in Figure 1 below. Ten thousands Thousands Hundreds Tens Ones (or units) Tenths Hundredths Thousandths Ten-thousandths Hundred-thousandths. 10, , 1 100, 000 Decimal point Figure 1 There are three classifications of decimal numbers. 1. finite (or terminating) decimal numbers,. infinite repeating decimal numbers, and 3. infinite nonrepeating decimal numbers. In arithmetic, we are used to dealing with finite (or terminating) decimal numbers, and these are the numbers we will discuss and operate with in Sections 3.1 through 3.3. In Section 3.5, we will show how fractions are related to infinite repeating decimals. 31 Chapter 3 Decimal Numbers, Percents, and Square Roots

4 147 In reading a fraction such as, we read the numerator as a whole number 1000 ( one hundred forty- seven ) and then attach the name of the denominator ( thousandths ). Note that ths (or th) is used to indicate the fraction. This same procedure is followed with numbers written in decimal notation. 147 = Read one hundred forty-seven thousandths = 36. Read two and thirty-six hundredths. 100 If there is no whole number part, as in the first example just shown, then 0 is commonly written to the left of the decimal point. Writing the 0 is not always necessary. However, writing the 0 does sometimes avoid confusion with periods at the end of sentences when decimal numbers are written in sentences. In general, decimal numbers are read and written according to the following convention. To Read or Write a Decimal Number 1. Read or write the whole number.. Read or write and in place of the decimal point. 3. Read or write the fraction part as a whole number with the name of the place of the last digit on the right. Example 1 Reading and Writing Decimal Numbers Write 7 8 in decimal notation and in words in decimal notation Teaching Note: When there is no whole number part, writing 0 for the whole number part is the standard format. This does help in aligning the decimal points for addition and subtraction and can help avoid confusion when decimal numbers are written in word problems. However, we do not read or write the word zero with the decimal fraction. Also, in multiplication and division with decimal numbers, the 0 can be a distraction and more confusing than helpful. You might consider treating the 0 as optional for the whole number part. Even though students understand the use of and to indicate the decimal point, whole numbers are so commonly misread in daily life that students have a tendency to forget the importance and proper use of and. You might ask them to listen (to announcers and advertisements on television) and look for misuses of and in numbers and report them in class. seventy-two and eight tenths in words And indicates the decimal point; the digit 8 is in the tenths position. Reading, Writing, and Rounding Decimal Numbers Section 3.1 3

5 Write the following mixed number in decimal notation and in words ; fifty-two and fortytwo hundredths ; twelve and three thousandths Example Reading and Writing Decimal Numbers Write 9 63 in decimal notation and in words One 0 must be inserted as a placeholder. in decimal notation nine and sixty-three thousandths in words And indicates the decimal point; the digit 3 is in the thousandths position. Now work margin exercises 1 and. notes 1. The ths (or th) at the end of a word indicates a fraction part (a part to the right of the decimal point). seven hundred = 700 seven hundredths = The hyphen (-) indicates one word. three hundred thousand = 300,000 three hundred-thousandths = Example 3 Reading and Writing Decimal Numbers Write fifteen hundredths in decimal notation Note that the digit 5 is in the hundredths position. 33 Chapter 3 Decimal Numbers, Percents, and Square Roots

6 Example 4 Reading and Writing Decimal Numbers Write four hundred and two thousandths in decimal notation. Two 0 s are inserted as placeholders The digit is in the thousandths position. Example 5 Reading and Writing Decimal Numbers Write four hundred two thousandths in decimal notation Note carefully how the use of and in the phrase in Example 4 gives it a completely different meaning from the phrase in this example. Now work margin exercises 3 through 5. Objective B Rounding Decimal Numbers Write the following words in decimal numbers. 3. ten and four thousandths 4. six hundred and five tenths 5. seven hundred and seven thousandths Measuring devices, such as rulers, meter sticks, speedometers, micrometers, and surveying transits, give only approximate measurements. Whether the units are large (such as miles and kilometers) or small (such as inches and centimeters), there are always smaller, more accurate units (such as eighths of an inch and millimeters) that could be used to indicate a measurement. We are constantly dealing with approximate (or rounded) numbers in our daily lives. If a recipe calls for 1.5 cups of flour and the cook puts in 1.53 cups (or 1.47 cups), the result will still be reasonably tasty. In fact, the measures of all ingredients will have been approximations. There are several rules for rounding decimal numbers. The IRS, for example, allows rounding to the nearest dollar on income tax forms. A technique sometimes used in cases when many numbers are involved is to round to the nearest even digit at some particular place value. The rule chosen in a particular situation depends on the use of the numbers and whether there might be some sort of penalty for an error. In this text, we will use the following rules for rounding. Reading, Writing, and Rounding Decimal Numbers Section

