Download Chapter 4: Decimals

Chapter 5
Decimals
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All rights reserved.
5.1 Reading and Writing Decimals
Objectives
1. Write parts of a whole using decimals.
2. Identify the place value of a digit.
3. Read and write decimals in words.
4. Write decimals as fractions or mixed numbers.
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Slide 4.1- 2
Decimals are used when a whole is divided into 10
equivalent parts, or into 100 or 1000 or 10,000
equivalent parts.
The dot in 0.1 is called the decimal point.
0.1
decimal point
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Slide 4.1- 3
Parallel
Example 1
Fraction
Decimal
Read As
0.6
six tenths
0.11
eleven hundredths
100
0.64
sixty-four hundredths
352
1000
0.352
three hundred fifty-two
thousandths
a. 6
10
b. 11
100
c. 64
d.
Using the Decimal Forms of Fractions
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Slide 4.1- 4
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 4.1- 5
Parallel
Example 2
Identify the Place Value of a Digit
Identify the place value of each digit.
a. 234.89
4: ones
3: tens
2: hundreds
8: tenths
9: hundredths
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Slide 4.1- 6
Parallel
Example 2
Identify the Place Value of a Digit
Identify the place value of each digit.
b. 0.12547
1: tenths
2: hundredths
5: thousandths
4: ten-thousandths
7: hundred-thousandths
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 4.1- 7
Parallel
Example 3
Reading Decimal Numbers
Tell how to read each decimal in words.
a. 0.4
four tenths
b. 0.52
fifty-two hundredths
c. 0.03
three hundredths
d. 0.826 eight hundred twenty-six thousandths
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Slide 4.1- 8
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 4.1- 9
Parallel
Example 4
Reading Decimals
Read each decimal.
7 is in the tenths place.
a. 12.7
12.7
twelve and seven tenths
12.7 is read “twelve and seven tenths.”
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Slide 4.1- 10
Parallel
Example 4
Reading Decimals
Read each decimal.
1 is in the hundredth place.
b. 352.71
352.71
three hundred fifty-two and seven-one hundredths
352.71 is read “three hundred fifty-two and
seventy-one hundredths.”
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Slide 4.1- 11
Parallel
Example 4
Reading Decimals
Read each decimal.
c. 0.025
“twenty-five thousandths”
d. 12.2059
“twelve and two thousand fifty-nine tenthousandths”
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Slide 4.1- 12
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 4.1- 13
Parallel
Example 5
Writing Decimals as Fractions or
Mixed Numbers
Write each decimal as a fraction or mixed number.
a. 0.21
The digits to the right of the decimal point, 21, are the
numerator of the fraction. The denominator is 100 for
hundredths because the rightmost digit is in the
hundredths place.
0.21 
21
100
100 for hundredths
Hundredths place
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Slide 4.1- 14
Parallel
Example 5
continued
Writing Decimals as Fractions or
Mixed Numbers
Write each decimal as a fraction or mixed number.
b. 0.743
743
0.743 
1000
1000 for thousandths
Thousandths place
The whole number part stays the same.
c. 5.0087
87
5.0087  5
10,000
10,000 for ten-thousandths
Ten-thousandths place
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Slide 4.1- 15
Parallel
Example 6
Writing Decimals as Fractions or
Mixed Numbers in Lowest Terms
Write each decimal as a fraction or mixed number
in lowest terms.
a.
4
0.4 
10
Write
10 for tenths
4
in lowest terms.
10
4
42 2


10 10  2 5
45
45  5
9


b. 0.45 
100
1 00  5
20
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Slide 4.1- 16
Parallel
Example 6
continued
Writing Decimals as Fractions or
Mixed Numbers in Lowest Terms
Write each decimal as a fraction or mixed number
in lowest terms.
The whole number part stays the same.
210
210  10
21
 22
 22
c. 22.210  22
1000
1000  10
100
7075  25  54 283
7075
 54
d. 54.7075  54
400
10,000  25
10,000
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Slide 4.1- 17
5.2 Rounding Decimals
Objectives
1. Learn the rules for rounding decimals.
2. Round decimals to any given place.
3. Round money amounts to the nearest cent
or nearest dollar.
