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Chapter 3 - Decimals

Chapter 3 - Decimals. Math Skills – Week 4. Introduction to Decimals – Section 3.1 Addition of Decimals – Section 3.2 Subtraction of Decimals – Section 3.3 Multiplication of Decimals – Section 3.4 Division of Decimals – Section 3.5

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Chapter 3 - Decimals

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  1. Chapter 3 - Decimals Math Skills – Week 4

  2. Introduction to Decimals – Section 3.1 • Addition of Decimals – Section 3.2 • Subtraction of Decimals – Section 3.3 • Multiplication of Decimals – Section 3.4 • Division of Decimals – Section 3.5 • Comparing and Converting Fractions and Decimals – Section 3.6 Outline

  3. Reduce all fractional answers to simplest form, and convert improper fractions to mixed numbers • MIDTERM Next Class • Chapters 1, 2, and 3 • Study tips • review slides • your notes • read sections in the book • look at example problems in book • Pay attention to what question is asking • Prime factorization vs. Finding all factors • On homework/quizzes, clearly circle your answer • Class Project Handout Stuff to Remember (forget???)…

  4. This is a number in decimal notation • The decimal part represents a number less than one • Just like… $61.88, 88 represents 88 cents, which is less than $1. 61.88 Decimal part Whole Number part Decimal Point Introduction to Decimals

  5. Just as with whole numbers, decimal numbers have place values: • The position of a digit in a decimal determines the digits place value • 0 is in the hundredths, 3 is in the tenths • 9 is in the _______ place • 4 is in the _______ place hundreds tens ones Ten-thousandths hundred-thousandths tenths hundredths thousandths millionths . 0 7 1 4 5 8 3 2 9 millionths hundredths Introduction to Decimals

  6. Rounding decimals is similar to rounding whole numbers. • Approximate the decimal to any place value • Steps • Write out the number to be rounded in a place value chart • Look at the number to the right of the place value you are rounding to. • If the number is > or = 5, increase the digit in the place value by 1, and remove all digits to the right of it • If the number is < 5, remove it and all of the digits to the right of it. • Examples • Round 0.46972 to the nearest thousandth • 0.470 • Round 0.635457 to nearest hundred thousandths • 0.63546 Introduction to Decimals

  7. Class Examples • Round 48.907 to the nearest tenth • 48.9 • Round 31.8562 to the nearest whole number • 32 • Round 3.675849 to the nearest ten-thousandth • 3.6758 Introduction to Decimals

  8. Adding and subtracting decimal numbers is the similar as adding and subtracting whole numbers • Catch: first align the decimal points of each number on a vertical line. • Assures us that we are adding/subtracting digits that are in the same place value 4290.3 000 16290.903 0 + 65.0729 20646.2759 Addition/Subtraction of Decimals

  9. Examples (Addition) • Add:0.83 + 7.942 + 15 • = 23.772 • Add: 23.037 + 16.7892 • = 39.8262 • Class Examples (Addition) • Find the sum of 4.62, 27.9, and 0.62054 • = 33.14054 • Add: 6.05 + 12 + 0.374 • = 18.424 Addition/Subtraction of Decimals

  10. Examples (Subtraction) • Subtract: 39.047 – 7.96 • = 31.087 • Find 9.23 less than 29 • = 19.77 • Class Examples (Subtraction) • Subtract 72.039 – 8.47 • = 63.569 • Subtract 35 – 9.67 • = 25.33 Addition/Subtraction of Decimals

  11. Multiplication of decimals is similar to multiplication of whole numbers. • Question: Where does decimal go? • Check this… • 0.3 x 5 = 1.5 • Start with 1 decimal place, answer has 1 decimal place • 0.3 x 0.5 = 0.15 • Start with a total of 2 decimal places, answer has 2 decimal places • 0.3 x 0.05 = 0.015 • Start with a total of 3 decimal places, answer has 3 decimal places Multiplication of Decimals

  12. Multiplication Steps • Do the multiplication as if it were whole numbers • To place the decimal in the right location • Count the total number of decimal places in all of the factors • Starting from the right of the product, count the total number of decimal places towards the left, and place the decimal point there. 21.4 x 0.36 3 total decimal places 7 704 . Multiplication of Decimals

  13. Examples • 920 x 3.7 • = 3404.0 • 0.00079 x 0.025 • = 0.00001975 • Class Examples • 870 x 4.6 • = 4002.0 • 0.000086 x 0.057 • = 0.000004902 Multiplication of Decimals

  14. To multiply a decimal by a power of 10 (for example 10, 100, 1,000 etc.) move the decimal to the right the same number of times as there are zeros. • 3.8925 x 10 • = 38.925 • 3.8925 x 100 • = 389.25 • 3.8925 x 1000 • = 3892.5 • 3.8925 x 10000 • = 38925.0 • 3.8925 x 100000 • =389250.0 (Note: we added a zero before the decimal) Multiplication of Decimals

