Chapter 1 / Whole Numbers and Introduction to Algebra Chapter Five Decimals 5.1 Introductions to Decimals 5.2 Adding & Subtracting Decimals 5.3 Multiplying Decimals & Circumference of a Circle 5.4 Dividing Decimals 5.5 Fractions, Decimals, & Order of Operations 5.6 Equations Containing Decimals

Introduction to Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Section 5.1 Introduction to Decimals

Chapter 1 / Whole Numbers and Introduction to Algebra Like fractional notation, decimal notation is used to denote a part of a whole. Numbers written in decimal notation are called decimal numbers, or simply decimals. The decimal 16.734 has three parts. 16.743 Whole- number part Decimal part Decimal point

The position of each digit in a number determines its place value. hundred-thousandths hundreds tens tenths hundredths thousandths ten-thousandths ones 1 6 7 3 4 1 10 1 1000 1 100,000 Place Value 100 10 1 1 100 1 10,000 decimal point

of the value of the place to its left. Notice that the value of each place is of the value of the place to its left. 1 10 Martin-Gay, Prealgebra, 5ed

Martin-Gay, Prealgebra, 5ed 16.734 The digit 3 is in the hundredths place, so its value is 3 hundredths or . 3 100 Martin-Gay, Prealgebra, 5ed

Writing (or Reading) a Decimal in Words Chapter 1 / Whole Numbers and Introduction to Algebra Writing (or Reading) a Decimal in Words Step 1. Write the whole-number part in words. Step 2. Write “and” for the decimal point. Step 3. Write the decimal part in words as though it were a whole number, followed by the place value of the last digit. Martin-Gay, Prealgebra, 5ed

Writing a Decimal in Words Write the decimal 143.056 in words. 143.056 whole-number part decimal part one hundred forty-three and fifty-six thousandths Martin-Gay, Prealgebra, 5ed

Writing Decimals in Standard Form A decimal written in words can be written in standard form by reversing the procedure. Write one hundred six and five hundredths in standard form. one hundred six and five hundredths whole-number part decimal decimal part 5 must be in the hundredths place 106 . 05

three and fifty-four thousandths is 3.054 Martin-Gay, Prealgebra, 5ed Helpful Hint When writing a decimal from words to decimal notation, make sure the last digit is in the correct place by inserting 0s after the decimal point if necessary. For example, three and fifty-four thousandths is 3.054 thousandths place Martin-Gay, Prealgebra, 5ed

Chapter 1 / Whole Numbers and Introduction to Algebra Writing Decimals as Fractions Once you master writing and reading decimals correctly, then you write a decimal as a fraction using the fractions associated with the words you use when you read it. 0.9 is read “nine tenths” and written as a fraction as Martin-Gay, Prealgebra, 5ed

Martin-Gay, Prealgebra, 5ed twenty-one hundredths and written as a fraction as 0.21 is read as 21 100 0.011 is read as eleven thousandths and written as a fraction as 11 1000 Martin-Gay, Prealgebra, 5ed

37 100 . = 029 29 1000 . = 2 decimal places 2 zeros 3 decimal places 3 zeros Notice that the number of decimal places in a decimal number is the same as the number of zeros in the denominator of the equivalent fraction. We can use this fact to write decimals as fractions.

Martin-Gay, Prealgebra, 5ed Comparing Decimals One way to compare decimals is to compare their graphs on a number line. Recall that for any two numbers on a number line, the number to the left is smaller and the number to the right is larger. To compare 0.3 and 0.7 look at their graphs. 0.3 0.7 1 0.3 < 0.7 or 0.7 > 0.3 3 10 7 10 Martin-Gay, Prealgebra, 5ed

Chapter 1 / Whole Numbers and Introduction to Algebra Comparing decimals by comparing their graphs on a number line can be time consuming, so we compare the size of decimals by comparing digits in corresponding places. Martin-Gay, Prealgebra, 5ed

