Transcript Chapter 3 - Decimals - ArbitraryY | Stochastics

Chapter 3 - Decimals
Math Skills – Week 4
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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
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Reduce all fractional answers to simplest form,
and convert improper fractions to mixed
numbers
MIDTERM Next Class
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Study tips
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◦ Chapters 1, 2, and 3
◦ review slides
◦ your notes
◦ read sections in the book
 look at example problems in book
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Pay attention to what question is asking
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On homework/quizzes, clearly circle your answer
Class Project Handout
◦ Prime factorization vs. Finding all factors
Stuff to Remember (forget???)…
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This is a number in decimal notation
61.88
Decimal
part
Whole
Number part
Decimal Point
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The decimal part represents a number less
than one
Just like… $61.88, 88 represents 88 cents,
which is less than $1.
Introduction to Decimals
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hundredthousandths
millionths
3
Tenthousandths
tenths
ones
8 .
thousandths
5
hundredths
4
tens
Just as with whole numbers, decimal
numbers have place values:
hundreds
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0
2
7
1
9
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 _______
millionths place
◦ 4 is in the hundredths
_______ place
Introduction to Decimals
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Rounding decimals is similar to rounding whole
numbers.
Approximate the decimal to any place value
◦ Steps
1. Write out the number to be rounded in a place value
chart
2. Look at the number to the right of the place value you
are rounding to.
1.
2.
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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
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0.470
Round 0.635457 to nearest hundred thousandths
0.63546
Introduction to Decimals
Class Examples
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Round 48.907 to the nearest tenth
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48.9
Round 31.8562 to the nearest whole number
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32
Round 3.675849 to the nearest tenthousandth
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3.6758
Introduction to Decimals
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.
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◦ 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
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Examples (Addition)
◦ Add:0.83 + 7.942 + 15
 = 23.772
◦ Add: 23.037 + 16.7892
 = 39.8262
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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
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Examples (Subtraction)
◦ Subtract: 39.047 – 7.96
 = 31.087
◦ Find 9.23 less than 29
 = 19.77
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Class Examples (Subtraction)
◦ Subtract 72.039 – 8.47
 = 63.569
◦ Subtract 35 – 9.67
 = 25.33
Addition/Subtraction of Decimals
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Multiplication of decimals is similar to
multiplication of whole numbers.
◦ Question: Where does decimal go?
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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
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Multiplication Steps
1. Do the multiplication as if it were whole
numbers
2. To place the decimal in the right location
1. Count the total number of decimal places in all
of the factors
2. 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
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Examples
◦ 920 x 3.7
 = 3404.0
◦ 0.00079 x 0.025
 = 0.00001975
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Class Examples
◦ 870 x 4.6
 = 4002.0
◦ 0.000086 x 0.057
 = 0.000004902
Multiplication of Decimals
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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
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Dividing decimals is similar to dividing whole
numbers.
◦ Same question…what about the decimal place?
Where does that go?
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Steps
1. Make the divisor a whole number by shifting the
decimal to the right as many times as necessary.
2. 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
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Dividing decimals…contd
◦ Steps
1. 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
706 42090.00
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Round quotient to nearest thousandth  need 4 zeros
after the decimal
706 42090.0000
Division of Decimals
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Dividing decimals…contd
◦ Steps
1. Do the division as if it were whole numbers
2. 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
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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
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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
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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
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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
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Comparing & Converting Fractions
& Decimals
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To convert a fraction  decimal
◦ Steps
1. Divide the numerator of the fraction by the
denominator
2. Round the quotient to a desired place value
◦
Example
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Convert 3/7 to a decimal and round to nearest
Hundredth and Thousandth
= 0.42857
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Nearest Hundredth: 0.43
Nearest Thousandth: 0.429
Comparing & Converting Fractions
& Decimals
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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
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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
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To convert a decimal  fraction
◦ Steps
1. Count the number of decimal places
2. Remove the decimal point (and any leading zeros)
3. 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)
4. Put the fraction in simplest form
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Example
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Convert 0.47 to a fraction
= 47/100
Convert 0.275 to a fraction
275/1000 = 11/40
Comparing & Converting Fractions
& Decimals
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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
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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
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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
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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
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Examples
◦ Find the order relation between 3/8 and 0.38
 3/8 = 0.375 < 0.380  3/8 < 0.38
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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