7 Rules for Rounding Decimal Numbers 1. Look at the single digit just to the right of the place of desired accuracy.. If this digit is 5 or greater, make the digit in the desired place of accuracy one larger and replace all digits to the right with zeros. All digits to the left remain unchanged unless a 9 is made one larger, and then the next digit to the left is increased by If this digit is less than 5, leave the digit in the desired place of accuracy as it is and replace all digits to the right with zeros. All digits to the left remain unchanged. 4. Trailing 0 s to the right of the place of accuracy must be dropped so that the place of accuracy is clearly understood. If a rounded number has a 0 in the desired place of accuracy, then that 0 remains. Example 6 Rounding Decimal Numbers Round to the nearest tenth. The next digit to the right is is in the tenths position. Since 4 is less than 5, leave the 6 and replace 4 and 9 with 0 s rounds to 18.6 to the nearest tenth. Note that the trailing 0 s in are dropped to indicate the position of accuracy. Example 7 Rounding Decimal Numbers Round to the nearest thousandth. The next digit to the right is is in the thousandths position. 35 Chapter 3 Decimal Numbers, Percents, and Square Roots

8 Since 7 is greater than 5, make 9 one larger and replace 7 and 1 with 0 s. (Making 9 one larger gives 10, which affects the digit 3, too.) rounds to to the nearest thousandth, and two trailing 0 s are dropped. Completion Example 8 Rounding Decimal Numbers Round to the nearest ten-thousandth. The digit in the ten-thousandths position is. The next digit to the right is. Since is less than 5, leave as it is and replace with a rounds to to the nearest. Completion Example 9 Rounding Decimal Numbers Round 9653 to the nearest hundred. The decimal point is understood to be to the right of. The digit in the hundreds position is. The next digit to the right is. Since is equal to 5, change the to and replace and with 0 s. So, 9653 rounds to to the nearest hundred. Now work margin exercises 6 though Round to the nearest tenth. 7. Round to the nearest hundredth. 8. Round to the nearest thousandth. 9. Round to the nearest ten Completion Example Answers 8. 4; 3; Since 3 is less than 5, leave 4 as it is and replace 3 with a 0.; rounds to.0064 to the nearest ten-thousandth. 9. 3; 5; Since 5 is equal to 5, change the 6 to a 7 and replace 5 and 3 with 0 s.; So 9653 rounds to 9700 to the nearest hundred. Reading, Writing, and Rounding Decimal Numbers Section

9 notes The 0 s must not be dropped in a whole number. Every 0 to the right of the desired place of accuracy to the right of the decimal point must be dropped. 37 Chapter 3 Decimal Numbers, Percents, and Square Roots

10 Exercises 3.1 Write the following mixed numbers in decimal notation. See Examples 1 and Write the following decimal number in mixed number form. Do not reduce the factional part Write the following words in decimal notation. See Examples 3 through four tenths fifteen thousandths twenty-three hundredths five and twenty-eight hundredths five and twenty-eight thousandths seventy-three and three hundred forty-one thousandths six hundred and sixty-six hundredths six hundred and sixty-six thousandths three thousand four hundred ninety-five and three hundred forty-two thousandths seven thousand five hundred and eighty-three ten-thousandths Write the following decimal numbers in words nine tenths fifty-three hundredths six and five hundredths six and four thousandths fifty and seven thousandths nineteen and one hundred two thousandths eight hundred and nine thousandths eight hundred nine thousandths five thousand and five thousandths twenty-five and four thousand five hundred thirty-eight ten-thousandths Fill in the blanks to correctly complete each statement. See Examples 8 and Round to the nearest tenth. a. The digit in the tenths position is 7. b. The next digit to the right is 8. c. Since 8 is greater than 5, change 7 to 8 and replace 8 with 0. d. So rounds to 34.8 to the nearest tenth. 3. Round to the nearest ten-thousandth. a. The digit in the ten-thousandths position is 5. b. The next digit to the right is. c. Since is less than 5, leave 5 as it is and replace with 0. d. So rounds to to the nearest ten-thousandth. Reading, Writing, and Rounding Decimal Numbers Section

11 Round each of the following decimal numbers to the nearest tenth Round each of the following decimal numbers to the nearest hundredth Round each of the following decimal numbers to the nearest thousandth Round each of the following decimal numbers to the nearest whole number (or nearest unit) Round each of the following decimal numbers to the nearest hundred ,53. 76, Round each of the following decimal numbers to the nearest thousand ,375 6, ,445 75, , , ,500,766 4,501, ,305,438 7,305, , , (nearest hundred-thousandth) ,419 (nearest ten thousand) 80,000 In the following exercises, write the decimal numbers that are not whole numbers in words. 71. Units of Length: One yard is equal to 36 inches. One yard is also approximately equal to meter. One meter is approximately equal to 1.09 yards. One meter is also approximately equal to inches. (Thus a meter is longer than a yard by about 3.37 inches.) nine hundred fourteen thousandths; one and nine hundredths; thirty-nine and thirty-seven hundredths; three and thirtyseven hundredths 7. Units of Length: One foot is equal to 1 inches. One foot is also equal to centimeters. One square foot is approximately square meter. thirty and forty-eight hundredths; ninety-three thousandths 39 Chapter 3 Decimal Numbers, Percents, and Square Roots