Comparing Two Positive Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Two Positive Decimals Compare digits in the same places from left to right. When two digits are not equal, the number with the larger digit is the larger decimal. If necessary, insert 0s after the last digit to the right of the decimal point to continue comparing. Compare hundredths place digits 35.638 35.657 3 < 5 35.638 < 35.657 Martin-Gay, Prealgebra, 5ed

Chapter 1 / Whole Numbers and Introduction to Algebra Helpful Hint For any decimal, writing 0s after the last digit to the right of the decimal point does not change the value of the number. 8.5 = 8.50 = 8.500, and so on When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit. 15 = 15.0 = 15.00, and so on Martin-Gay, Prealgebra, 5ed

Chapter 1 / Whole Numbers and Introduction to Algebra Rounding Decimals We round the decimal part of a decimal number in nearly the same way as we round whole numbers. The only difference is that we drop digits to the right of the rounding place, instead of replacing these digits by 0s. For example, 63.782 rounded to the nearest hundredth is 63.78 Martin-Gay, Prealgebra, 5ed

Rounding Decimals To a Place Value to the Right of the Decimal Point Chapter 1 / Whole Numbers and Introduction to Algebra Rounding Decimals To a Place Value to the Right of the Decimal Point Step 1. Locate the digit to the right of the given place value. Step 2. If this digit is 5 or greater, add 1 to the digit in the given place value and drop all digits to the right. If this digit is less than 5, drop all digits to the right of the given place. Martin-Gay, Prealgebra, 5ed

Rounding Decimals to a Place Value Round 326.4386 to the nearest tenth. Locate the digit to the right of the tenths place. tenths place 326.4386 digit to the right Since the digit to the right is less than 5, drop it and all digits to its right. 326.4386 rounded to the nearest tenths is 326.4 Martin-Gay, Prealgebra, 5ed

Adding and Subtracting Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Section 5.2 Adding and Subtracting Decimals 21

Adding or Subtracting Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Adding or Subtracting Decimals Step 1. Write the decimals so that the decimal points line up vertically. Step 2. Add or subtract as for whole numbers. Step 3. Place the decimal point in the sum or difference so that it lines up vertically with the decimal points in the problem. 22 Martin-Gay, Prealgebra, 5ed 22

Helpful Hint Recall that 0s may be inserted to the right of the decimal point after the last digit without changing the value of the decimal. This may be used to help line up place values when adding or subtracting decimals. 85 - 13.26 becomes 85.00 - 13.26 two 0s inserted 71.74 23

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Helpful Hint Don’t forget that the decimal point in a whole number is after the last digit. 24 Martin-Gay, Prealgebra, 5ed 24

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Estimating Operations on Decimals Estimating sums, differences, products, and quotients of decimal numbers is an important skill whether you use a calculator or perform decimal operations by hand. Martin-Gay, Prealgebra, 5ed 25

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Estimating When Adding Decimals Add 23.8 + 32.1. Exact Estimate rounds to rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed 26

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Helpful Hint When rounding to check a calculation, you may want to round the numbers to a place value of your choosing so that your estimates are easy to compute mentally. Martin-Gay, Prealgebra, 5ed 27

Evaluating with Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Evaluating with Decimals Evaluate x + y for x = 5.5 and y = 2.8. Replace x with 5.5 and y with 2.8 in x + y. x + y = ( ) + ( ) 5.5 2.8 = 8.3 28 Martin-Gay, Prealgebra, 5ed 28

Multiplying Decimals and Circumference of a Circle Chapter 1 / Whole Numbers and Introduction to Algebra Section 5.3 Multiplying Decimals and Circumference of a Circle 29

Martin-Gay, Prealgebra, 5ed Multiplying Decimals Multiplying decimals is similar to multiplying whole numbers. The difference is that we place a decimal point in the product. 7 10 3 100 x 21 1000 0.7 x 0.03 = = 1 decimal place 2 decimal places = 0.021 3 decimal places 30 Martin-Gay, Prealgebra, 5ed

