Decimals - NJCTLcontent.njctl.org/courses/math/archived-courses... · 5/13/2009 · Comparing Decimals · Line up all of the decimal points · Put zeros in wherever digits are missing

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Decimals

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Define & Review Vocabulary

Identify Place Values

Read and Write Decimals

Compare & Order Decimals

Round Numbers to Designated Place Values

Relate Fractions, Decimals & Percents

Decimal Operations

Real Life Applications

Table of Contents

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Return to Table of Contents

Define & Review Vocabulary

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A Decimal Number (based on the number 10) contains

a Decimal Point.

Move me to find out!

What is a decimal number?

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What is the value of a yellow rod?

Blue Block

Red Block

How many yellow rods are needed to fill the blue block?

Yellow Rod

How many red blocks are needed to fill the blue block?(Hint: Fill a yellow rod with red blocks)

What is the value of a red block?

One Unit

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Identify Place Values

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...........

PU

LLN

OT

E:

CA

N B

ES

TU

DE

N

Place Value Chart

million

hundred thousandten thousand

thousand

hundred

ten

one

and

tenth

hundredth

thousandthten thousandth

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Remember the place values after the decimal point start with tenths

mill

ion

hund

red

thou

sand

ten

thou

sand

thou

sand

hund

red

ten

one

and

tent

h

hund

redt

h

thou

sand

thte

n th

ousa

ndth

9, 8 7 5, 6 1 0 . 4 7 9 4

This number is read:

Nine million, eight hundred seventy-five thousand, six hundred ten AND four thousand, seven hundred ninety-four ten thousandths

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1

A 7

What digit is in the thousandths place?

4,567,098.2318

B 1

C 3

D 8

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2 What digit is in the hundreds place?

6,789.043

A 7

B 4

C 3

D 0

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3 What digit is in the millionths place? 6,754,123.945678

A 6

B 7

C 8

D 4

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4 What digit is in the tens place?

5,432.981

A 4

B 1

C 9

D 3

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5

A 3

B 4

C 2

D 5

What digit is in the ten thousands place?

543,127.9832

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0 . 0 7

5 4 6 . 2 5 9

8 . 3

0 . 8 9

0 . 0 1 1

2 3 5 4 . 6

9 . 7

Can you read the number?

thou

sand

s

hund

reds

tens

ones

and

tent

hs

hund

redt

hs

thou

sand

ths

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Read & Write Decimals

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1. Look to see if there is a number to the left of the decimal; if so write it out. If there is no number to the left of the decimal, skip to step 3. 2. Write an and for the decimal point. 3. Write the number in the decimal part. 4. Write the word for the place value of the rightmost digit.

How do you write a decimal in words?

Write 13.24 in words

Thirteen and twenty-four hundredths

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How does a comma help when writing a decimal number?

The place value before the comma is always stated.

For example:

1,547 is writtenOne THOUSAND, five hundred forty-seven

6,547,100 is written6 MILLION, five hundred forty-seven THOUSAND, one hundred

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Write the decimal in words

1) 5.04 Five and four hundredths

2) 146.457 One hundred forty-six and four hundred fifty-seven thousandths

3) .0009 Nine ten-thousandths

4) 6,345.1498 Six thousand, three hundred forty-five and one thousand, four hundred ninety-eight ten thousandths

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Writing a decimal numerically

1. If there is a ths, there will be a decimal

2. If there is an and, there will be a whole number to the left of the decimal

3. Note the ending - this is the place where the decimal number will end

4. Use the decimal chart to help you!

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Use the chart as a guide to write the decimal numbers

1. four hundred nineteen and five hundredths

2. sixty-five and nine thousandths

3. seven thousand, four hundred thirty-three and twenty hundredths

4. six thousand and eight tenths

thou

sand

s

hund

reds

tens

ones

and

tent

hs

hund

redt

hs

thou

sand

ths

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Compare & OrderDecimals

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Comparing Decimals

· Line up all of the decimal points· Put zeros in wherever digits are missing· Compare the digits, from left to right· The largest digit, having the same place value, names the largest number

Example:

910.800085.007910.801911.900085.070

What digit tells us this is the largest number?

1

Why isthis the smallest number?

Compare the digits

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Order from least to the greatest

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12.405Order from least to greatest

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6 The number with the least value is:

A 12.001

B 12.011

C 12.0009

D 12.0019

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7 The number with the least value is:

A 0.01

B 0.011

C 0.001

D 0.00101

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8 The number with the greatest value is:

A 2.9

B 2.99

C 2.899

D 2.8999

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9 The number with the greatest value is:

A 597.65

B 598.09

C 598.12

D 597.605

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10 How much money would you like to win?