12 73. Water Weight: One quart of water weighs approximately.085 pounds. two and eight hundred twenty-five ten-thousandths.085 lbs. 74. States: The largest state in the United States is Alaska, which covers approximately thousand square miles. The second largest state is Texas, with approximately 68.6 thousand square miles. Alaska is more than 10 times the size of Wisconsin (twenty-third in size), with about 65.5 thousand square miles. six hundred fifty-six and four tenths; two hundred sixty-eight and six tenths; sixty-five and five tenths thousand sq mi 68.6 thousand sq mi 65.5 thousand sq mi 75. Pi: The number π is approximately equal to three and fourteen thousand, one hundred fifty-nine hundred-thousandths 76. Euler s Number: The number e (used in higher-level mathematics) is approximately equal to two and seventy-one thousand, eight hundred twenty-eight hundred-thousandths 77. Aging: An interesting fact about aging is that the longer you live, the longer you can expect to live. A white male of age 40 can expect to live 35.8 more years; of age 50, can expect to live 6.9 more years; of age 60 can expect to live 18.9 more years; of age 70 can expect to live 1.3 more years; and of age 80 can expect to live 7. more years. (This same phenomenon is true of men and women of all races.) thirty-five and eight tenths; twentysix and nine tenths; eighteen and nine tenths; twelve and three tenths; seven and two tenths 78. The Sun: The mean distance from the Sun to Earth is about 9.9 million miles and from the Sun to Venus is about 67.4 million miles. One period of revolution of the Earth about the Sun takes 365. days, and one period of revolution of Venus about the Sun takes 4.7 days. ninety-two and nine tenths; sixty-seven and twenty-four hundredths; three hundred sixty-five and two tenths; two hundred twentyfour and seven tenths 67.4 million mi 9.9 million mi Venus Earth 79. Unicycle: The tallest unicycle ever ridden was feet tall, and was ridden by Sem Abrahams (with a safety wire suspended from an overhead crane) for a distance of 8 feet in Pontiac, Michigan on January 9, 004. Source: one hundred fourteen and eight tenths 80. World Records: 9.58 seconds for 100 meters (by Usain Bolt, Jamaica, 009); seconds for 00 meters (by Usain Bolt, Jamaica, 009); seconds for 400 meters (by Michael Johnson, USA, 1999). Source: Wikipedia.com nine and fifty-eight hundredths; nineteen and nineteen hundredths; forty-three and eighteen hundredths Reading, Writing, and Rounding Decimal Numbers Section

13 Objectives A Know how to add with decimal numbers. 3. Addition and Subtraction with Decimal Numbers Objective A Addition with Decimal Numbers B C D Know how to subtract with decimal numbers. Learn how to add and subtract with positive and negative decimal numbers. Be able to estimate sums and differences by using rounded decimal numbers. Addition with decimal numbers can be accomplished by writing the decimal numbers one under the other and keeping the decimal points aligned vertically. In this way, the whole numbers will be added to whole numbers, tenths added to tenths, hundredths to hundredths, and so on. The decimal point in the sum is in line with the decimal points in the addends. Thus, we have the format shown here In the number 6.14, 0 s may be written to the right of the last digit in the part to help keep the digits in the correct line. This will not change the value of any number or the sum. To Add Decimal Numbers 1. Write the addends in a vertical column.. Keep the decimal points aligned vertically. 3. Keep digits with the same position value aligned. (Zeros may be filled in as aids.) 4. Add the numbers, just as with whole numbers, keeping the decimal point in the sum aligned with the other decimal points. 1. Find the sum: Example 1 Adding Decimal Numbers Find the sum: Align decimal points vertically The decimal point is understood to be to the right of s are filled in to help keep the digits in line. sum Now work margin exercise Chapter 3 Decimal Numbers, Percents, and Square Roots

