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# Which of the following fractions has a decimal equivalent

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Director
Joined: 19 Nov 2004
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Location: SF Bay Area, USA
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Which of the following fractions has a decimal equivalent [#permalink]  04 Feb 2005, 21:42
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Which of the following fractions has a decimal equivalent that is a terminating decimal?

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

Is there an easy way to solve this one?
VP
Joined: 25 Nov 2004
Posts: 1495
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Re: PS - Terminating Decimal [#permalink]  04 Feb 2005, 22:13
terminating decimal is the decimal number which terminates at a certain point. the fraction whose denominator is ends at 5 or in multiple of 5 are terminating decimals as compared with the fractions ending at integers other than 5 or its multiples.

the OA should be C, but E terminates earlier than C.
SVP
Joined: 03 Jan 2005
Posts: 2250
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189=9*21
196=4*7^2
225=25*9
144=4^2*3^2
128=2^7

We know that 1/3 and 1/7 are not terminant decimals. But 1/2 is.

So (E)
VP
Joined: 13 Jun 2004
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Director
Joined: 19 Nov 2004
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I think I know why E is the correct answer

The denominator can be expressed as a power of 2 and 5 alone i.e 2^7x5^0 => it is a terminating number

From http://mathforum.org/library/drmath/view/58174.html

The fraction will terminate if and only if the denominator has for
prime divisors only 2 and 5, that is, if and only if the denominator
has the form 2^a * 5^b for some exponents a >= 0 and b >= 0. The
number of decimal places until it terminates is the larger of a
and b.

The proof of this lies in the fact that every terminating decimal
has the form n/10^e, for some e >= 0 (e is the number of places to
the right that the decimal point must be moved to give you an integer,
and n is that integer), and every fraction of that form has a
terminating decimal found by writing down n and moving the decimal
point e places to the left. Now when you cancel common factors from
n/10^e = n/(2*5)^e = n/(2^e*5^e), it may reduce the exponents
in the denominator, but that is all that can happen.
SVP
Joined: 03 Jan 2005
Posts: 2250
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Yes, that is good explanation.

Basically you turn 1/128 to (1/2)^7, you know that 1/2 is a terminating decimal, therefore (1/2)^7 must also be. The denominator of the others has factors of 3 or 7 and cannot be terminating decimals.
Manager
Joined: 01 Jan 2005
Posts: 167
Location: NJ
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Good explaination HongHu and Nocilis.
Thanks.
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