A 1600.50

B 1599.09

C 1601.01

D 1610.02

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Round Numbers to Designated Place

Values

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Let's learn how to round decimals using the following examples:

Round 513.287 to the nearest tenths place.

When rounding decimals, put your pencil point under the place value that you are rounding to

DO NOT MOVE IT

513.287

Rounding Decimals

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Look to the right.

Is the digit 5 or more? Yes OR No

What happens to the 2? Increases by 1 OR remains the same

What happens to everything to the left of the tenths place? Those digits always remain the same.

513.287

Put your pencil point under the digit in the tenths place.

Final Answer: 513.3

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What happened to the digits to the right of the tenths place?

Remember, we are rounding.

The place values to the right of the tenths place have zero value, so now the digits 8 and 7 become zeroes.

513.300 = 513.3

Try the next example...

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Round 3.14159265 to the nearest hundredths place.

First, choose the digit that represents the hundredths place.

11 Answer?

A B C

First, put your pencil point under the 4 and look at the digit to the right. Ask yourself: "Is this digit 5 or more?" Since it's a 1, the 4 remains a 4, and everything after it becomes a zero.

3.14000000 = 3.14

D

Choose which letter is under the hundredths place.

3.14159265

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When rounding to a specific place, your answer MUST have a digit in that place.

Example:Round 14.95 to the nearest tenth

Answer: 15.0 There must be a digit in the tenths place, since we were to round to the tenths place.

Try this:Round 345.6982 to the nearest hundredths place

Answer: 345.70

Caution!

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12 Round

9876.543219

to the nearest hundredths

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13 Round

9876.543219

to the nearest hundred

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14 Round

9876.543219

to the nearest tenths

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15 Round

9876.543219

to the nearest thousandths

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16 Round

9876.543219

to the nearest unit

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decimals

fractions

Writing Decimals as Fractions

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Writing decimals as fractions...

Move the decimal part of the original number into the numerator (without the decimal point)

Example 1:

=0.25

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0.25

Example 1:

=

Then count the number of places to the right of the decimal point.

This place value will become your denominator...

25

100


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Finally, simplify the fraction.


Example 1:

0.25 =25

100=

4

1

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Example 2:

=0.032


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0.032

Example 2:

=

Then count the number of places to the right of the decimal point.

This place value will become your denominator...

32

1,000


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Finally, simplify the fraction.


Example 2:

0.032 =32

1000=

125

4

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If there is a whole part to the original number, move the whole part to front of the fraction.


Example 3:

=4.1

What will the place value be?


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Can this fraction be reduced?


Example 3:

4.1 =1

104

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Let's review how to write a decimal as a fraction!

· Numerator is decimal part of the original number

· Denominator is place value of rightmost digit

· Simplify the fraction

· Whole numbers are written in front of the fraction

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17 Write the decimal as a fraction or mixed number in simplest form:

0.85

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4.375

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1.0025

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20 Write the decimal as a fraction or mixed number in simplest form: 14.12

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8.002

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Common Equivalents you should know like the back of your hand!

0.25 0.75 0.5 0.2

14

34

12

15

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Some repeating decimals can be converted to fractions. These are common equivalents you should also know!

0.3333... 0.6666... 0.1666... 0.8333...

13

23

16

56

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decimals

fractions

Writing Fractions as Decimals

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The Pit!

Divide the numerator by the denominator...Or kick the numerator into the pit!

Writing fractions as decimals...

Example 1:

7

20=

7

20

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Then do the math


Example 1:

7

20=

7.0020=0.35 0.35

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PULLIt is less work to put the 3 in front of the decimal point.What happens if we change the mixed number to an improper fraction? Do we eventually get the same answer?

The Pit!



Example 2:

5

8=3

5

83

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What happens if you use

Then do the math


Example 2:

0.6255

8=3

5.00083 = 3.625

298

?

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The Pit!



Example 3:

5

6=

5

6

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Then do the math


Example 3:

5

6=

5.00060.833

= 0.83

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Definitions:Repeating decimal: A decimal in which a pattern of one or more digits is repeated indefinitely, such as 0.353535...To show that a decimal is repeating, a bar is often put above the repeating digits: .35 means .353535...A repeating decimal does not end.

Terminating decimal: A decimal that has only a finite number of nonzero digits to the right of the decimal point, such as 0.56

Challenge Create a decimal that is non-terminating

and non-repeating.

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The Pit!



Example 4:

5

3=

5

3

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Then do the math


Example 4:

5

3=

5.00031.666

= 1.6

Repeatingdecimal!

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Then do the math


Example 5:

-74

=-7.004-1.75

= -1.75

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22 Write the fraction as a decimal:

3 10

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23 Write the mixed number as a decimal: 3

4

5

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31 22

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25 Write the fraction as a decimal and round to the nearest hundredth:

37

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-12

5

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1 2

1 3

1 4

0.5 0.3333... 0.25


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1 5

2 3

3 4

0.2 0.666... 0.75

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0

deci

mal

s percentsWriting Decimals as Percents

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What does percent mean?

