CHAPTER OUTLINE 5 Decimals Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5.1Decimal Notation and Rounding 5.2Addition and Subtraction of Decimals 5.3Multiplication of Decimals and Applications with Circles 5.4Division of Decimals 5.5Fractions, Decimals, and the Order of Operations 5.6Solving Equations Containing Decimals 5.7Mean, Median, and Mode Section Objectives 5.1 Decimal Notation and Rounding Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Decimal Notation 2.Writing Decimals as Mixed Numbers or Fractions 3.Ordering Decimal Numbers 4.Rounding Decimals Section 5.1 Decimal Notation and Rounding 1.Decimal Notation Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A decimal fraction is a fraction whose denominator is a power of 10. Section 5.1 Decimal Notation and Rounding 1.Decimal Notation Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 1Identifying Place Values Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Identifying Place Values Slide 7 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.1 Decimal Notation and Rounding 1.Decimal Notation Slide 8 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The decimal point is interpreted as the word and. PROCEDUREReading a Decimal Number Slide 9 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 The part of the number to the left of the decimal point is read as a whole number. Note: If there is no whole-number part, skip to step 3. Step 2 The decimal point is read and. Step 3 The part of the number to the right of the decimal point is read as a whole number but is followed by the name of the place position of the digit farthest to the right. Example 2Reading Decimal Numbers Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Reading Decimal Numbers Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 3Writing a Numeral from a Word Name Slide 12 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the word name as a numeral. a. Four hundred eight and fifteen ten-thousandths b. Negative five thousand eight hundred and twenty- three hundredths Example Solution: 3Writing a Numeral from a Word Name Slide 13 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.1 Decimal Notation and Rounding 2.Writing Decimals as Mixed Numbers or Fractions Slide 14 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A fractional part of a whole may be written as a fraction or as a decimal. To convert a decimal to an equivalent fraction, it is helpful to think of the decimal in words. PROCEDUREConverting a Decimal to a Mixed Number or Proper Fraction Slide 15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 The digits to the right of the decimal point are written as the numerator of the fraction. Step 2 The place value of the digit farthest to the right of the decimal point determines the denominator. Step 3 The whole-number part of the number is left unchanged. Step 4 Once the number is converted to a fraction or mixed number, simplify the fraction to lowest terms, if possible. Example 4Writing Decimals as Proper Fractions or Mixed Numbers Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the decimals as proper fractions or mixed numbers and simplify. Example Solution: 4Writing Decimals as Proper Fractions or Mixed Numbers (continued) Slide 17 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Writing Decimals as Proper Fractions or Mixed Numbers Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.1 Decimal Notation and Rounding 2.Writing Decimals as Mixed Numbers or Fractions Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A decimal number greater than 1 can be written as a mixed number or as an improper fraction. PROCEDUREWriting a Decimal Number Greater Than 1 as an Improper Fraction Slide 20 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 The denominator is determined by the place position of the digit farthest to the right of the decimal point. Step 2 The numerator is obtained by removing the decimal point of the original number. The resulting whole number is then written over the denominator. Step 3 Simplify the improper fraction to lowest terms, if possible. Example 5Writing Decimals as Improper Fractions Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the decimals as improper fractions and simplify. Example Solution: 5Writing Decimals as Improper Fractions Slide 22 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. PROCEDUREComparing Two Positive Decimal Numbers Slide 23 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Starting at the left (and moving toward the right), compare the digits in each corresponding place position. Step 2 As we move from left to right, the first instance in which the digits differ determines the order of the numbers. The number having the greater digit is greater overall. Example 6Ordering Decimals Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 6Ordering Decimals Slide 25 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 7Ordering Negative Decimals Slide 26 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 7Ordering Negative Decimals Slide 27 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. Insert a zero in the hundredths place for the number on the left. The digits in the tenths place are different. b. Insert a zero in the hundred-thousandths place for the number on the right. The digits in the hundred- thousandths place are different. PROCEDURERounding Decimals to a Place Value to the Right of the Decimal Point Slide 28 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Identify the digit one position to the right of the given place value. Step 2 If the digit in step 1 is 5 or greater, add 1 to the digit in the given place value. Then discard the digits to its right. Step 3 If the digit in step 1 is less than 5, discard it and any digits to its right. Example 8Rounding Decimal Numbers Slide 29 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 8Rounding Decimal Numbers Slide 30 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 9Rounding Decimal Numbers Slide 31 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 