j9ng-05_WB.qxd 11/14/03 3:41 PM Page 78
5.1
Rational Numbers
Goal: Write fractions as decimals and vice versa.
Vocabulary
Rational A rational number is a number that can be written as
number: the quotient of two integers.
Terminating
decimal:
Repeating
decimal:
Example 1
A decimal that has a final digit
A decimal that has one or more digits that repeat
without end
Identifying Rational Numbers
Show that the number is rational by writing it as a quotient of two
integers.
2
3
b. 12
a. 3
1
4
d. 2 ᎏᎏ
c. 4 ᎏᎏ
Solution
3
a. Write the integer 3 as ᎏ1ᎏ .
⫺12
12
ᎏ . These fractions
b. Write the integer 12 as ᎏ1ᎏ or ᎏ
⫺1
are equivalent .
2
14
c. Write the mixed number 4 ᎏᎏ as the improper fraction ᎏ3ᎏ .
3
1
1
1
9
d. Think of 2 ᎏᎏ as the opposite of 2 ᎏ4ᎏ . First write 2 ᎏ4ᎏ as ᎏ4ᎏ .
4
1
9
9
Then you can write 2 ᎏᎏ as ⫺ᎏ4ᎏ . To write ⫺ᎏ4ᎏ as a quotient
4
of two integers, you can assign the negative sign to either the
⫺9
4
numerator or the denominator . You can write ᎏᎏ or
9
ᎏᎏ .
⫺4
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Chapter 5
Notetaking Guide
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Writing Fractions as Decimals
Example 2
4
9
5
16
a. Write ᎏᎏ as a decimal.
b. Write ᎏᎏ as a decimal.
a.
b.
0.3125
苶0
苶0
苶
16冄苶.0
5苶0
4
ᎏᎏ8
ᎏ
20
16
ᎏᎏ
40
32
ᎏᎏ
80
80
ᎏᎏ
0
0.44. . .
9冄苶.0
4苶0
苶
3
ᎏᎏ6
ᎏ
40
36
ᎏᎏ
Answer: Use a bar to
show the repeating digit
in the repeating decimal:
4
ᎏᎏ 0.4
苶.
9
Answer: The remainder
is 0 , so the decimal is a
terminating decimal:
5
ᎏᎏ 0.3125.
16
Checkpoint Write the fraction or mixed number as a decimal.
3
5
7
20
2. ᎏᎏ
1. ᎏᎏ
⫺0.6
0.35
12
25
23
40
3. 2 ᎏᎏ
4. ᎏᎏ
2.48
0.575
Using Decimals to Compare Fractions
Example 3
42
48
27
30
Compare ᎏᎏ and ᎏᎏ.
42
ᎏᎏ 0.875
48
and
27
ᎏᎏ 30
0.9
Answer: Compare the decimals.
Divide.
0.9
> 0.875 ,
27
42
so ᎏᎏ > ᎏᎏ.
30
48
Lesson 5.1
Rational Numbers
79
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Example 4
Writing Terminating Decimals as Fractions
3 is in hundredths’ place,
3
ᎏ
a. 0.03 ᎏ
100
so denominator is 100 .
4
b. 9.4 ⫺9 ᎏ10ᎏ
2
⫺9 ᎏ5ᎏ
Example 5
4 is in tenths’ place,
so denominator is 10 .
Simplify fraction.
Writing a Repeating Decimal as a Fraction
To write 0.8
苶1
苶 as a fraction, let x 0.8
苶1
苶.
1. Because 0.8
苶1
苶 has 2 repeating digits, multiply each side of
苶1
苶 .
苶1
苶 by 102 , or 100 . Then 100 x 81.8
x 0.8
苶1
苶
100 x 81.8
2. Subtract x from 100 x.
⫺
( x 0.8
苶1
苶)
99 x 81
3. Solve for x and simplify.
99
81
x
ᎏᎏ ᎏᎏ
99
99
x ᎏ9ᎏ
11
Answer: The decimal 0.8
苶1
苶 is equivalent to the fraction ᎏ9ᎏ .
11
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Chapter 5
Notetaking Guide
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Checkpoint Write the decimal as a fraction or mixed number.
6. 3.75
5. 0.8
4
ᎏᎏ
5
3
4
⫺3 ᎏᎏ
7. 0.1
苶2
苶
8. 5.3
苶
1
3
4
ᎏᎏ
33
Example 6
5 ᎏᎏ
Ordering Rational Numbers
17
11
13
Order the numbers ᎏᎏ, 1.35, 5.67, 6, ᎏᎏ, ᎏᎏ from least
2
3
4
to greatest.
Graph the numbers on the number line. You may want to write
improper fractions as mixed numbers.
17
2
⫺8
6
⫺6
13
4
⫺4
11
3
1.35
⫺2
0
2
4
5.67
6
8
Answer: Read the numbers graphed on the number line from
17
13
11
left to right: ⫺ᎏ2ᎏ , ⫺6 , ⫺ᎏ4ᎏ , ⫺1.35 , ᎏ3ᎏ , 5.67 .
Lesson 5.1
Rational Numbers
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