Decimals are absolutely amazing We have only 10 symbols, yet can represent any number, large or small We use zero (0) as a place holder to allow us

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1 Decimals 1

2 Decimals are absolutely amazing We have only 10 symbols, yet can represent any number, large or small We use zero (0) as a place holder to allow us to do this 2

3 Some Older Number Systems 3

4 Can you imagine doing arithmetic with Roman Numerals? 29 plus 28 becomes: XXIX+ XXVII =? How do you put together the X, V, and I? 4

5 5

6 We Use Place Value vs : What does the position of the 2 say about the value of the number? For numbers greater than one we have: ones tens hundreds For numbers less than one we have: tenths hundredths We separate those less than one by using a decimal point; one is the focus point, not zero! 6

7 One tenth is 1/10 is 0.1 For Numbers Less Than One One thousandth is 1/1000 is What is 0.321? 7

8 We can think in terms of adding is 3 hundreds tens 40 5 ones 5 2 tenths 2/10 6 hundredths 6/ /10 + 6/100 =

9 Examples 9

10 Decimals to Fractions We can write decimals as fractions (and/or mixed numbers) 0.51 = 51/ = 126/1000 We could also write as 1/10 + 2/ /1000, find a common denominator, and add the fractions! 10

11 Mixed Numbers 5.7 = 5 7/ = /

12 Fractions as Decimals 6/10 = /100 = /10 = /100 = 3.06 How do I know that there is something to the left of the decimal point? If the numerator is larger than the denominator 12

13 Examples/Review Three and nine tenths in decimals = Eight hundred four and three hundred ninety-nine thousandths in decimals = in words = 3601/1000 in decimals = 13

14 Solution Three and nine tenths in decimals = 3.9 Eight hundred four and three hundred ninety-nine thousandths in decimals = in words = forty seven and 65 hundredths 3601/1000 in decimals =

15 Place Value Our base-ten place-value numeration system is adequate for expressing all whole numbers and decimal numbers. The value of each digit in a numeral is determined by the position of the digit in the numeral. Digits in different places have different values. Finally, the number 1, not the decimal point, serves as the focal point of decimal numbers. 15

16 Order and Rounding 16

17 Which is larger? or ? or 3.405? 25.1 or 25.13? 0.61 or 0.076? or ? 17

18 Solution or ? or 3.405? 25.1 or 25.13? 0.61 or 0.076? or ? It is less negative, so larger 18

19 Note: 76 = 76.0 = , etc. Similarly 0.35 =.35, etc. and 02 (which makes no sense) is still 2 19

20 Rounding Remember If we round 56 to the nearest ten, we have 60 If we round 3465 to the nearest thousand we have 3000 Rule: if the number in the place to the right of where we are rounding to is <5, drop it: 343 rounded to the nearest 10 is 340 If the number in the place to the left of where we are rounding to is 5, go up: 345 rounded to the nearest 10 is

21 Rounding Similarly, rounded to the nearest hundredth is 2.95 What is rounded to the nearest tenth??? Nearest one? 21

22 Solution What is rounded to the nearest tenth??? 3.0 Nearest one? 3 (or 3.0, they are the same) 22

23 Examples Suzie has an annual income of $41, She is paying her taxes, and the tax table use amounts to the nearest dollar. What income value should she use in determining her tax? Jane has an income of $154, For her income bracket, the tax tables use amounts to the nearest hundred dollars. What income value should she use in determining her tax? 23

24 Solution $41, to the nearest dollar is $41,569 $154, to the nearest hundred dollars is $154,300 24

25 More Examples Which number(s) round to 0.06?

26 Solution and

27 Adding and Subtracting Decimals 27

28 How? Can think of in two ways: need a common denominator, or line up the decimal points = 2385/ /1000; common denominator is 1000, ( )/1000 = 25454/1000 =

29 Estimating =? We know that this has to be close to or 530. This helps us figure out where the decimal point goes If the number seems wrong, check where your decimal point is. The exact answer is , which is near 530 This is especially useful for division! 29