9Rounding Decimal Numbers (continued) Slide 32 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 9Rounding Decimal Numbers Slide 33 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section Objectives 5.2 Addition and Subtraction of Decimals Slide 34 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Addition and Subtraction of Decimals 2.Applications of Addition and Subtraction of Decimals 3.Algebraic Expressions PROCEDUREAdding and Subtracting Decimals Slide 35 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Write the numbers in a column with the decimal points and corresponding place values lined up. (You may insert additional zeros as placeholders after the last digit to the right of the decimal point.) Step 2 Add or subtract the digits in columns from right to left, as you would whole numbers. The decimal point in the answer should be lined up with the decimal points from the original numbers. Example 1Adding Decimals Slide 36 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Adding Decimals Slide 37 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.2 Addition and Subtraction of Decimals 1.Addition and Subtraction of Decimals Slide 38 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. With operations on decimals it is important to locate the correct position of the decimal point. Example 2Adding Decimals Slide 39 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Adding Decimals Slide 40 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 3Subtracting Decimals Slide 41 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 3Subtracting Decimals Slide 42 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 4Adding and Subtracting Negative Decimal Numbers Slide 43 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Adding and Subtracting Negative Decimal Numbers (continued) Slide 44 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Adding and Subtracting Negative Decimal Numbers (continued) Slide 45 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Adding and Subtracting Negative Decimal Numbers Slide 46 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 5Applying Addition and Subtraction of Decimals in a Checkbook Slide 47 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fill in the balance for each line in the checkbook register, shown in the figure. What is the ending balance? Example Solution: 5Applying Addition and Subtraction of Decimals in a Checkbook Slide 48 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We begin with $ in the checking account. For each debit, we subtract. For each credit, we add. The ending balance is $ Example 6Applying Decimals to Perimeter Slide 49 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. Find the length of the side labeled x. b. Find the length of the side labeled y. c. Find the perimeter of the figure. Example Solution: 6Applying Decimals to Perimeter (continued) Slide 50 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. If we extend the line segment labeled x with the dashed line as shown below, we see that the sum of side x and the dashed line must equal 14 m. Therefore, subtract 14 2.9 to find the length of side x. Length of side x: Side x is 11.1 m long. Example Solution: 6Applying Decimals to Perimeter (continued) Slide 51 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. b. The dashed line in the figure below has the same length as side y. We also know that y must equal Since = 10.0, y = 15.4 10.0 = 5.4 The length of side y is 5.4 m. Example Solution: 6Applying Decimals to Perimeter Slide 52 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. c. Now that we have the lengths of all sides, add them to get the perimeter. The perimeter is 58.8 m. Example 7Combining Like Terms Slide 53 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 7Combining Like Terms Slide 54 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section Objectives 5.3 Multiplication of Decimals and Applications with Circles Slide 55 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Multiplication of Decimals 2.Multiplication by a Power of 10 and by a Power of Applications Involving Multiplication of Decimals 4.Circumference and Area of a Circle PROCEDUREMultiplying Two Decimals Slide 56 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Multiply as you would integers. Step 2 Place the decimal point in the product so that the number of decimal places equals the combined number of decimal places of both factors. Note: You may need to insert zeros to the left of the whole-number product to get the correct number of decimal places in the answer. Example 1Multiplying Decimals Slide 57 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Multiplying Decimals Slide 58 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 2Multiplying Decimals Slide 59 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Multiply. Then use estimation to check the location of the decimal point. Example Solution: 2Multiplying Decimals (continued) Slide 60 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Multiplying Decimals (continued) Slide 61 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To check the answer, we can round the factors and estimate the product. The purpose of the estimate is primarily to determine whether we have placed the decimal point correctly. Therefore, it is usually sufficient to round each factor to the left-most nonzero digit. This is called front-end rounding. Example Solution: 2Multiplying Decimals Slide 62 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Thus, The first digit for the actual product and the first digit for the estimate 90 is the tens place. Therefore, we are reasonably sure that we have located the decimal point correctly. The estimate 90 is close to PROCEDUREMultiplying a Decimal by a Power of 10 Slide 63 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Move the decimal point to the right the same number of decimal places as the number of zeros in the power of 10. Example 4Multiplying by Powers of 10 Slide 64 