  15. Dividing decimals is similar to dividing whole numbers. • Same question…what about the decimal place? Where does that go? • Steps • Make the divisor a whole number by shifting the decimal to the right as many times as necessary. • Move the decimal in the dividend the same number of times that we moved it in the divisor 7 0 6 4 2 0 9 . 0 ?????? . Division of Decimals

  16. Dividing decimals…contd • Steps • Add zeros to the end of the dividend so that we can round to the desired place value • Example: Round quotient to nearest tenth  write 2 zeros after the decimal • Round quotient to nearest thousandth  need 4 zeros after the decimal 706 42090.00 706 42090.0000 Division of Decimals

  17. Dividing decimals…contd • Steps • Do the division as if it were whole numbers • Put the decimal place in the quotient directly over the decimal point in the dividend 00059.61 ≈ 59.6 706 42090.00 Division of Decimals

  18. Examples • Divide 58.092 ÷ 82 round to the nearest thousandth • = 0.7084 ≈ 0.708 • Divide: 420.9 ÷ 7.06, round to the nearest tenth • = 59.61 ≈ 59.6 • Divide: 2.178 ÷ 0.039, round to the nearest hundredth • ≈ 55.85 Division of Decimals

  19. Class Examples • Divide 37.042 ÷ 76 round to the nearest thousandth • = 0.4873 ≈ 0.487 • Divide: 370.2 ÷ 5.09, round to the nearest tenth • = 72.73 ≈ 72.7 Division of Decimals

  20. To divide a decimal by a power of 10 (for example 10, 100, 1,000 etc.) move the decimal to the left the same number of times as there are zeros. Fill in the blank spaces with zeros. • 34.65 ÷ 10 or 101 • = 3.465 • 34.65 ÷ 100 or 102 • = 0.3465 • 34.65 ÷ 1000 or 103 • = 0.03465 • 34.65 ÷ 10000 or 104 • = 0.003465 Division of Decimals

  21. Fractions and decimals are two ways of representing parts of a whole number. • ¼ is a portion of 1 whole • 0.345 is a portion of 1 whole • Every fraction can be written as a decimal • Every decimal can be written as a fraction Comparing & Converting Fractions & Decimals

  22. To convert a fraction  decimal • Steps • Divide the numerator of the fraction by the denominator • Round the quotient to a desired place value • Example • Convert 3/7 to a decimal and round to nearest Hundredth and Thousandth • = 0.42857 • Nearest Hundredth: 0.43 • Nearest Thousandth: 0.429 Comparing & Converting Fractions & Decimals

  23. Examples • Convert 3/8 to a decimal; round to nearest hundredth • = 0.375 ≈ 0.38 • Convert 2 ¾ to a decimal; round to nearest tenth • = 2.75 ≈ 2.8 • Class Examples • Convert 9/16 to a decimal; round to nearest tenth • = 0.6 • Convert 4 1/6 to a decimal; round to nearest hundredth • = 4.17 Comparing & Converting Fractions & Decimals

  24. To convert a decimal  fraction • Steps • Count the number of decimal places • Remove the decimal point (and any leading zeros) • Put the decimal part over a denominator, • The denominator is a factor of 10 that has the same number of zeros as decimal places (from step 1) • Put the fraction in simplest form • Example • Convert 0.47 to a fraction • = 47/100 • Convert 0.275 to a fraction • 275/1000 = 11/40 Comparing & Converting Fractions & Decimals

  25. Examples: • Convert 0.82 to a fraction • = 82/100 = 2·41 / 2·50 = 41/50 • Convert 4.75 to a fraction • = 4 75/100 = 4 3·25/4·25 = 4 3/4 • Class Examples • Convert 0.56 to a fraction • = 56/100 = 4·14 / 4·25 • Convert 5.35 to a fraction • = 5 35/100 = 5 7·5 / 5·20 = 5 7/20 Comparing & Converting Fractions & Decimals

  26. The order relation between two decimals tells us which decimal is larger than the other • Example: Which is larger 0.88 or 0.088? • 0.88 • Think of this like money • 0.88 is like $0.88 = 88 cents • 0.088 is ≈ $0.09 = 9 cents • Comparing decimals is easy, what about comparing a decimal to a fraction? • Which is larger 5/6 or 0.625? • Question: What to do? • Convert 5/6  Decimal OR • Convert 0.625  fraction Comparing & Converting Fractions & Decimals

  27. Examples • Find the order relation between 3/8 and 0.38 • 3/8 = 0.375 < 0.380  3/8 < 0.38 • Class Example • Find the order relation between 5/16 and 0.32 • 5/16 ≈ 0.313 < 0.32  5/16 < 0.32 Comparing & Converting Fractions & Decimals

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