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terminating decimal, pls help

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by mals24 » Mon Apr 20, 2009 3:55 am
this is an official question and should be studied.
Yes Sir!!! :)

Answer should be E.

A fraction is a terminating decimal if the denominator is of the form: 2^n or 5^m or 2^n*5^m.

A, C, D are out straight away because their denominators have 3 as one of the factors.

Between B and E

128 = 2^7

So E is a terminating decimal.
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by Ian Stewart » Mon Apr 20, 2009 3:47 pm
mals24 wrote:
A fraction is a terminating decimal if the denominator is of the form: 2^n or 5^m or 2^n*5^m.
Absolutely correct, with one important note: your fraction must be *reduced* before you apply this test. For example, if you try to apply the above to the fraction 27/48, you might mistakenly think that this is not a terminating decimal (because the denominator is divisible by a prime different from 2 or 5), but you need to reduce the fraction first: 27/48 = 9/16 = 9/(2^4), which must terminate.
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by tanviet » Tue Apr 21, 2009 1:31 am
how do you know that if the denominator is in form 2^n, 5^m, or 2^n*5^m , the fractiom is terminating.

pls. explain this
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by Ian Stewart » Tue Apr 21, 2009 4:11 am
duongthang wrote:how do you know that if the denominator is in form 2^n, 5^m, or 2^n*5^m , the fractiom is terminating.
If you have, say, the fraction a/2^n, you can multiply on the top and bottom by 5^n to get

(a*5^n)/(2^n * 5^n) = (a*5^n)/(10^n)

Since you have a power of 10 (so a number like 10, or 100, or 1000, etc) in the denominator, this must be a terminating decimal. Similarly, if you have a fraction like

21/125 = 21/5^3

we can calculate the decimal of this very quickly by multiplying the top and bottom by 2^3

21/5^3 = (2^3 * 21)/(2^3 * 5^3) = 162/10^3 = 162/1000 = 0.162

As long as the only primes in the denominator are 2, 5, or both, you can get 10^x in the denominator by multiplying on the top and bottom by a well-chosen power of 2 or 5. Since you can get a denominator that is a power of 10, the decimal must terminate.
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by mals24 » Tue Apr 21, 2009 4:34 am
If you have, say, the fraction a/2^n, you can multiply on the top and bottom by 5^n to get

(a*5^n)/(2^n * 5^n) = (a*5^n)/(10^n)

Since you have a power of 10 (so a number like 10, or 100, or 1000, etc) in the denominator, this must be a terminating decimal. Similarly, if you have a fraction like

21/125 = 21/5^3

we can calculate the decimal of this very quickly by multiplying the top and bottom by 2^3

21/5^3 = (2^3 * 21)/(2^3 * 5^3) = 162/10^3 = 162/1000 = 0.162

As long as the only primes in the denominator are 2, 5, or both, you can get 10^x in the denominator by multiplying on the top and bottom by a well-chosen power of 2 or 5. Since you can get a denominator that is a power of 10, the decimal must terminate.
Thanks a lot Ian for taking the time out and posting such great explanations. :)
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