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Multiplying by Powers of 10 Slide 65 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. PROCEDUREMultiplying a Decimal by Powers of 0.1 Slide 66 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Move the decimal point to the left the same number of places as there are decimal places in the power of 0.1. Example 5Multiplying by Powers of 0.1 Slide 67 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 5Multiplying by Powers of 0.1 Slide 68 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 6Naming Large Numbers Slide 69 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Write the decimal number representing each word name. a. The distance between the Earth and Sun is approximately 92.9 million miles. b. The number of deaths in the United States due to heart disease in 2010 was projected to be 8 hundred thousand. c. A recent estimate claimed that collectively Americans throw away 472 billion pounds of garbage each year. Example Solution: 6Naming Large Numbers Slide 70 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 7Applying Decimal Multiplication Slide 71 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Jane Marie bought eight cans of tennis balls for $1.98 each. She paid $1.03 in tax. What was the total bill? Example Solution: 7Applying Decimal Multiplication Slide 72 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 8Finding the Area of a Rectangle Slide 73 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Mona Lisa is perhaps the most famous painting in the world. It was painted by Leonardo da Vinci somewhere between 1503 and 1506 and now hangs in the Louvre in Paris, France. The dimensions of the painting are 30 in. by in. What is the total area? Example Solution: 8Finding the Area of a Rectangle Slide 74 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.3 Multiplication of Decimals and Applications with Circles 4.Circumference and Area of a Circle (continued) Slide 75 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A circle is a figure consisting of all points in a flat surface located the same distance from a fixed point called the center. Section 5.3 Multiplication of Decimals and Applications with Circles 4.Circumference and Area of a Circle Slide 76 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 9Finding Diameter and Radius Slide 77 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 9Finding Diameter and Radius Slide 78 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.3 Multiplication of Decimals and Applications with Circles 4.Circumference and Area of a Circle Slide 79 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The distance around a circle is called the circumference. Slide 80 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. FORMULA Circumference of a Circle Example 10Determining Circumference of a Circle Slide 81 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine the circumference. Use 3.14 for Example Solution: 10Determining Circumference of a Circle Slide 82 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The radius is given, r = 4 cm. The distance around the circle is approximately cm. Slide 83 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. FORMULA Area of a Circle Example Determine the area of a circle that has radius 0.3 ft. Approximate the answer by using 3.14 for Round to two decimal places. 11Determining the Area of a Circle Slide 84 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 11Determining the Area of a Circle Slide 85 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section Objectives 5.4 Division of Decimals Slide 86 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Division of Decimals 2.Rounding a Quotient 3.Applications of Decimal Division PROCEDUREDividing a Decimal by a Whole Number Slide 87 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To divide by a whole number: Step 1 Place the decimal point in the quotient directly above the decimal point in the dividend. Step 2 Divide as you would whole numbers. Example 1Dividing by a Whole Number Slide 88 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Divide and check the answer by multiplying. Example Solution: 1Dividing by a Whole Number Slide 89 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.4 Division of Decimals 1.Division of Decimals Slide 90 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. When dividing decimals, we do not use a remainder. Instead we insert zeros to the right of the dividend and continue dividing. Example 2Dividing by an Integer Slide 91 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Divide and check the answer by multiplying. Example Solution: 2Dividing by an Integer (continued) Slide 92 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The quotient will be positive because the divisor and dividend have the same sign. We will perform the division without regard to sign, and then write the quotient as a positive number. Locate the decimal point in the quotient. Rather than using a remainder, we insert zeros in the dividend and continue dividing. Example Solution: 2Dividing by an Integer Slide 93 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Rather than using a remainder, we insert zeros in the dividend and continue dividing. Check by multiplying: The quotient is Example 3Dividing by an Integer Slide 94 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Divide and check the answer by multiplying. Example Solution: 3Dividing by an Integer Slide 95 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The dividend is a whole number, and the decimal point is understood to be to its right. Insert the decimal point above it in the quotient. Since 40 is greater than 5, we need to insert zeros to the right of the dividend. Check by multiplying. The quotient is Section 5.4 Division of Decimals 1.Division of Decimals Slide 96 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sometimes when dividing decimals, the quotient follows a repeated pattern. The result is called a repeating decimal. Example 4Dividing Where