30 Examples: Estimate and Solve = = = What is wrong with:

31 = 6.6; estimate = 6 Solutions = ; estimate = = = ; estimate = = 115 What is wrong with: The decimal points were not kept straight! 31

32 Subtraction Same as addition, but may need to borrow: Get a common denominator or Line up the decimal points = change the 82 to 81 and make the one into 10/

33 Examples: Estimate and Solve = = = 33

34 Solutions = 5.5; estimate = = 24.7; estimate 28 3 = = 3.36; estimate =

35 Example Find the total cost of owning this car: monthly car payment $ monthly car insurance $47.52 average gas per month $

36 Solution To find the total monthly cost, we need to add up all the monthly expenses =? estimate: = 400 actual $

37 Example The average temperature throughout the US, for many years is F. In 1998, the warmest year, the average was F. How much warmer than usual was it in 1998? 37

38 Solution = 2.23 F 38

39 Example To send a 2 lb package by parcel post cost $4.90. To send it express mail costs $ How much more does it cost to send the package by express mail than by regular mail? 39

40 Solution To send a 2 lb package by parcel post cost $4.90. To send it express mail costs $ How much more does it cost to send the package by express mail than by regular mail? = $

41 Example; Mean Chocolate Consumption Which country has the most chocolate consumption per person? The least? How much more chocolate do Austrians eat than Germans? Pounds of Chocolate/yr Austria Germany Norway Switzerland Ireland Country 41

42 Solution Which country has the most chocolate consumption per person? Switzerland The least? Norway How much more chocolate do Austrians eat than Germans? 2.09 lb 42

43 Example The screen of the 2007 iphone was 4.5 by 2.4. What was its perimeter 43

44 Solution The screen of the 2007 iphone was 4.5 by 2.4. What was its perimeter Perimeter = 2 L + 2 W = 2 (4.5) + 2( 2.4) = =

45 Multiplying and Dividing Decimals 45

46 It is like multiplying fractions: 0.6 x 0.03 = 6/10 x 3/100 = 18/1000 = Or a rule: Count the number of digits to the right of the decimal point in the problem, and you need that many in the answer; just multiply the numbers as if they are whole 0.6 x 0.03: 1 place plus 2 places = 3 places; 6 x 3 = 18, but need three decimal places, so

47 Estimation Figuring out how big the number should be is a helpful check x 0.6 = You have 1/10 time 6/10, so you know it is near 6/ x 6 = 672, but it has to be near 6/100, so Or, 112/1000 x 6/100 Or, there are four decimals in the problem, four in the answer 47

48 x 10 = Multiplying by Power of Ten x 0.1 = Just move the decimal point to the right or left, depending on how many decimal places there are in the power of ten 48

49 Examples 23.7 x 10 = x 100 = 7.62 x 0.1 = 1.9 x 0.01 = 7682 x = 49

50 Solutions 23.7 x 10 = x 100 = 20, x 0.1 = x 0.01 = x =

51 Examples 0.7 x 0.9 = 6.8 x 0.3 = x 5.5 = x 3.71 = 51

52 Solutions 0.7 x 0.9 = x 0.3 = 2.04 (estimate 7 x 0.3) x 5.5 = (estimate 2 x 5.5) x 3.71 = (estimate 31 x 4) 52

53 Example The circumference of a circle is 2π times the radius, where π = Approximating π as 3.14, find the circumference of a circle that has a diameter of 10 cm. 53

54 Solution If the diameter is 10, the radius is 5 2 π r = 2 x π x 5 = 10 π =

55 Example In 2010 the average US airplane passenger paid $ per mile to fly. If it is 905 mils from Atlanta to Minneapolis, find the fare. (Round to the nearest cent) 55

56 Solution In 2010 the average US airplane passenger paid $ per mile to fly. If it is 905 mils from Atlanta to Minneapolis, find the fare. (Round to the nearest cent) x 905 = $ Estimate: 0.14 x