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Decimals Step 1. Multiply the decimals as though they were whole numbers. Step 2. The decimal point in the product is placed so the number of decimal places in the product is equal to the sum of the number of decimal places in the factors. 31 Martin-Gay, Prealgebra, 5ed 31

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Estimating when Multiplying Decimals Multiply 32.3 x 1.9. Exact Estimate rounds to rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed 32

Multiplying Decimals by Powers of 10 Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Decimals by Powers of 10 There are some patterns that occur when we multiply a number by a power of ten, such as 10, 100, 1000, 10,000, and so on. 33 Martin-Gay, Prealgebra, 5ed 33

Multiplying Decimals by Powers of 10 Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying Decimals by Powers of 10 76.543 x 10 = 765.43 76.543 x 100 = 7654.3 76.543 x 100,000 = 7,654,300 Decimal point moved 1 place to the right. 1 zero Decimal point moved 2 places to the right. 2 zeros Decimal point moved 5 places to the right. 5 zeros The decimal point is moved the same number of places as there are zeros in the power of 10. 34 Martin-Gay, Prealgebra, 5ed 34

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying by Powers of 10 such as 10, 100, 1000 or 10,000, . . . Move the decimal point to the right the same number of places as there are zeros in the power of 10. Multiply: 3.4305 x 100 Since there are two zeros in 100, move the decimal place two places to the right. 3.4305 x 100 = 3.4305 = 343.05 35 Martin-Gay, Prealgebra, 5ed 35

Notice that zeros had to be inserted. Chapter 1 / Whole Numbers and Introduction to Algebra Multiplying by Powers of 10 such as 0.1, 0.01, 0.001, 0.0001, . . . Move the decimal point to the left the same number of places as there are decimal places in the power of 10. Multiply: 8.57 x 0.01 Since there are two decimal places in 0.01, move the decimal place two places to the left. 8.57 x 0.01 = 008.57 = 0.0857 Notice that zeros had to be inserted. 36 36

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Finding the Circumference of a Circle The distance around a polygon is called its perimeter. The distance around a circle is called the circumference. This distance depends on the radius or the diameter of the circle. 37 Martin-Gay, Prealgebra, 5ed 37

Circumference of a Circle Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Circumference of a Circle r d Circumference = 2·p ·radius or Circumference = p ·diameter C = 2 p r or C = p d 38 Martin-Gay, Prealgebra, 5ed 38

Martin-Gay, Prealgebra, 5ed The symbol p is the Greek letter pi, pronounced “pie.” It is a constant between 3 and 4. A decimal approximation for p is 3.14. A fraction approximation for p is . 22 7 39 Martin-Gay, Prealgebra, 5ed

Find the circumference of a circle whose radius is 4 inches. Chapter 1 / Whole Numbers and Introduction to Algebra 4 inches Find the circumference of a circle whose radius is 4 inches. C = 2pr = 2p ·4 = 8p inches 8p inches is the exact circumference of this circle. If we replace  with the approximation 3.14, C = 8  8(3.14) = 25.12 inches. 25.12 inches is the approximate circumference of the circle. 40 40

Dividing Decimals Section 5.4 Chapter 1 / Whole Numbers and Introduction to Algebra Section 5.4 Dividing Decimals 41

Division of decimal numbers is similar to division of whole numbers. The only difference is the placement of a decimal point in the quotient. If the divisor is a whole number, divide as for whole numbers; then place the decimal point in the quotient directly above the decimal point in the dividend. 8 4 quotient divisor 63 52.92 dividend - 5 0 4 2 5 2 -2 52 42

Martin-Gay, Prealgebra, 5ed If the divisor is not a whole number, we need to move the decimal point to the right until the divisor is a whole number before we divide. divisor 6 3 52 92 . dividend 63 529 2 . 8 4 63 52 9.2 - 504 25 2 -252 43 Martin-Gay, Prealgebra, 5ed

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Dividing by a Decimal Step 1. Move the decimal point in the divisor to the right until the divisor is a whole number. Step 2. Move the decimal point in the dividend to the right the same number of places as the decimal point was moved in Step 1. Step 3. Divide. Place the decimal point in the quotient directly over the moved decimal point in the dividend. 44 Martin-Gay, Prealgebra, 5ed 44