the Quotient Is a Repeating Decimal Slide 97 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Dividing Where the Quotient Is a Repeating Decimal (continued) Slide 98 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Notice that as we continue to divide, we get the same values for each successive step. This causes a pattern of repeated digits in the quotient. Therefore, the quotient is a repeating decimal. Example Solution: 4Dividing Where the Quotient Is a Repeating Decimal (continued) Slide 99 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The quotient is To denote the repeated pattern, we often use a bar over the first occurrence of the repeat cycle to the right of the decimal point. That is, Avoiding Mistakes Slide 100 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In Example 4, notice that the repeat bar goes over only the 6. The 5 is not being repeated. Section 5.4 Division of Decimals 1.Division of Decimals Slide 101 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The numbers and are examples of repeating decimals. A decimal that stops is called a terminating decimal. PROCEDUREDividing When the Divisor Is Not a Whole Number Slide 102 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Move the decimal point in the divisor to the right to make it a whole number. Step 2 Move the decimal point in the dividend to the right the same number of places as in step 1. Step 3 Place the decimal point in the quotient directly above the decimal point in the dividend. Step 4 Divide as you would whole numbers. Then apply the correct sign to the quotient. Example 6Dividing Decimals Slide 103 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 6Dividing Decimals (continued) Slide 104 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. The quotient will be negative because the divisor and dividend have different signs. We will perform the division without regard to sign and then apply the negative sign to the quotient. Move the decimal point in the divisor and dividend one place to the right. Example Solution: 6Dividing Decimals (continued) Slide 105 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Line up the decimal point in the quotient. The quotient is 0.8. Example Solution: 6Dividing Decimals Slide 106 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The quotient is 620. PROCEDUREDividing by a Power of 10 Slide 107 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To divide a number by a power of 10, move the decimal point to the left the same number of places as there are zeros in the power of 10. Example 8Dividing by a Power of 10 Slide 108 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 8Dividing by a Power of 10 Slide 109 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 9Rounding a Repeating Decimal Slide 110 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: To round the number we must write out enough of the repeated pattern so that we can view the digit to the right of the rounding place. In this case, we must write out the number to the thousandths place. 9Rounding a Repeating Decimal (continued) Slide 111 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. This digit is less than 5. Discard it and all others to its right. Example Solution: 9Rounding a Repeating Decimal Slide 112 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Now multiply the rounded value by 1.1. This value is close to 50. Section 5.4 Division of Decimals 2.Rounding a Quotient Slide 113 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sometimes we may want to round a quotient to a given place value. To do so, divide until you get a digit in the quotient one place value to the right of the rounding place. At this point, you can stop dividing and round the quotient. Example 10Rounding a Quotient Slide 114 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Round the quotient to the tenths place. Example Solution: 10Rounding a Quotient Slide 115 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To round the quotient to the tenths place, we must determine the hundredths-place digit and use it to base our decision on rounding. The hundredths-place digit is 5. Therefore, we round the quotient to the tenths place by increasing the tenths-place digit by 1 and discarding all digits to its right. The quotient is approximately 8.8. Example 11Applying Division of Decimals Slide 116 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A dinner costs $45.80, and the bill is to be split equally among 5 people. How much must each person pay? Example Solution: 11Applying Division of Decimals Slide 117 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Each person must pay $9.16. Example 12Using Division to Find a Rate of Speed Slide 118 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In a recent year, the world-record time in the mens 400-m run was 43.2 sec. What is the speed in meters per second? Round to one decimal place. (Source: International Association of Athletics Federations) Example Solution: 12Using Division to Find a Rate of Speed (continued) Slide 119 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To find the rate of speed in meters per second, we must divide the distance in meters by the time in seconds. Example Solution: 12Using Division to Find a Rate of Speed Slide 120 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The speed is approximately 9.3 m/sec. Section Objectives 5.5 Fractions, Decimals, and the Order of Operations Slide 121 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Writing Fractions as Decimals 2.Writing Decimals as Fractions 3.Decimals and the Number Line 4.Order of Operations Involving Decimals and Fractions 5.Applications of Decimals and Fractions Section 5.5 Fractions, Decimals, and the Order of Operations 1.Writing Fractions as Decimals Slide 122 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sometimes it is possible to convert a fraction to its equivalent decimal form by rewriting the fraction as a decimal fraction. That is, try to multiply the numerator and denominator by