57 Dividing Decimals and Order of Operations 57

58 Dividing by a Whole Number The key is to fix the decimal point! Estimation can help / 7 Estimate 270/7 ~ 40; you can check your result against this Write the problem, but fix the decimal point above where it should be! xxx.xxxxx 7)270.6 Do the long division. 7 doesn t divide 2, 7 divides 27, 3 times, 7 divides 60, 8 times, THEN PUT THE DECIMAL POINT / 7 =

59 Examples 8.32 / 32 = / 48 = 59

60 Solutions 8.32 / 32 = 0.26; estimate 8/32 ~ / 48 = 0.71; estimate 34/48 ~ 3/4 =

61 Dividing by a Decimal DON T!!!! Multiply the numerator and denominator by the same power of 10 so that the denominator is no longer a decimal / 2.3 =? Multiply top and bottom (numerator and denominator) by / 23 Then divide a decimal by a whole number 61

62 Examples 17.5 / 0.48 = 1750 / 48 = 23.4 / 0.57= / 28.4 = 62

63 Solution 17.5 / 0.48 = 1750 / 48 = 36.46; estimate 1750/50 = 175/5 = / 0.57= 41.05; rewrite 2340 / 57; estimate 2400/60 = 240/ / 28.4 = 9.59; rewrite / 284; estimate 3000/300 = 30/3 = 10 63

64 Order of Operations PEMDAS: Please Excuse My Dear Aunt Sally P: Parentheses; simplify quantities in parentheses, inside to outside, and remove the parentheses E: Exponents; evaluate any expressions with exponents M and D: Multiplication and Division; perform any multiplication and division from left to right A and S: Addition and Subtraction; perform any addition and subtraction from left to right M and D are done at the same time, as are A and S; My and Dear are adjectives, Aunt and Sally are nouns; they stay together. 64

65 Examples / 1000 x 10 = No P, no E, M & D, working from left to right Step 1: 723.6/1000 = Step 2: x 10 = What happens if we mix the order? if we multiply 1000 x 10 = 10, / 10,000 = : WRONG!!! 65

66 Example 0.5 ( ) = P first: = 7.4 No E M and D: 0.5 x 7.4 =

67 [ (0.9) 2 / 100 ] / 0.2 = Example 67

68 Solution [ (0.9) 2 / 100 ] / 0.2 = P: can t simplify [ ] until we do some work Inside the [ ] first E: (0.9) 2 = 0.81 M & D: 0.81 / 100 = A & S: = Now we have [ ] evaluated [ ] / 0.2 = / 0.2 = / 2 =

69 Example / 100 x 10 = 8.69 ( ) = [ (1.2) 2 / 10 ] / 0.02 = 69

70 897.8 / 100 x 10 = Solution 8.69 ( ) = [ (1.2) 2 / 10 ] / 0.02 =

71 Fractions as Decimals Writing decimals as fractions is easy: denominator is a power of 10 Writing fractions as decimals is a bit harder may not have a finite number of digits. To make a fraction a decimal, just divide: 1/4 = 1 4 = /5 = 4 5 =

72 When it Doesn t Come out Even 2/3 =? : The means keep writing 6 for a long as you want; it will never end _ You can also write 0.6 =

73 Example Write as a decimal, round to nearest thousandth: 28/13 = 2 3/16 = 3 5/16 = 7/9 = 73

74 Solution 28/13 = /16 = /16 = /9 = =

75 Which is Larger? 1/2 or 0.54? 0.44 or 4/9? 5/7 or 0.72? 75

76 Solution 1/2 or 0.54? 0.44 or 4/9? Same 5/7 or 0.72? 76

77 Summary Decimal Arithmetic: Estimate the result Multiplication: You can make decimals fractions and multiply, or Count the number of decimal places (riskier) Division: When dividing by a hole number, mark the decimal point When dividing by decimals, GET RID OF THE DECIMALS IN THE DENOMINATOR! 77

78 Summary Cont. Order of operations Clear parentheses, inside to out Clear exponents Multiplication and Division, from left to right Addition and Subtraction, from left to right PEMDAS: Please Excuse My Dear Aunt Sally M and D go together (adjectives) Aunt and Sally go together (nouns) 78

79 Solution 8.32 / / 48 79

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