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Slide 4.2- 18
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Slide 4.2- 19
Parallel
Example 1
Rounding a Decimal Number
Round 16.98467 to the nearest thousandth.
Step 1 Draw a “cut off “ line after the
thousandths place.
16. 984
67
Look only
at the 6.
Ignore the
7.
Thousandths
Step 2 Look only at the first digit you are cutting off.
Ignore the other digits you are cutting off.
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Slide 4.2- 20
Parallel
Example 1
continued
Rounding a Decimal Number
Round 16.98467 to the nearest thousandth.
Step 3
If the first digit you are cutting off is 5
or more, round up the part of the
number you are keeping.
16. 984
+ 0. 0 0 1
16.985
67
First digit cut is more
than 5, so round up by
adding 1 thousandth
to the part you are
keeping.
So, 16.98467 rounded to the nearest thousandth is
16.985. We can write 16.98467 ≈ 16.985
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Slide 4.2- 21
Parallel
Example 2
Rounding Decimals to Different Places
Round 0.8912 to the nearest hundredth.
Step 1
0.89
12
You are cutting off the 1 and 2.
they will be dropped.
Hundredths
Look only at the 1.
Step 2
0.89
Step 3
12
The first digit is 4 or less, so the part
you are keeping stays the same.
0.8912 rounded to the nearest hundredth is 0.89. We
can write 0.8912 ≈ 0.89
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Slide 4.2- 22
In many everyday situations, such as shopping in
a store, money amounts are rounded to the
nearest cent. There are 100 cents in a dollar.
1
Each cent is
of a dollar.
100
1
Another way of writing 100 is 0.01. So rounding
to the nearest cent is the same as rounding to
the nearest hundredth of a dollar.
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Slide 4.2- 23
Parallel
Example 3
Rounding to the Nearest Cent
How much money will you pay in each
shopping situation? Round each money amount
to the nearest cent.
a.
$7.6241 (Is it closer to $7.62 or to $7.63?)
$7.62
First digit cut is 4 or
less, so the part you
are keeping stays the
same.
41
You pay $7.62.
b. $3.649 (Is it closer to $3.64 or to $3.65?)
$3.64
First digit cut is 5 or
more, so you round up.
9
$3.64
+ $0.01
$3.65
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You pay.
Slide 4.2- 24
Parallel
Example 4
Rounding to the Nearest Dollar
Round to the nearest dollar.
a.
$84.59 (Is it closer to $84 or to $85?)
First digit cut is 5 or
more, so you round up.
$84
$84. 59
+ 1
So $84.59 rounded to the nearest dollar $85
is $85.
b.
$599.79 (Is it closer to $599 or $600?)
So $599.79
$599
$599. 79
rounded to the
+ 1
First digit cut is 5 or
nearest dollar is
$600
more, so you round up.
$600.
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Slide 4.2- 25
5.3 Adding and Subtracting Decimals
Objectives
1. Add decimals.
2. Subtract decimals.
3. Estimate the answer when adding or
subtracting decimals.
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Slide 4.3- 26
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Slide 4.3- 27
Parallel
Example 1
Adding Decimal Numbers
Find each sum.
a. 17.23 + 56.94
tenths
hundredths
tens
ones
Step 1 Write the numbers in columns with the
decimal points lined up.
17.23
+56.94
Decimal points are lined up.
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Slide 4.3- 28
Parallel
Example 1
continued
Adding Decimal Numbers
Find each sum.
a. 17.23 + 56.94
Step 2 Add as if these were whole numbers.
1 1
Step 3
17.23
+56.94
7 4. 1 7
Decimal point in answer is
lined up under decimal points
in problem.
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Slide 4.3- 29
Parallel
Example 1
continued
Adding Decimal Numbers
Find each sum.
b. 3.271 + 5.819 + 16.248
Write the numbers in columns with the
decimal points lined up.
1 1 11
3.271
5.819
+ 16.248
2 5.3 3 8
Decimal points are lined up.
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 4.3- 30
Parallel
Example 2
Writing Zeros as Placeholders
before Adding
Find each sum.
a. 6.5 + 0.72
There are two decimal places in 0.72, so write a 0 in
the hundredths place in 6.5 so that it has two
decimal places also.