a number that will make the denominator a power of 10. Section 5.5 Fractions, Decimals, and the Order of Operations 1.Writing Fractions as Decimals Slide 123 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. However, some fractions cannot be converted to a fraction with a denominator that is a power of 10. For this reason, we recommend dividing the numerator by the denominator, Example 1Writing Fractions as Decimals Slide 124 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Writing Fractions as Decimals (continued) Slide 125 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Divide the numerator by the denominator. Example Solution: 1Writing Fractions as Decimals (continued) Slide 126 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Writing Fractions as Decimals Slide 127 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 2Converting Fractions to Repeating Decimals Slide 128 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Converting Fractions to Repeating Decimals (continued) Slide 129 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Converting Fractions to Repeating Decimals Slide 130 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.5 Fractions, Decimals, and the Order of Operations 1.Writing Fractions as Decimals Slide 131 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Several fractions are used quite often. Their decimal forms are worth memorizing. Example 3Converting Fractions to Decimals with Rounding Slide 132 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convert the fraction to a decimal rounded to the indicated place value. Example Solution: 3Converting Fractions to Decimals with Rounding (continued) Slide 133 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 3Converting Fractions to Decimals with Rounding Slide 134 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.5 Fractions, Decimals, and the Order of Operations 2.Writing Decimals as Fractions (continued) Slide 135 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Convert terminating decimals to fractions by writing the decimal as a decimal fraction and then reducing the fraction to lowest terms. We do not yet have the tools to convert a repeating decimal to its equivalent fraction form. However, we can make use of our knowledge of the common fractions and their repeating decimal forms from the following table. Section 5.5 Fractions, Decimals, and the Order of Operations 2.Writing Decimals as Fractions Slide 136 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 4Writing Decimals as Fractions Slide 137 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Writing Decimals as Fractions Slide 138 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.5 Fractions, Decimals, and the Order of Operations 3.Decimals and the Number Line Slide 139 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A rational number is a fraction whose numerator is an integer and whose denominator is a nonzero integer. Rational numbers consist of all numbers that can be expressed as terminating decimals or as repeating decimals. Section 5.5 Fractions, Decimals, and the Order of Operations 3.Decimals and the Number Line Slide 140 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A number that cannot be expressed as a repeating or terminating decimal is not a rational number. These are called irrational numbers. The rational numbers and the irrational numbers together make up the set of real numbers. Furthermore, every real number corresponds to a point on the number line. Example 5Ordering Decimals and Fractions Slide 141 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Rank the numbers from least to greatest. Then approximate the position of the points on the number line. Example Solution: 5Ordering Decimals and Fractions (continued) Slide 142 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 5Ordering Decimals and Fractions Slide 143 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The position of these numbers can be seen on the number line. Note that we have expanded the segment of the number line between 0.4 and 0.5 to see more place values to the right of the decimal point. PROCEDUREApplying the Order of Operations Slide 144 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 First perform all operations inside parentheses or other grouping symbols. Step 2 Simplify expressions containing exponents, square roots, or absolute values. Step 3 Perform multiplication or division in the order that they appear from left to right. Step 4 Perform addition or subtraction in the order that they appear from left to right. Example 6Applying the Order of Operations Slide 145 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 6Applying the Order of Operations Slide 146 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 7Applying the Order of Operations Slide 147 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 7Applying the Order of Operations (continued) Slide 148 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Approach 1 Convert all numbers to fractional form. Example Solution: 7Applying the Order of Operations Slide 149 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Approach 2 Convert all numbers to decimal form. Section 5.5 Fractions, Decimals, and the Order of Operations 4.Order of Operations Involving Decimals and Fractions Slide 150 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fractions should be kept in fractional form as long as possible as you simplify an expression. Then perform division and rounding in the last step. Example 8Dividing a Fraction and Decimal Slide 151 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: If we attempt to write as a decimal, we find that it is the repeating decimal Rather than rounding this number, we choose to change 3.6 to fractional form: 3.6 = 8Dividing a Fraction and Decimal (continued) Slide 152 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 8Dividing a Fraction and Decimal Slide 153 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We must write the answer in decimal form, rounded