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Estimating When Dividing Decimals Divide 258.3 ÷ 2.8 Exact Estimate rounds to This is a reasonable answer. Martin-Gay, Prealgebra, 5ed 45

Martin-Gay, Prealgebra, 5ed There are patterns that occur when dividing by powers of 10, such as 10, 100, 1000, and so on. . 45 62 10 = 456.2 The decimal point moved 1 place to the left. 1 zero 1 000 4562 , . = 456.2 The decimal point moved 3 places to the left. 3 zeros The pattern suggests the following rule. 46 Martin-Gay, Prealgebra, 5ed

To divide by a number is the same as multiplying by its reciprocal. Dividing Decimals by Powers of 10 such as 10, 100, or 1000, . . . Move the decimal point of the dividend to the left the same number of places as there are zeros in the power of 10. Notice that this is the same pattern as multiplying by powers of 10 such as 0.1, 0.01, or 0.001. Because dividing by a power of 10 such as 100 is the same as multiplying by its reciprocal , or 0.01. To divide by a number is the same as multiplying by its reciprocal. 47

Fractions, Decimals, and Order of Operations Section 5.5 Fractions, Decimals, and Order of Operations

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Writing Fractions as Decimals To write a fraction as a decimal, divide the numerator by the denominator. 49 Martin-Gay, Prealgebra, 5ed 49

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Fractions and Decimals To compare decimals and fractions, write the fraction as an equivalent decimal. Compare 0.125 and Therefore, 0.125 < 0.25 50 Martin-Gay, Prealgebra, 5ed 50

Martin-Gay, Prealgebra, 5ed Chapter 1 / Whole Numbers and Introduction to Algebra Order of Operations 1. Do all operations within grouping symbols such as parentheses or brackets. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right. 51 Martin-Gay, Prealgebra, 5ed 51

Using the Order of Operations Martin-Gay, Prealgebra, 5ed Simplify ( – 2.3)2 + 4.1(2.2 + 3.1) ( – 2.3)2 + 4.1(2.2 + 3.1) = ( – 2.3)2 + 4.1(5.3) Simplify inside parentheses. Objective D Continued = 5.29 + 4.1(5.3) Write ( – 2.3)2 as 5.29. = 5.29 + 21.73 Multiply. = 27.02 Add. Martin-Gay, Prealgebra, 5ed 52

Martin-Gay, Prealgebra, 5ed Finding the Area of a Triangle height A base • height = 1 2 base A bh = 1 2 53 Martin-Gay, Prealgebra, 5ed

Equations Containing Decimals Chapter 1 / Whole Numbers and Introduction to Algebra Section 5.6 Equations Containing Decimals 54

Steps for Solving an Equation in x Chapter 1 / Whole Numbers and Introduction to Algebra Steps for Solving an Equation in x Step 1. If fractions are present, multiply both sides of the equation by the LCD of the fractions. Step 2. If parentheses are present, use the distributive property. Step 3. Combine any like terms on each side of the equation. 55 Martin-Gay, Prealgebra, 5ed 55

Steps for Solving an Equation . . . Chapter 1 / Whole Numbers and Introduction to Algebra Steps for Solving an Equation . . . Step 4. Use the addition property of equality to rewrite the equation so that variable terms are on one side of the equation and constant terms are on the other side. Step 5. Divide both sides by the numerical coefficient of x to solve. Step 6. Check the answer in the original equation. 56 Martin-Gay, Prealgebra, 5ed 56

Solving Equations with Decimals – 0.01(5a + 4) = 0.04 – 0.01(a + 4) – 1(5a + 4) = 4 – 1(a + 4) Multiply both sides by 100. – 5a – 4 = 4 – a – 4 Apply the distributive property. – 4a – 4 = 4 – 4 Add a to both sides. – 4a = 4 Add 4 to both sides and simplify. a = – 1 Divide both sides by 4.