6.50
+ 0.72
7.22
6.50
50
6
100
is equivalent to
6.5
in lowest terms is
5
6
10
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Slide 4.3- 31
Parallel
Example 2
continued
Writing Zeros as Placeholders
before Adding
Find each sum.
b. 9.21 + 8 + 1.378
Write in zeros so all the addends have three decimal
places.
9.210
8.000
+ 1.378
18.588
One 0 is written in.
8 is a whole number, decimal point and
3 zeros are written in.
No zeros are needed.
Decimal points are lined up.
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Slide 4.3- 32
Parallel
Example 3
Subtracting Decimal Numbers
Find the difference. Check your answer.
13.25 from 49.68
Step 1
Step 2
Step 3
49.68
− 13.25
36.43
Decimal points are lined up.
Both numbers have two decimal
places, no need to write in zeros.
Decimal points are lined up.
Check the answer by adding 36.43 and 13.25. If the
subtraction is done correctly, the sum will be 49.68.
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Slide 4.3- 33
Parallel
Example 4
Writing Zeros as Placeholders
before Subtracting
Find each difference.
a. 21.8 from 74.341
Line up decimal points
74.341
− 21.800
52.541
Check the
answer by
adding.
Write two zeros
Subtract as usual.
21.800
+ 52.541
74.341
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Matches minuend in
original problem.
Slide 4.3- 34
Parallel
Example 4
continued
Writing Zeros as Placeholders
before Subtracting
Find each difference.
b. 28.154 from 64.9
Line up decimal points
64.900
− 28.154
36.746
Write two zeros
Subtract as usual.
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Slide 4.3- 35
A common error in working decimal problems
by hand is to misplace the decimal point in the
answer. Or, when using a calculator, you may
accidentally press the wrong key. Estimating
the answer will help you avoid these mistakes.
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Slide 4.3- 36
Parallel
Example 5
Estimating Decimal Answers
Use front end rounding to round each number.
Then add or subtract the rounded numbers to get an
estimated answer. Finally, find the exact answer.
a. Find the sum of 283.5 and 7.913.
Exact:
Estimate:
283.5
300
Rounds to
+ 7.913
+ 8
291.413
308
The estimate goes out to the hundreds and so does the
exact answer. Therefore, the decimal point is probably in
the correct place.
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Slide 4.3- 37
Parallel
Example 5
continued
Estimating Decimal Answers
Use front end rounding to round each number.
Then add or subtract the rounded numbers to get an
estimated answer. Finally, find the exact answer.
b.
Find the difference between 0.75 inches
and 11 inches.
Estimate:
11
− 1
10
Rounds to
Exact:
Write a decimal
11.00
point and three
zeros.
− 0.75
10.25 inches
Exact answer is close to estimate.
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Slide 4.3- 38
5.5 Multiplying Decimals
Objectives
1. Multiply decimals.
2. Estimate the answer when multiplying
decimals.
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Slide 4.4- 39
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Slide 4.4- 40
Parallel
Example 1
Multiplying Decimal Numbers
Find the product of 5.13 and 7.4.
Step 1
Multiply the numbers as if they were whole
numbers.
5.13
You do not have to line up the
× 7.4
decimal points when multiplying.
2052
3591
37962
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Slide 4.4- 41
Parallel
Example 1
continued
Multiplying Decimal Numbers
Find the product of 5.13 and 7.4.
Step 2
Count the total number of decimal places in
both factors.
2 decimal places
5.13
1 decimal places
× 7.4
3 total decimal places
2052
3591
37962
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Slide 4.4- 42
Parallel
Example 1
continued
Multiplying Decimal Numbers
Find the product of 5.13 and 7.4.
Step 3
Count over 3 places in the product and write
the decimal point. Count from right to left.
2 decimal places
5.13
1 decimal places
× 7.4
2052
3591
3 decimal places in
3 7. 9 6 2
product.
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Slide 4.4- 43
Parallel
Example 2
Multiplying Decimal Numbers
Find the product (0.056)(0.09).
Start by multiplying, then count decimal places.