to the nearest hundredth. Example 9Evaluating an Algebraic Expression Slide 154 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine the area of the triangle. Example Solution: 9Evaluating an Algebraic Expression. Slide 155 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In this triangle, the base is 6.8 ft. The height is 4.2 ft and is drawn outside the triangle. Example Joanne filled the gas tank in her car and noted that the odometer read 22,341.9 mi. Ten days later she filled the tank again with 11 gal of gas. Her odometer reading at that time was 22,622.5 mi. a. How many miles had she driven between fill-ups? b. How many miles per gallon did she get? 10Using Decimals and Fractions in a Consumer Application Slide 156 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 10Using Decimals and Fractions in a Consumer Application (continued) Slide 157 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. To find the number of miles driven, we need to subtract the initial odometer reading from the final reading. Joanne had driven mi between fill-ups. Example Solution: 10Using Decimals and Fractions in a Consumer Application Slide 158 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. b. To find the number of miles per gallon (mi/gal), we divide the number of miles driven by the number of gallons. Joanne got 24.4 mi/gal. Section Objectives 5.6 Solving Equations Containing Decimals Slide 159 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Solving Equations Containing Decimals 2.Solving Equations by Clearing Decimals 3.Applications and Problem Solving Example 1Applying the Addition and Subtraction Properties of Equality Slide 160 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Applying the Addition and Subtraction Properties of Equality (continued) Slide 161 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Applying the Addition and Subtraction Properties of Equality Slide 162 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 2Applying the Multiplication and Division Properties of Equality Slide 163 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Applying the Multiplication and Division Properties of Equality (continued) Slide 164 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 2Applying the Multiplication and Division Properties of Equality Slide 165 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 4Solving Equations Involving Multiple Steps Slide 166 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 4Solving Equations Involving Multiple Steps Slide 167 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.6 Solving Equations Containing Decimals 2.Solving Equations by Clearing Decimals Slide 168 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. When solving an equation with decimals, clear the decimals first. To do this, multiply both sides of the equation by a power of 10 (10, 100, 1000, etc.). This will move the decimal point to the right in the coefficient on each term in the equation. Example 5Solving an Equation by Clearing Decimals Slide 169 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 5Solving an Equation by Clearing Decimals (continued) Slide 170 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To determine a power of 10 to use to clear fractions, identify the term with the most digits to the right of the decimal point. In this case, the terms 0.05x and 1.45 each have two digits to the right of the decimal point. Therefore, to clear decimals, we must multiply by 100. This will move the decimal point to the right two places. Example Solution: 5Solving an Equation by Clearing Decimals Slide 171 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The solution is 85. Example 6Translating to an Algebraic Expression Slide 172 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The sum of a number and 7.5 is three times the number. Find the number. Example Solution: 6Translating to an Algebraic Expression (continued) Slide 173 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 6Translating to an Algebraic Expression Slide 174 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 7Solving an Application Involving Geometry Slide 175 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The perimeter of a triangle is 22.8 cm. The longest side is 8.4 cm more than the shortest side. The middle side is twice the shortest side. Find the lengths of the three sides. Example Solution: 7Solving an Application Involving Geometry (continued) Slide 176 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 7Solving an Application Involving Geometry (continued) Slide 177 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 7Solving an Application Involving Geometry Slide 178 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example 8Using a Linear Equation in a Consumer Application Slide 179 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Joanne has a cellular phone plan in which she pays $39.95 per month for 450 min of air time. Additional minutes beyond 450 are charged at a rate of $0.40 per minute. If Joannes bill comes to $87.95, how many minutes did she use beyond 450 min? Example Solution: 8Using a Linear Equation in a Consumer Application (continued) Slide 180 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 8Using a Linear Equation in a Consumer Application Slide 181 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section Objectives 5.7 Mean, Median, and Mode Slide 182 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Mean 2.Median 3.Mode 4.Weighted Mean Section 5.7 Mean, Median, and Mode 1.Mean Slide 183 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. When given a list of numerical data, it is often desirable to obtain a single number that represents the central value of the data. Three such values are called the mean, median, and mode. DEFINITION Mean Slide 184 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The mean (or average) of a set of numbers is the sum of the values divided by the number of values. We can write this as a formula. Example 1Finding the Mean of