14 Example Combining Like Terms Simplify by combining like terms: 8.3x + 9.4x x.. Combine like terms. 45.x +.08x + 3.5x 50.78x The method for combining like terms is to use the distributive property with decimal number coefficients just as with integer coefficients. 83. x+ 94. x x = x = 4. 79x Now work margin exercise. ( ) Objective B Subtraction with Decimal Numbers To Subtract Decimal Numbers 1. Write the numbers in a vertical column.. Keep the decimal points aligned vertically. 3. Keep digits with the same position value aligned. (Zeros may be filled in as aids.) 4. Subtract, just as with whole numbers, keeping the decimal point in the difference aligned with the other decimal points. Example 3 Subtracting Decimal Numbers Find the difference: difference Example 4 Subtracting Decimal Numbers At the bookstore, Mrs. Gonzalez bought a text for $55, art supplies for $3.50, and computer supplies for $9.5. If tax was $9.34, how much change did she receive from a gift certificate worth $150? Addition and Subtraction with Decimal Numbers Section 3. 33

15 3. Find the difference: a. Find the total of her expenses including tax. 4. Mr. Jones had $1000. He wrote checks for $80.5, $640.3, $54.19, and $ How much he have left? $8.68 $ $ total b. Subtract the answer in part a. from $150. $ $ She received $3.91 in change. Now work margin exercises 3 and 4. Objective C Positive and Negative Decimal Numbers Decimal numbers can be positive and negative, just as integers and other rational numbers can be positive and negative. Some positive and negative decimal numbers are illustrated on the number line in Figure Figure 1 The rules for operating with positive and negative decimal numbers are the same as those for operating with integers (Chapter 1) and fractions and mixed numbers (Chapters ). Examples 5 and 6 illustrate these ideas. 5. Find the difference: ( 75.3) Example 5 Subtracting Signed Decimal Numbers Find the difference: ( 45.87) ( 45.87) = Now work margin exercise Chapter 3 Decimal Numbers, Percents, and Square Roots

16 Example 6 Solving Equations involving Decimal Numbers Solve the equation: x 5.67 = Solve the equation: x = x = x = x = 105. x = x = 4.6 Now work margin exercise 6. Objective D Estimating Sums and Differences By rounding each number to the place of the last nonzero digit on the left and adding (or subtracting) these rounded numbers, we can estimate (or approximate) the answer before the actual calculations are done. (See Section 1. for estimating answers with whole numbers.) This technique of estimating is especially helpful when working with decimal numbers, where the incorrect placement of a decimal point can change an answer dramatically. Example 7 Estimating the Sum of Decimal Numbers First estimate the sum ; then find the sum. 7. First, estimate the sum ( 35.47); then find the sum. 14; a. Estimate by adding rounded numbers. (Each number is rounded to the place of the leftmost nonzero digit in that number.) Actual value Rounded value estimate b. Find the actual sum Now work margin exercise 7. Teaching Note: You might remind the students that any form of estimation involves a basic understanding of numbers and some judgment on their part. Regardless of the method used for estimating, there will be cases when the estimate is not particularly accurate. Addition and Subtraction with Decimal Numbers Section

17 In Example 7, the estimated sum and the actual sum are reasonably close (the difference is about 6). This leads us to have confidence in the answer. If, for example, 84 had been written as 0.84 (instead of 84.00), then addition would have given a sum as follows. decimal in the wrong place Here, the difference between the estimate of 144 and the wrong sum of is over 70 (a relatively large amount), and an error should be suspected. 8. Estimate the difference ; then find the difference. 960; Completion Example 8 Estimating the Difference of Decimal Numbers Estimate the difference ; then find the difference. a. Estimate: b. Actual difference: estimate actual difference Now work margin exercise 8. Completion Example 9 Subtracting Decimal Numbers Samantha bought a pair of shoes for $47.50, a jacket for $5.60 (on sale), and a pair of slacks for $ (Mentally estimate both answers before you actually calculate them.) $5.60 $39.75 $47.50 Completion Example Answers 8. a. Estimate: b. Actual difference: estimate actual difference 335 Chapter 3 Decimal Numbers, Percents, and Square Roots

18 a. How much did she spend? (Tax was included in the prices.) 9. Fred bought cheese, crackers, and a soft drink for $.5, $3.49, and $0.35, respectively. How much was his total? What was his change from $10? (Mentally estimate both answers before you actually calculate them.) $ expenses b. What was her change from $10? $6.09; $3.91 $10.00 expenses change Did you estimate her expenses as $10 and her change as $0? Now work margin exercise 9. Example Use a calculator to Using a Calculator perform the following operations. Use a calculator to perform the following operations. a a b b a b a. To add the values given, press the keys. Then press. The display will read b. To subtract the values given, press the keys. Then press. The display will read Now work margin exercise 10. Completion Example Answers 9. a. b. $ $11.85 $ $7.15 expenses change expenses Addition and Subtraction with Decimal Numbers PIA Chapter 3.indd 336 Section /7/011 3:14:3 PM

