Section 3.5
Decimal Numbers and Fractions
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Objectives
o Change decimal numbers to fractions.
o Change fractions to decimal numbers.
o Operate with both fractions and decimal numbers.
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Changing Decimal Numbers to Fractions
To Change from Decimal Numbers to Fractions
A terminating decimal number can be written in
fraction form by writing a fraction with the following:
1. a numerator that consists of the whole number
formed by all the digits of the decimal number, and
2. a denominator that is the power of ten that names
the position of the rightmost digit.
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Example 1: Changing Decimal Numbers to
Fractions
25
5 51
1
0.25 


100 2  5  2  5 4
hundredths
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Example 2: Changing Decimal Numbers to
Fractions
32
48
8
0.32 


100 4  25 25
hundredths
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Example 3: Changing Decimal Numbers to
Fractions
131
0.131 
1000
thousandths
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Example 4: Changing Decimal Numbers to
Fractions
75
25  3
3
0.075 


1000 25  40 40
thousandths
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Example 5: Changing Decimal Numbers to
Fractions
26 2  13 13
2.6 


10 2  5
5
tenths
or, as a mixed number,
23
6
3
2.6  2  2
2
25
10
5
tenths
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Example 6: Changing Fractions to Decimal
Numbers
3
8
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0.3 75
8 3.000
24
60
56
40
40
0
3
 0.375
8
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Example 7: Changing Fractions to Decimal
Numbers
5
4
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1. 2 5
4 5.00
4
10
8
20
20
0
5
 1.25
4
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Example 8: Changing Fractions to Decimal
Number
7
12
0.5833
12 7.0000
60
1 00
96
40
36
40
36
4
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The 3 will repeat without end.
Continuing to divide will give
a remainder of 4 each time.
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Example 8: Changing Fractions to Decimal
Number (Cont.)
7
= 0.58333... where the three dots (called
We write
12
an ellipsis) mean “and so on” or to continue the
pattern without stopping. Or, if we agree to round, we
7
can write
= 0.583 (to the nearest thousandth).
12
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Example 9: Changing Fractions to Decimal
Number
1
7
0. 142857
7 1.000000
7
30
28
20
14
60
56
40
35
50
49
1
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The six digits will repeat in the
same pattern without end.
We write
1
= 0.142857142857142857...
7
The remainder will repeat in sequence
1,3,2,6,4,5,1, and so on. Therefore, the digits
in the quotient will also repeat in sequence.
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Operating with Both Fractions and Decimal
Numbers
Notes
In some cases changing fractions to decimal form may
involve rounding the decimal form of a number and
settling for an approximate answer.
To have a more accurate answer, we may need to
change the decimals to fraction form and then perform
the operations.
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Example 10: Operating with Both Fractions and
Decimal Numbers
1
3
Find the sum 10 + 7.32 + 5 in decimal form.
2
5
Solution
1

1
=
0.50
= 10.50
10


2
2
7.32 = 7.32
3
+5
5
=
5.60
23.42
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3

 = 0.60
5
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Example 11: Operating with Both Fractions and
Decimal Numbers
3
3
Determine whether 16 is larger than 0.18 by changing
16
to decimal form and then comparing the two numbers.
Find the difference.
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Example 11: Operating with Both Fractions and
Decimal Numbers (cont.)
Solution
Divide first.
0.1875 So 3  0.1875.
16 3.0000
16
16
1 40
1 28
120
112
80
80
0
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Example 11: Operating with Both Fractions and
Decimal Numbers (cont.)
Now subtract.
0.1875
 0.1800
0.0075
difference
3
Thus
is larger than 0.18 and their difference is
16
0.0075.
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Practice Problems
Convert the following to fraction form.
1. 0.475
2. 2.9
3. 0.003
4. 1.17
Convert the following to decimal form. Write repeating
decimals using bar notation.
1
13
2
18
5.
6.
7.
8.
20
16
7
37
1
4
9. Find the sum 3 + 5.16 + 6 in decimal form.
4
5
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Practice Problem Answers
19
1.
40
117
4.
100
29
2.
10
3
3.
1000
5. 0.8125
6. 0.05
7. 0.285714
8. 0.486
9. 15.21
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