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Manager  Joined: 22 Feb 2006
Posts: 77
Which of the following fractions has a decimal equivalent th  [#permalink]

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4 00:00

Difficulty:   35% (medium)

Question Stats: 65% (01:03) correct 35% (01:15) wrong based on 318 sessions

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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
Manager  Joined: 10 Aug 2009
Posts: 96

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1
Is it E? If I remember correctly, the fraction will terminate only if the denominator has only 2s or 5s in its prime factorization.
Manager  Joined: 10 Aug 2009
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3
I do not think you have to calculate a lot here. The rule is that the fraction terminates if the denominator has ONLY 2 or/and 5 in its prime factorization.

A 10/189...no common factors can be canceled out, and 189 is divisible by 3. The denominator does not have ONLY 2's and 5's since it has 3 in it as well. You can stop here, concluding the fraction does not terminate...
B 15/196 - no common factors can be canceled out. 196=2*2*7*7...there is a prime number 7...so the fraction does not terminate..
C 16/225...no common factors can be caceled out. decomposing the denominator 225=25*9=5^2*3^2. Does not terminate since it has 3 in its denominator
D 25/144 no common factors can be canceled out. 144= 2*72, and 72 is divisible by 3...so the fraction has 3 in its denominator...does not terminate.
E you do not have to check. you know it should be E since it is your last option. But if you want to be sure decompose the denominator and you will get only 2's in it
Manager  Joined: 14 Aug 2009
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if the denominator has 3 or 7 in it and can't be reduct, it must be un-terminating.

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E. Great question.
Be suspicious about 3,7 (more suspicious if the numerator has a factor of 3 or 7). Do not blindly select the answer if the denominator alone has a 3 or 7.
If it is 8,4,2,6 then they are usually terminating.
Math Expert V
Joined: 02 Sep 2009
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TriColor wrote:
-----------------------------------------

Q15:
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

THEORY:

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be terminating decimal.

(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.)

Questions testing this concept:
700-question-94641.html?hilit=terminating%20decimal
is-r-s2-is-a-terminating-decimal-91360.html?hilit=terminating%20decimal
pl-explain-89566.html?hilit=terminating%20decimal
which-of-the-following-fractions-88937.html?hilit=terminating%20decimal

BACK TO THE ORIGINAL QUESTION:

As $$128=2^7$$, then $$\frac{39}{128}$$ will be terminating decimal.

Hope it helps.
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Re: Touch Question Qithout a Calculator  [#permalink]

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Ans = E

the necessary and sufficient condition of terminating decimals is that the denominator's prime factors should only be 2 or 5 or both (form $$2^x * 5^y$$)
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Re: Which of the following fractions has a decimal equivalent th  [#permalink]

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_________________ Re: Which of the following fractions has a decimal equivalent th   [#permalink] 14 Dec 2016, 03:25
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