19 Exercises 3. Find each of the indicated sums. Estimate your answers, before performing the actual calculations. Check to see that your sums are close to the estimated values. See Example Find each of the indicated differences. First estimate the difference mentally. See Example Simplify each expression by combining like terms. See Example x + x.7x 13.4x. 9.54x x 1.8x 4.8x y.3y 17.9y 45.5y y 34.1y 56.3y 1.9y x x 0.7x x + x 0.93x t.7t t 6.3t s 00s s 1.4s 9..1x + 8.x y 3.1y 10.3x 4.1y x 3.7x + y 0.1y 1.1x + 1.9y Solve each of the following equations. See Example x = x = x 7.9 = x = y = y = y = y = z 8.65 = z = z 7.64 = z = w = w = w = w = Chapter 3 Decimal Numbers, Percents, and Square Roots

20 39. New Car: Martin wants to buy a new car for $18,000. He has talked to the loan officer at his credit union and knows that they will loan him $1,600. He must also pay $350 for a license fee and $100 for taxes. What amount of cash will he need to buy the car? $ Architect: An architect s scale drawing shows a rectangle that measures.5 inches on one side and 3.75 inches on another side. What is the perimeter (distance around) of the rectangle in the drawing? 1 in. 40. Haircut: Terri wants to have a haircut and a manicure. She knows that the haircut will cost $38.00 and the manicure will cost $ If she plans to tip the hair stylist $8.00, how much change should she receive from $70? $ Grain: In 010, U.S. farmers produced million bushels of barley, million bushels of corn, and million bushels of wheat. (Source: US Department of Agriculture, National Agricultural Statistics Survey) a. What was the total U.S. production of these three grains in 010? million bushels b. How many more bushels of wheat were produced than barley? million bushels 43. Livestock: In 010, U.S. farmers owned the following amounts of livestock: million calves, million hogs and sows, and 3.6 million sheep. (Source: US Department of Agriculture, National Agricultural Statistics Survey) a. What was the total amount of these livestock in 010? million b. How many more calves than sheep were there? million 44. Orbit: The eccentricity of a planet s orbit is the measure of how much the orbit varies from a perfectly circular pattern. Earth s orbit has an eccentricity of 0.017, and Pluto s orbit has an eccentricity of How much greater is Pluto s eccentricity of orbit than Earth s? Rain: Albany, New York, receives a mean rainfall of inches and 65.5 inches of snow. Charleston, South Carolina, receives a mean rainfall of inches and 0.6 inches of snow. What is the difference between the mean rainfalls in Charleston and Albany? What is the difference between the mean snow falls in Charleston and Albany? in.; 64.9 in. 46. Check Book: Suppose your checking account shows a balance of $38.35 at the beginning of the month. During the month, you make deposits of $580.00, $300.00, $18.50, and $45.00, and you write checks for $85.35, $10.50, $43.75, and $650. Find the end-of-themonth balance in your account. $500.5 YOUR CHECKBOOK REGISTER Albany, NY Check No. Payment Deposit Date Transaction Description ( ) ( ) (+) Balance 11-4 Deposit $ Balance brought forward Charleston, SC = 10 inches of rain = 10 inches of snow Electric Bill Wally's Computer Shop 11-7 Deposit Groceries 11- Deposit Deposit BJ's Auto Repair True Balance: Addition and Subtraction with Decimal Numbers Section

21 47. If the sum of 33.7 and is subtracted from the sum of 9.61 and 35.7, what is the difference? What is the sum if is added to the difference between 16.3 and 0.1? -1.3 Use a calculator to find the value of each expression ( ) ( ) ( ) + ( ) ( ) ( ) Chapter 3 Decimal Numbers, Percents, and Square Roots

22 3.3 Multiplication and Division with Decimal Numbers Objective A Multiplication with Decimal Numbers When decimal numbers are added or subtracted, the decimal points are aligned so that digits in the tenths position are added (or subtracted) to digits in the tenths position, digits in the hundredths position are added (or subtracted) to digits in the hundredths position, and so on. In this way fractions with a common denominator are added (or subtracted). However, when fractions are multiplied, there is no need to have a common denominator. For example, in adding 1 3 +, we need the common denominator 8: = + = + =.. But, in multiplying, there is no need for a common denominator: = 16. Thus, since there is no concern about common denominators when multiplying fractions, the same is true for multiplying decimals. Therefore, there is no need to keep the decimal points in line for multiplication with decimals. Decimal numbers are multiplied in the same manner as whole numbers, but with the added concern of the correct placement of the decimal point in the product. Two examples are shown here, in both fraction form and decimal form, to illustrate how the decimal point is to be placed in the product. Products in Fraction Form = = , 000 Products in Decimal Form. 4 1 place total of 3 places places (thousandths) places total of 5 places places (hundred-thousandths) Objectives A B C D E F Know how to multiply decimal numbers and place the decimal point correctly in the product. Learn how to multiply decimal numbers mentally by powers of 10. Know how to divide decimal numbers and place the decimal point correctly in the product. Learn how to divide decimal numbers mentally by powers of 10. Be able to estimate products and quotients with rounded decimal numbers. Understand how to solve word problems involving decimal numbers. The following rule states how to multiply positive decimal numbers and place the decimal point in the product. (For negative decimal numbers, the rule is the same except for the determination of the sign.) To Multiply Positive Decimal Numbers 1. Multiply the two numbers as if they were whole numbers.. Count the total number of places to the right of the decimal points in both numbers being multiplied. 3. Place the decimal point in the product so that the number of places to the right is the same as that found in step. Multiplication and Division with Decimal Numbers Section