Percent means parts per 100The symbol is %

Example: 50% means 50 per 100

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Multiply by 100 and add the percent symbol.

0.75

Example 1:

=1000.75 =

RememberTo multiply a # by 100, move the decimal two places to the RIGHT. 0.75

Writing decimals as percents...

75%

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0.09

Example 2:

=1000.09 =



9%

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0.007

Example 3:

=1000.007 =



0.7%

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0.4

Example 4:

=1000.4 =



40%

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1.49

Example 5:

=1001.49 =



149%

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8

Example 6:

=1008 =

RememberTo multiply a # by 100, move the decimal two places to the RIGHT. 8


800%

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27 Write the decimal as a percent:

0.45

45%

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1.3

130%

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0.008

.8%

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5

500%

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.2

20%

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perc

ent

decimal

Writing Percents as Decimals

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Divide by 100 and remove the percent symbol.

28%

Example 1:

=10028% =

RememberTo divide a # by 100, move the decimal two places to the LEFT. 28%

Writing percents as decimals...

0.28

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8%

Example 2:

=1008% =



0.08

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0.4%

Example 3:

=1000.4% =

RememberTo divide a # by 100, move the decimal two places to the LEFT. 0.4%


0.004

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375%

Example 4:

=100375% =



3.75

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32 Write the percent as a decimal:

2%

.02

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658%

6.58

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0.019%

.00019

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.043


4.3%

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0.005


.5%

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fractions

percents

Writing Fractions as Percents

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Write an equivalent fraction with a denominator of 100.

Example 1:

34

=100

x 34

=100

x25

10075

Writing fractions as percents...

75%

25

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Example 2:

45

=100

x 45

=100

x20

10080


80%

20

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Example3:

94

=100

x 94

=100

x25

100225


225%

25

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Example 4:

3600

=100

x 3600

=100

x6

1000.5


0.5%

6

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37 Write the fraction as a percent:

1920

95%

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9 1000

.9%

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8 100

8%

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5 2

250%

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3 500

.6%

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To write any fraction as a percent:

Express the fraction as a decimaland then express the decimal as a percent.

Example 5:

78

0.8757.000

87.5%

80.875 100

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Example 6:

129

1.33312.000

133.3%

91.333 100

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Example 7:

58

0.6255.000

362.5%

83.625 1003 3

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58

62.5%

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43 Write the fraction as a percent. Round to the nearest whole percent.

47

57%

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3 5

360%

3

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2 500

.4%

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92

450%

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percents

fractions

Writing Percents as Fractions

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Writing percents as fractions...

Express the % as a fraction with a denominator of 100, then simplify.

Example 1:

75% =75

=100

3

4

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Example 2:

120% =120

=100

1

51

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Multiply by 10to get rid of the decimal



Example 3:

0.3% =0.3

=100

3

1000

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Multiply by 100to get rid of the decimal

Writing percents as fractions...Express the % as a fraction with a denominator of 100, then simplify.

Example 4:

% = =100

1

42

9

4

100

2.25=

10,000

225=

400

9

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Writing percents as fractions...another wayExpress the % as a fraction with a denominator of 100, then simplify.

Example 4:

Convert the percent to a fraction. Divide the numerator by the denominator (100). Simplify.

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2 5

47 Write the percent as a fraction in simplest form:

40%

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110%

1110

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0.5%

1200

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8%

2 25

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5 %

4 75

13

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...Tying it all together!

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Helping you remember...

Fill in each box below with an example of the process described.

% to a fraction % to a decimal

fraction to a % decimal to a %

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52 Find the lowest value

A 5%

B 1/2

C .5%

D .05

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53 Find the greatest value

A 120%

B 1.02

C .2%

D 1.19

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54 Find the greatest value

A 6%

B .6

C 60

D 6

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A 2%

B .2

C .02

D .2%

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A 50%

B 500%

C 50.0

D 50.01

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Reorder the amount starting with the lowest value

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Decimal Operations

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Add decimals

0.25 0.25 0.25 0.25

Four quarters equal one dollar

= $1.00

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When adding or subtracting decimals, always remember to align the decimals vertically...

0.250.250.250.251.00+

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Now, try this - Don't forget - LINE 'EM UP

0.1 + 1.25 + 0.04 + 1.39

0.101.250.041.39+

2.78

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If you know how to add whole numbers then you can add decimals. Just follow these few steps.

Step 1: Put the numbers in a vertical column, aligning the decimal points.

Step 2: Add each column of digits, starting on the right and working left.