a Data Set Slide 185 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A small business employs five workers. Their yearly salaries are $42,000 $36,000 $45,000 $35,000 $38,000 a. Find the mean yearly salary for the five employees. b. Suppose the owner of the business makes $218,000 per year. Find the mean salary for all six individuals (that is, include the owners salary). Example Solution: 1Finding the Mean of a Data Set (continued) Slide 186 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Solution: 1Finding the Mean of a Data Set Slide 187 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Section 5.7 Mean, Median, and Mode 2.Median Slide 188 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The median is the middle number in an ordered list of numbers. PROCEDUREFinding the Median Slide 189 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To compute the median of a list of numbers, first arrange the numbers in order from least to greatest. If the number of data values in the list is odd, then the median is the middle number in the list. If the number of data values is even, there is no single middle number. Therefore, the median is the mean (average) of the two middle numbers in the list. Example 2Finding the Median of a Data Set Slide 190 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Consider the salaries of five workers. $42,000 $36,000 $45,000 $35,000 $38,000 a. Find the median salary for the five workers. b. Find the median salary including the owners salary of $218,000. Example Solution: 2Finding the Median of a Data Set (continued) Slide 191 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Arrange the data in order. Because there are five data values (an odd number), the median is the middle number. The median is $38,000. Example Solution: 2Finding the Median of a Data Set Slide 192 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. b. Now consider the scores of all six individuals (including the owner). Arrange the data in order. The median of all six salaries is $40,000. There are six data values (an even number). The median is the average of the two middle numbers. Section 5.7 Mean, Median, and Mode 2.Median Slide 193 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The median is a better representation for a central value when the data list has an unusually high (or low) value. Section 5.7 Mean, Median, and Mode 3.Mode Slide 194 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A third representative value for a list of data is called the mode. DEFINITION Mode Slide 195 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The mode of a set of data is the value or values that occur most often. If two values occur most often we say the data are bimodal. If more than two values occur most often, we say there is no mode. Example 4Finding the Mode of a Data Set Slide 196 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The student-to-teacher ratio is given for elementary schools for ten selected states. For example, California has a student-to-teacher ratio of This means that there are approximately 20.6 students per teacher in California elementary schools. (Source: National Center for Education Statistics) Find the mode of the student-to-teacher ratio for these states. Example Solution: 4Finding the Mode of a Data Set Slide 197 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The data value 16.1 appears the most often. Therefore, the mode is 16.1 students per teacher. Example 5Finding the Mode of a Data Set Slide 198 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the mode of the list of average monthly temperatures for Albany, New York. Values are in F. Example Solution: 5Finding the Mode of a Data Set Slide 199 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. No data value occurs most often. There is no mode for this set of data. Example 6Finding the Mode of a Data Set Slide 200 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The grades for a quiz in college algebra are as follows. The scores are out of a possible 10 points. Find the mode of the scores. Example Solution: 6Finding the Mode of a Data Set Slide 201 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sometimes arranging the data in order makes it easier to find the repeated values. The score of 9 occurs 6 times. The score of 7 occurs 6 times. There are two modes, 9 and 7, because these scores both occur more than any other score. We say that these data are bimodal. TIP: Slide 202 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To remember the difference between median and mode, think of the median of a highway that goes down the middle. Think of the word mode as sounding similar to the word most. Section 5.7 Mean, Median, and Mode 4.Weighted Mean Slide 203 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sometimes data values in a list appear multiple times. In such a case, we can compute a weighted mean. Example 7Using a Weighted Mean to Compute GPA (continued) Slide 204 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. At a certain college, the grades AF are assigned numerical values as follows. Example 7Using a Weighted Mean to Compute GPA Slide 205 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Elmer takes the following classes with the grades as shown. Determine Elmers GPA. Example Solution: 7Using a Weighted Mean to Compute GPA (continued) Slide 206 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The data in the table can be visualized as follows. Example Solution: 7Using a Weighted Mean to Compute GPA (continued) Slide 207 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The number of grade points earned for each course is the product of the grade for the course and the number of credit-hours for the course. Example Solution: 7Using a Weighted Mean to Compute GPA Slide 208 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To determine GPA, we will add the number of grade points earned for each course and then divide by the total number of credit hours taken. Elmers GPA for this term is 2.5.