23 1. Multiply: ( )( 5. 01) Example 1 Multiplying Positive Decimal Numbers Multiply: places 1 place 4 places in the product total of 4 places (ten-thousandths) Now work margin exercise 1.. Find the product: ( 3. )( 00. ) Teaching Note: Note that in the introductory examples, the 0 whole number part has been omitted. You might allow your students to follow the style shown here and insert the 0, if needed, in their answers for both multiplication and division with decimal numbers. Example Multiplying Signed Decimal Numbers Find the product: ( 0.16)(0.003). Since the signs are not alike, the product will be negative places total of 6 places places places in the product Note that three 0 s had to be inserted between the 3 and the decimal point in the product to get a total of 6 decimal places. Now work margin exercise. Objective B Multiplication by Powers of 10 The following general guidelines can be used to multiply decimal numbers (including whole numbers) by powers of Chapter 3 Decimal Numbers, Percents, and Square Roots

24 To Multiply a Decimal Number by a Power of Move the decimal point to the right.. Move it the same number of places as the number of 0 s in the power of 10. Multiplication by 10 moves the decimal point one place to the right; Multiplication by 100 moves the decimal point two places to the right; Multiplication by 1000 moves the decimal point three places to the right; and so on. Example 3 Multiplying by Powers of 10 The following products illustrate multiplication by powers of 10. a. 10( 935. )= 935. b. 100( 9. 35)= 935 c. 100( 163)= 16, 300 d. 1000( )= Move decimal point 1 place to the right. Move decimal point places to the right. Move decimal point places to the right. Move decimal point 3 places to the right. 3. Find each product by performing the operation mentally. a. 10( 836. ) b. 100( ) c ( 14. 8) 14, 80 a b c. 14, 80 e. 10 ( 87. 5)= f. 10 ( )= Exponent tells how many places to move the decimal point. Move decimal point places. Move decimal point 3 places to the right. Now work margin exercise 3. Objective C Division with Decimal Numbers The process of division with decimal numbers is, in effect, the same as division with whole numbers with the added concern of where to place the decimal point in the quotient. This is reasonable because whole numbers are decimal numbers, and we would not expect a great change in a process as important as division. Division with whole numbers gives a quotient and possibly a remainder. 3. divisor 40) quotient dividend remainder Multiplication and Division with Decimal Numbers Section

25 By adding 0 s onto the dividend, we can continue to divide and get a decimal quotient other than a whole number divisor 40) quotient is a decimal number 0 s added on If the divisor is a decimal number rather than a whole number, multiply both the divisor and dividend by a power of 10 so that the new divisor is a whole number. For example, we can write This means that Similarly, we can write ) as = 6. ) ) is the same as ) as = ) or To Divide Decimal Numbers 1. Move the decimal point in the divisor to the right so that the divisor is a whole number.. Move the decimal point in the dividend the same number of places to the right. 3. Place the decimal point in the quotient directly above the new decimal point in the dividend. 4. Divide just as with whole numbers. notes 1. In moving the decimal point, you are multiplying by a power of 10.. Be sure to place the decimal point in the quotient before actually dividing. 343 Chapter 3 Decimal Numbers, Percents, and Square Roots

26 Example 4 Dividing Decimal Numbers Find the quotient: a. Write down the numbers. 6. ) b. Move both decimal points one place to the right so that the divisor becomes a whole number. Then place the decimal point in the quotient.. Decimal point in quotient ) c. Proceed to divide as with whole numbers ) Example 5 Dividing Decimal Numbers Find the quotient: a. Write down the numbers ). Multiplication and Division with Decimal Numbers Section

27 Find the following quotients b. Move two decimal points two places to the right so that the divisor becomes a whole number. Then place the decimal point in the quotient.. decimal point in quotient ) c. Proceed to divide as with whole numbers ) Now work margin exercises 4 and 5. If the remainder is eventually 0, then the quotient is a terminating decimal number. We will see in Section 3.4 that if the remainder is never 0, then the quotient is an infinite repeating decimal number. That is, the quotient will be a repeating pattern of digits. To avoid an infinite number of steps, which can never be done anyway, we generally agree to some place of accuracy for the quotient before the division is performed. If the remainder is not 0 by the time this place of accuracy is reached in the quotient, then we divide one more place and round the quotient. When the Remainder is Not Zero 1. Decide first how many decimal places are to be in the quotient.. Divide until the quotient is one digit to the right of the place of desired accuracy. 3. Using this last digit, round the quotient to the desired place of accuracy. 345 Chapter 3 Decimal Numbers, Percents, and Square Roots