0.056
× 0.09
504
3 decimal places
2 decimal places
5 decimal places needed in product
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Slide 4.4- 44
Parallel
Example 2
continued
Multiplying Decimal Numbers
0.056
× 0.09
.0 0 5 0 4
3 decimal places
2 decimal places
5 decimal places needed in product
After multiplying, the answer has only three decimal
places, but five are needed, so write two zeros on the
left side of the answer.
Now count over 5 places and write the decimal point.
The final product is 0.00504, which has five decimal
places.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 4.4- 45
Parallel
Example 3
Estimate before Multiplying
First estimate the answer to (71.47)(12.8) using
front end rounding. Then find the exact answer.
Estimate:
70
× 10
700
3 decimal places are
needed in the product.
Exact:
71.47
× 12.8
57176
14294
7147
9 1 4. 8 1 6
Both the estimate and the exact answer go out to the
hundreds place, so the decimal point in 914.816 is probably
in the correct place.
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Slide 4.4- 46
5.6 Dividing Decimals
Objectives
1. Divide a decimal by a whole number.
2. Divide a number by a decimal.
3. Estimate the answer when dividing decimals.
4. Use the order of operations with decimals.
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Slide 4.5- 47
There are two kinds of decimal division problems:
those in which a decimal is divided by a whole
number, and those in which a number is divided
by a decimal. Recall the parts of a division
problem.
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Slide 4.5- 48
Parallel
Example 1
Dividing Decimals by Whole Numbers
Find each quotient. Check the quotients by
multiplying.
a. 18.46 by 2
Dividend
Divisor
Rewrite the division problem.
Step 1
2 18.46
Write the decimal point in the quotient
directly above the decimal point in the
dividend.
.
2 18.46
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Slide 4.5- 49
Parallel
Example 1
continued
Step 2
Dividing Decimals by Whole Numbers
Divide as if the numbers were whole
numbers.
9.23
2 18.46
Matches, so 9.23 is correct.
Check by multiplying
the quotient times the
divisor.
9.23
×
2
18.46
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Slide 4.5- 50
Parallel
Example 1
continued
Dividing Decimals by Whole Numbers
b. 7 172.2
Dividend
Divisor
2 4. 6
7 172.2
14
32
28
42
42
0
Write the decimal point in the quotient
above the decimal point in the
dividend. Then divide as if the
numbers were whole numbers.
Matches
24.6
×
7
172.2
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Slide 4.5- 51
Parallel
Example 2
Writing Extra Zeros to Complete a
Division
Divide 7.5 by 8. Check the quotient by multiplying.
Keep dividing until the remainder is 0, or until the
digits in the quotient begin to repeat in a pattern.
0.9 375
8 7.5000
0.9375
72
×
8
30
7.5000
24
60
Matches dividend, so 0.9375 is correct.
56
40
40
Stop dividing when the remainder is 0.
0
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Slide 4.5- 52
Parallel
Example 3
Rounding a Decimal Quotient
Divide 5.8 by 6. Round the quotient to the nearest
thousandth. Write extra zeros in the dividend so you
can continue dividing.
0.9 666
6 5.8000
54
Notice that the digit 6 in
40
the answer is repeating. It
36
will continue to do so. The
40
remainder will never be 0.
36
40
36
4
Remainder is still not 0.
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Slide 4.5- 53
Parallel
Example 3
continued
Rounding a Decimal Quotient
There are two ways to show that the answer is a
repeating decimal. You can write three dots after
the answer, or you can write a bar above the
digits that repeat. 0.9666… or
0.96
When repeating decimals occur, round the
quotient to the directions in the problem. In this
problem, to round to the thousandths, divide out
one more place, to the ten-thousandths.
5.8  6  0.9666
rounds to 0.967
Check: (0.967)(6) = 5.802
Not exact because 0.967 was rounded
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Slide 4.5- 54
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Slide 4.5- 55
Parallel
Example 4
Dividing by a Decimal
0.004 36.84
Move the decimal point in the divisor three places to
the right so 0.004 becomes the whole number 4.
Move the decimal point in the dividend the same
number of places, write in an extra zero.
0.004 36.840.
Moving the decimal point three
places to the right is like multiplying
by 1000.
