Equivalent Fractions: Simplifying And Building With Answers Page 11
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16Now look at the prime factorizations of the denominators:
17
17
=
= 0.425
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2 • 2 • 2 • 5
19
19
=
= 0.63
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2 • 3 • 5
Recall that the decimal system has denominators which are powers of 10 = 2 • 5. Suppose we
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want to build the first fraction
up to a power of 10 in the denominator. Since each 10 = 2 • 5,
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we will need to multiply by two additional factors of 5:
17
17
5 • 5
17 • 25
425
=
=
=
= 0.425
•
40
2 • 2 • 2 • 5
5 • 5
10 • 10 • 10
1000
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However, with the second fraction
, the prime factor of 3 will always be in the prime
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factorization of the denominator. Thus we can never build its denominator to a power of 10, and
thus it can’t be represented as a terminating decimal. In summary, only fractions whose
denominators have prime factors of 2 and 5 can be converted to terminating decimals. If a
denominator of a fraction has prime factors other that 2 or 5, it will result in a repeating decimal
(assuming the fraction is simplified). Thus the vast majority of fractions have repeating, rather
than terminating, decimal forms.
Example 6
Determine whether the decimal form of each rational number will be terminating
or repeating. Do not actually convert the fraction to decimal!
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a.
85
23
!
b.
400
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c.
320
97
!
d.
440
155
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