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### 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
- Comparing and Converting Fractions and Decimals – Section 3.6

- 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

- This is a number in improper fractions to mixed numbersdecimal 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- Just as with whole numbers, decimal numbers have improper fractions to mixed numbersplace 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- Rounding decimals is similar to rounding whole numbers. improper fractions to mixed 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.

- Steps
- Examples
- Round 0.46972 to the nearest thousandth
- 0.470

- Round 0.635457 to nearest hundred thousandths
- 0.63546

- Round 0.46972 to the nearest thousandth

- Class Examples improper fractions to mixed numbers
- 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

- Round 48.907 to the nearest tenth

- Adding improper fractions to mixed numbersand 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- Examples improper fractions to mixed numbers(Addition)
- Add:0.83 + 7.942 + 15
- = 23.772

- Add: 23.037 + 16.7892
- = 39.8262

- Add:0.83 + 7.942 + 15
- Class Examples (Addition)
- Find the sum of 4.62, 27.9, and 0.62054
- = 33.14054

- Add: 6.05 + 12 + 0.374
- = 18.424

- Find the sum of 4.62, 27.9, and 0.62054

- Examples improper fractions to mixed numbers(Subtraction)
- Subtract: 39.047 – 7.96
- = 31.087

- Find 9.23 less than 29
- = 19.77

- Subtract: 39.047 – 7.96
- Class Examples (Subtraction)
- Subtract 72.039 – 8.47
- = 63.569

- Subtract 35 – 9.67
- = 25.33

- Subtract 72.039 – 8.47

- 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

- 0.3 x 5 = 1.5

- Multiplication Steps whole numbers.
- 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- Examples whole numbers.
- 920 x 3.7
- = 3404.0

- 0.00079 x 0.025
- = 0.00001975

- 920 x 3.7
- Class Examples
- 870 x 4.6
- = 4002.0

- 0.000086 x 0.057
- = 0.000004902

- 870 x 4.6

- 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)

- 3.8925 x 10

- Dividing decimals is similar to dividing whole numbers. 1,000 etc.) move the decimal to the right the same number of times as there are zeros.
- 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- Dividing decimals…contd 1,000 etc.) move the decimal to the right the same number of times as there are zeros.
- 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

- Add zeros to the end of the dividend so that we can round to the desired place value

- Steps

706

42090.00

706

42090.0000

Division of Decimals- Dividing decimals…contd 1,000 etc.) move the decimal to the right the same number of times as there are zeros.
- 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

- Steps

00059.61 ≈ 59.6

706

42090.00

Division of Decimals- Examples 1,000 etc.) move the decimal to the right the same number of times as there are zeros.
- 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

- Divide 58.092 ÷ 82 round to the nearest thousandth

- Class Examples 1,000 etc.) move the decimal to the right the same number of times as there are zeros.
- 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

- Divide 37.042 ÷ 76 round to the nearest thousandth

- 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

- 34.65 ÷ 10 or 101

- 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

- To convert a fraction a whole number. 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

- = 0.42857

- Convert 3/7 to a decimal and round to nearest Hundredth and Thousandth

- Steps

- Examples a whole number.
- 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

- Convert 3/8 to a decimal; round to nearest hundredth
- 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

- Convert 9/16 to a decimal; round to nearest tenth

- To convert a decimal a whole number. 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

- Convert 0.47 to a fraction

- Steps

- Examples: a whole number.
- 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

- Convert 0.82 to a fraction
- 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

- Convert 0.56 to a fraction

- The a whole number. 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

- Example: Which is larger 0.88 or 0.088?
- 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

- Question: What to do?

- Which is larger 5/6 or 0.625?

- Examples a whole number.
- Find the order relation between 3/8 and 0.38
- 3/8 = 0.375 < 0.380 3/8 < 0.38

- Find the order relation between 3/8 and 0.38
- Class Example
- Find the order relation between 5/16 and 0.32
- 5/16 ≈ 0.313 < 0.32 5/16 < 0.32

- Find the order relation between 5/16 and 0.32

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