Step 3: Place the decimal point in the answer directly below the decimal points that you lined up in Step 1.

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1.15

57 Add the following:

0.6 + 0.55

A 6.1

B .115

C 1.15

D 0.16

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58 Find the sum

1.025 + 0.03 + 14.0001

15.0551

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59 What is the sum of

12.034 and 0.0104?

A 12.1344

B 12.0444

C 12.138

D 1.20444

B

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20.34

60 Find the sum

8 + 0.04 + 12.3

A 13.14

B 19.34

C 20.34

D 20.14

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61 Find the sum:

5 + 100.14 + 57.89 + 2.3

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Let's go to Cool Math and practice addition:

Cool Math Link

http://www.coolmath.com/prealgebra/02-decimals/decimals-cruncher-addition.html

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Put the numbers in a vertical column aligning the decimal points.

1.1- 0.3

Subtract the numbers from right to left using the same rules as whole numbers.

1.1- 0.3

0.8

Subtract decimals

10

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What do we do if there aren't enough decimal places when we subtract?

4.3 - 2.05

Don't forget...Line 'em Up!

4.32.05

What goes here?

4.302.05

2.25

2

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Cool Math Link

Let's go to Cool Math and practice subtraction:

http://www.coolmath.com/prealgebra/02-decimals/decimals-cruncher-subtraction.htm

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625 - 0.23

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6312.809 - 4

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641897.11 - 0.647

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654.1 - 0.09

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6617 - 13.008

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Multiply & Divide Decimals

To multiply two decimals:

· Ignore the decimal points

· Multiply the numbers

· Count the total number of digits to the right of the decimal points

· Put that many digits to the right of the decimal point in your answer

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Multiply & Divide Decimals

3.21x .04

.1284

There are a total of four digits to the right of the decimal points.

There must be four digits to the right of the decimal point in the answer.

}

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23.2x 4.04

928

93.728

There are a total of three digits to the right of the decimal points.

There must be three digits to the right of the decimal point in the answer.

} 92800

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67 Multiply 0.4 x 0.03

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69 Multiply 53 x 0.08

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71 Multiply 0.012 x 0.21

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Divide decimals

To divide a decimal by a whole number:Use long division.Bring the decimal point up in the answer

56.08

28 04

2

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Try these!

112.045

56.0008

PULLPull5

2

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To divide a number by a decimal:

· Change the divisor to a whole number by multiplying by a power of 10

· Multiply the dividend by the same power of 10

· Divide

· Bring the decimal point up in the answer

DividendDivisor

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15.6 6.24

Multiply by 10, so that 15.6 becomes 1566.24 must also be multiplied by 10

156 62.4

.234 23.4

Multiply by 1000, so that .234 becomes 23423.4 must also be multiplied by 1000

234 23400

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By what power of 10 should the divisor and dividend be multiplied?

4.15 2.5 means

.008

0.9

68.342 means

4.2

678.921

2.2

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72 Divide

0.78 0.02

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73

0.6 4.866

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74

10 divided by 0.25

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75

12.03 0.04

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76

0.012 24.6

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Real Life Applications


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77 If you had six hundredths of a mile left to run, how would you write this distance as a decimal?

A 6.0

B .6

C .06

D 06.0

E .60

C

Pull

Pull

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78 Your weekly grocery bill averages $97.39. Round your total to the nearest 10 dollars to figure out approximately how much money to save for groceries per week.

A $100.00

B $90.00

C $97.00

D $95.00A

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79 Sally won $25.00 for her science fair project. Her project cost $12.57 to prepare. How much did Sally actually make as a profit?

A $37.57

B $12.43

C $13.57

D $12.00B

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80 Five students collected paper to be recycled. Shelly's stack was .008 cm. thick; Ken's stack was .125 cm. thick; Joe's stack was .150 cm. thick; Betty's stack was .185 cm. thick; Mary's stack was .005 cm. thick. What was the thickness of the papers collected to be recycled?

A .561 cm.

B .452 cm.

C .480 cm.

D .473 cm.

D

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81 The regular price of a pair of jeans is $29.99. Mrs. Jones has four children for whom she must buy new jeans. The jeans are on sale for $22.50.

What would the total cost be of four pairs of jeans on sale?

A $119.96

B $90.00

C $86.00

D $52.49

B

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82 The regular price of a pair of jeans is $29.99. Mrs. Jones has four children for whom she must buy new jeans. The jeans are on sale for $22.50.

How much money does she save by buying the jeans on sale?

A $7.49

B $30.96

C $29.96

D $89.96

C

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Decimals - NJCTLcontent.njctl.org/courses/math/archived-courses... · 5/13/2009 · Comparing Decimals · Line up all of the decimal points · Put zeros in wherever digits are missing