9210.
4 36840.
Divide as usual.
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Slide 4.5- 56
Estimating answers helps you catch errors.
Compare the estimate to your exact answer. If
they are very different, work the problem again.
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Slide 4.5- 57
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Slide 4.5- 58
Parallel
Example 6
Using Order of Operations
Use order of operations to simplify each
expression.
a. 3.9 + 5.42 + 8.16
3.9 + 29.16 + 8.16
Apply the exponent (5.4)(5.4) is 29.16.
Add from left to right.
33.06 + 8.16
41.22
b. 8.64 + (4.8 – 3.1)(2.7)
8.64 +
(1.7) (2.7)
8.64 +
4.59
Work inside the parenthesis.
Multiply next.
Add last.
13.23
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Slide 4.5- 59
5.8 Writing Fractions as Decimals
Objectives
1. Write fractions as equivalent decimals.
2. Compare the size of fractions and decimals.
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Slide 4.6- 60
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Slide 4.6- 61
Parallel
Example 1
Writing Fractions or Mixed Numbers
as Decimals
a. Write 3/8 as a decimal.
3/8 means 3 ÷ 8. Write it as 8 3. The decimal point
in the dividend is on the right side of 3. Write
extra zeros in the dividend so you can continue
0.375
dividing until the remainder is 0.
8 3.000
24
60
3
 0.375
8
56
40
40
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0
Slide 4.6- 62
Parallel
Example 1
Writing Fractions or Mixed Numbers
as Decimals
1
2
4
b. Write
as a decimal.
One method is to divide 1 by 4 to get 0.25 for the
fraction part. Then add the whole number part.
0.25
1
 4 1  4 1.00
4
8
20
2.00
20
0.25
0
A second method is to write the mixed
number as an improper fraction and
then divide numerator by
2.25
denominator.
1 9
2 
4 4
4 9.00
8
10
8
20
2.25
20
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0
Slide 4.6- 63
Parallel
Example 2
Writing a Fraction as a Decimal with
Rounding
1
3
Write as a decimal and round to the nearest
thousandth.
0.3333
1
 1  3  3 1  3 1.0000
3
9
10
9
Written as a repeating
decimal, 31  0.3
10
9
10
9
1
Rounded to the nearest
thousandth, 31  0.333
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Slide 4.6- 64
You can use a number line to compare fractions and
decimals. The number line below shows the space
between 0 and 1. The locations of commonly used
fractions are marked, along with their decimal
equivalents.
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Slide 4.6- 65
This number line shows the locations of some
commonly used fractions between 0 and 1 that are
equivalent to repeating decimals. The decimal
equivalents use a bar above repeating digits.
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Slide 4.6- 66
Parallel
Example 3
Using a Number Line to Compare
Numbers
Use the number lines to decide whether to write >,
< or = in the blank between each pair of numbers.
a.
0.375 ____ 0.3125
0.375 is to the right of 0.3125, use the > symbol.
0.375 is greater than 0.3125
> 0.3125
0.375 ____
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Slide 4.6- 67
Parallel
Example 3
Using a Number Line to Compare
Numbers
Use the number lines to decide whether to write >,
< or = in the blank between each pair of numbers.
7
b. 8 ____ 0.875
They are the same point on the number line.
7
They are equivalent.
= 0.875
____
8
c. 0.16 ____ 0.16
0.16 is less than 0.16
< 0.16
0.16 ____
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Slide 4.6- 68
Parallel
Example 4
Arranging Numbers in Order
Write each group of numbers in order, from least to
greatest.
0.52
0.513
0.5204
It is easiest to compare decimals if they are all
tenths, or all hundredths and so on. Because
0.5204 has four decimal places, write zeros to the
right of the other numbers so they also have four
decimal places.
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Slide 4.6- 69
Parallel
Example 4
0.52
Arranging Numbers in Order
0.513
0.5204
0.52
= 0.5200 = 5200 ten-thousandths
0.513 = 0.5130 = 5130 ten-thousandths
0.5204 = 0.5204 = 5204 ten-thousandths
From least to greatest, the correct order is shown.
0.513
0.52
0.5204
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Slide 4.6- 70