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# I am looking for an approach to determine if a fraction has

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I am looking for an approach to determine if a fraction has [#permalink]

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24 Sep 2008, 11:14
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I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:
which of the following has decimal equivalent to terminating decimal:

A- 10/189
B- 15/196
C- 16/225
D- 25/144
E- 39/128

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24 Sep 2008, 12:13
jugolo1 wrote:
I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:
which of the following has decimal equivalent to terminating decimal:

A- 10/189
B- 15/196
C- 16/225
D- 25/144
E- 39/128

If the denominator can be written in terms of only 2^x*5^y or 2^x or 5^y. Then the fraction is terminating. Here (E) is 2^7 and meets the above condition and hence terminating.
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24 Sep 2008, 12:14
jugolo1 wrote:
I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:
which of the following has decimal equivalent to terminating decimal:

A- 10/189
B- 15/196
C- 16/225
D- 25/144
E- 39/128

You need to prime factorize the denominator. If the denominator's prime factors are only 2 and/or 5, then the fraction is a terminating decimal

A: 189 = 7*3^4
B: 196 = 14^2 = 7^2*2^2
C: 225 = 15^2 = 3^2*5^2
D: 144 = 12^2 = 3^2*2^4
E: 128 = 2^7
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25 Sep 2008, 03:00
Thanks for clarifying.. excellent

zoinnk wrote:
jugolo1 wrote:
I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:
which of the following has decimal equivalent to terminating decimal:

A- 10/189
B- 15/196
C- 16/225
D- 25/144
E- 39/128

You need to prime factorize the denominator. If the denominator's prime factors are only 2 and/or 5, then the fraction is a terminating decimal

A: 189 = 7*3^4
B: 196 = 14^2 = 7^2*2^2
C: 225 = 15^2 = 3^2*5^2
D: 144 = 12^2 = 3^2*2^4
E: 128 = 2^7
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25 Sep 2008, 07:10
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zoinnk wrote:
jugolo1 wrote:
I am looking for an approach to determine if a fraction has a terminating decimal or not without calculating it, for example:
which of the following has decimal equivalent to terminating decimal:

A- 10/189
B- 15/196
C- 16/225
D- 25/144
E- 39/128

You need to prime factorize the denominator. If the denominator's prime factors are only 2 and/or 5, then the fraction is a terminating decimal

A: 189 = 7*3^4
B: 196 = 14^2 = 7^2*2^2
C: 225 = 15^2 = 3^2*5^2
D: 144 = 12^2 = 3^2*2^4
E: 128 = 2^7

I'd just add one thing here: you *must* make sure that your fraction is reduced before you apply this test. It's not important in the above question, because all of the fractions are already reduced, but it could be important in a different question. For example, you might see:

51/300

and if you just steam ahead and prime factorize that denominator, you'll get 3*2^2*5^2, which might make you think that it's not a terminating decimal. It is, however; when you reduce the fraction, the 3 in the denominator disappears:

51/300 = 17/100 = 0.17
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25 Sep 2008, 07:18
IanStewart wrote:
I'd just add one thing here: you *must* make sure that your fraction is reduced before you apply this test. It's not important in the above question, because all of the fractions are already reduced, but it could be important in a different question. For example, you might see:

51/300

and if you just steam ahead and prime factorize that denominator, you'll get 3*2^2*5^2, which might make you think that it's not a terminating decimal. It is, however; when you reduce the fraction, the 3 in the denominator disappears:

51/300 = 17/100 = 0.17

Good point IanStewart. I'd like to add one more thing.....
On a reduced fraction, when you see denominators like 300, 234, etc... which shows that the number is divisible by another number other than 2 and 5, you can toss it... this speeds up the process a little.
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To find what you seek in the road of life, the best proverb of all is that which says:
"Leave no stone unturned."
-Edward Bulwer Lytton

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25 Sep 2008, 14:00
Thank you..excellent clarification

leonidas wrote:
IanStewart wrote:
I'd just add one thing here: you *must* make sure that your fraction is reduced before you apply this test. It's not important in the above question, because all of the fractions are already reduced, but it could be important in a different question. For example, you might see:

51/300

and if you just steam ahead and prime factorize that denominator, you'll get 3*2^2*5^2, which might make you think that it's not a terminating decimal. It is, however; when you reduce the fraction, the 3 in the denominator disappears:

51/300 = 17/100 = 0.17

Good point IanStewart. I'd like to add one more thing.....
On a reduced fraction, when you see denominators like 300, 234, etc... which shows that the number is divisible by another number other than 2 and 5, you can toss it... this speeds up the process a little.
Re: Termintating decimal   [#permalink] 25 Sep 2008, 14:00
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