28 Example 6 Dividing Decimal Numbers Find the quotient to the nearest tenth. Divide until the quotient is in hundredths (one place more than tenths); then round to tenths. hundredths read approximately ) Add 0 s as needed accurate to the nearest tenth Example 7 Dividing Decimal Numbers Find the quotient to the nearest hundredth. 6. Find the quotient to the nearest tenth. 7. Find the quotient to the nearest hundredth ) thousandths Now work margin exercises 6 and 7. accurate to the nearest hundredth Multiplication and Division with Decimal Numbers Section

29 Objective D Division By Powers of 10 Earlier, we found that multiplication by powers of 10 can be accomplished by moving the decimal point to the right. Division by powers of 10 can be accomplished by moving the decimal point to the left. To Divide a Decimal Number by a Power of Move the decimal point to the left.. Move it the same number of places as the number of 0 s in the power of 10. Division by 10 moves the decimal point one place to the left; Division by 100 moves the decimal point two places to the left; Division by 1000 moves the decimal point three places to the left; and so on. Two general guidelines will help you to understand work with powers of Multiplication by a power of 10 will make a number larger, so move the decimal point to the right.. Division by a power of 10 will make a number smaller, so move the decimal point to the left. 8. Find each product by performing the operation mentally. a b c a. 1.6 b c Example 8 Dividing by Powers of 10 The following quotients illustrate division by powers of 10. a. b. c = = = = = d =. 865 Now work margin exercise 8. Objective E Move decimal point places to the left. Move decimal point 1 place to the left. Move decimal point 3 places to the left. The exponent tells how many places to move the decimal point. Move decimal point places to the left. Estimating Products and Quotients Estimating with multiplication and division can be used to help in placing the decimal point in the product and in the quotient to verify the reasonableness of the answer. The technique is to round all numbers to the place of the leftmost nonzero digit and then operate with these rounded numbers. 347 Chapter 3 Decimal Numbers, Percents, and Square Roots

30 Example 9 Estimating Products of Decimal Numbers For the product (0.358)(6.): a. Estimate the product. Estimate by multiplying rounded numbers rounded 6. rounded estimate 9. For the product (. )(. ) : a. Estimate the product. b. Find the product. a. 8. b b. Find the product. Find the actual product actual product The estimated product.4 helps place the decimal point correctly in the product.196. Thus an answer of or.196 would indicate an error in the placement of the decimal point, since the answer should be near.4 (or between and 3). Now work margin exercise 9. Example 10 Estimating Quotients of Decimal Numbers a. Estimate the quotient Using and , estimate the quotient. 0. estimate ) Multiplication and Division with Decimal Numbers Section

31 10. First estimate the quotient , then find the quotient to the nearest tenth. estimate: 4.0 quotient: 4.3 b. Find the quotient to the nearest tenth. Find the quotient to the nearest tenth ) Now work margin exercise 10. Objective F Applications Some word problems may involve several operations with decimal numbers. The words do not usually say directly to add, subtract, multiply, or divide. Experience and reasoning abilities are needed to decide which operations to perform with the given numbers. Example 11 illustrates a problem that involves several steps and how estimating can provide a check for a reasonable answer. Teaching Note: Example 11 is a good illustration of the need for judgement in any estimation. In this case, if 1 is rounded to 10, the resulting estimate will actually be a negative number. You might ask the students the implication of the negative result (paying cash would increase the cost) and why this does not make sense. Example 11 Buying a Car You can buy a car for $8500 cash, or you can make a down payment of $1700 and then pay $ each month for 1 months. How much can you save by paying cash? a. Find the amount paid in monthly payments by multiplying the amount of each payment by 1. In this case, judgement dictates that we do not want to lose two full monthly payments in our estimate, so we use 1 and do not round to Chapter 3 Decimal Numbers, Percents, and Square Roots

32 Estimate $ $ 700 Actual Amount $ $ paid in monthly payments 11. You can buy a lawnmower for $450 cash, or you can make a down payment of $50 and then pay $5 each month for two years. How much can you save by paying cash? b. Find the total amount paid by adding the down payment to the answer in part a. You can save $ 00. Estimate Actual Amount $ + $ down payment monthly payments total paid $ $ down payment monthly payments total paid c. Find the savings by subtracting $8500 from the answer in part b. Estimate $ $ $ Actual Amount $ $ $ savings by paying cash The $700 estimate is very close to the actual savings of $ Now work margin exercise 11. We know from a previous section that the mean of a set of numbers can be found by adding the numbers, and then dividing the sum by the number of addends. Another meaning of the term mean is in the sense of a mean speed of 5 miles per hour or the mean number of miles per gallon of gas. This kind of mean can be found by division. If we know the total amount of a quantity (distance, dollars, gallons of gas, etc.) and a number of units (time, items bought, miles, etc.), then we can find the mean amount per unit by dividing the total amount by the number of units. Think of per as indicating division. Thus miles per hour means miles divided by hours. Example 1 Fuel Efficiency The gas tank of a car holds 18 gallons of gasoline. Approximately how many miles per gallon does the car mean if it will go 470 miles on one tank of gas? Teaching Note: Keep reminding students that per means to divide. This idea will help them a great deal in deciding how to divide to find various means. For example, miles per gallon means miles divided by gallons. Once they understand this concept, you might want to show them the technique of operating with fractions made up of unit labels to help them decide what operations to perform. As in Example 13, miles hour hours=miles. Multiplication and Division with Decimal Numbers Section

33 1. A batter had 99 hits in 114 games. Estimate the batter s mean number of hits per game? about 3 hits per game Miles per gallon means miles divided by gallons. Since the question calls only for an approximate answer, rounded values can be used gal and miles 5 miles per gallon 0) The car gets a mean of about 5 miles per gallon. Now work margin exercise 1. If a mean amount per unit is known, then a corresponding total amount can be found by multiplying. For example, if you jog a mean of 5 miles per hour, then the distance you jog can be found by multiplying your mean speed by the time you spend jogging. 13. If your boat has a mean speed of 15.3 miles per hour, how far will you travel in 1.8 hours? The boat travels miles Example 13 Jogging If you jog at a mean speed of 4.8 miles per hour, how far will you jog in 3. hours? Multiply the mean speed by the number of hours miles per hour hours miles You will jog miles in 3. hours. Now work margin exercise 13. In some cases, a business will advertise the total price of its items, including sales tax. You will see a phrase such as tax included. If, for example, the sales tax is figured at 0.06 times the actual price, you can find the price you are paying for the item by dividing the total price by (The number 1.06 represents the actual price plus 0.06 times the actual price.) If you buy gas for your car, the price at the gas pump includes all types of taxes (state, federal, local, etc.). If these taxes are figured at, say 0.45 times the price of a gallon of gas, then the actual price of the gas to you can be found by dividing the price at the pump by the number Chapter 3 Decimal Numbers, Percents, and Square Roots

34 Example If a hot dog costs $.5 (before taxes) at a little league hockey game and taxes were 0.06 times the actual price, how much (to the nearest cent) does the hot dog cost? Price of Gasoline Suppose that the price of a gallon of gas is stated as $3.15 at the pump and the station owner tells you that the taxes you are paying are figured at 0.45 times the price he is actually charging for a gallon of gas. What price (to the nearest penny) is he actually charging for a gallon of gas? The hot dog costs $.39. To find the actual price of a gallon of gas, before taxes, divide the total price by ) So, the actual cost of the gas is about $.17 per gallon before taxes. Now work margin exercise 14. Example Use a calculator to Using a Calculator Use a calculator to perform the following operations and write the answer accurate to the nearest ten-thousandth. a (0.35)(.3) a. 7.5 ( )(0.73 ) b b a. b. a. To multiply the values given, press the keys. Then press ( 4.138)(0.73) The display will read b. To divide the values given, press the keys. Then press. The display will read Now work margin exercise 15. Multiplication and Division with Decimal Numbers PIA Chapter 3.indd 35 perform the following operations and write the answer accurate to the nearest ten-thousandth. Section /7/011 3:14:33 PM

35 Exercises 3.3 Estimate each of the following products mentally by using rounded values. 1. a. ( 09. )( ) 0.7 b. ( 33. 6)( ) 3.0 c. ( 064. )( ) 6.0 d. ( 0. )( 0. 6 ) 0.06 e. ( 47. )( 11.) 5.0. a. (. 16)( ) 0.06 b (. ) c (. ) 0.6 d ( ) 6.0 Estimate each of the following quotients by using rounded values. 3. a ). b ) c. 36. ) d. 18. ) e ) a ,000 b c d Find each of the indicated products. See Examples 1 and. 5. (0.5)(0.7) (0.)(0.8) (1.8) (3.5) (0.0)(0.0) (0.03)(0.03) ( ) ( 05. ) ( 0. 15) ( ) ( ) (. ) Find each of the indicated quotient. See Examples 4 and ( ) ( ) Chapter 3 Decimal Numbers, Percents, and Square Roots