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# Which of the following fractions has a decimal equivalent

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Manager
Joined: 22 Jul 2009
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Location: Manchester UK
Which of the following fractions has a decimal equivalent  [#permalink]

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08 Jan 2010, 13:22
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67% (01:11) correct 33% (01:13) wrong based on 1009 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
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Joined: 02 Sep 2009
Posts: 64249
Re: Which of the following fractions  [#permalink]

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08 Jan 2010, 13:59
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sagarsabnis wrote:
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

Can some one tell how to solve it in a faster way?

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 $$2*5^2$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

$$\frac{39}{128}=\frac{39}{2^7}$$, denominator has only prime factor 2 in its prime factorization, hence this fraction will be terminating decimal.

All other fractions (after reducing, if possible) have primes other than 2 and 5 in its prime factorization, hence they will be repeated decimals.

Hope it's clear.
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Re: Which of the following fractions  [#permalink]

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12 Jan 2010, 08:05
3
study wrote:
Bunuel:

numbers with terminating decimals basically should have 5 or 2 or both in its denominators, right? So any numerator with denominator 125 or 8 would be a terminating decimal?

Thanks.

Yes, as denominator 125=5^3 or 8=2^3, numerator can be any integer.
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Re: Which of the following fractions  [#permalink]

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14 Sep 2010, 04:51
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prashantbacchewar wrote:
Hi Bunuel
As per your explanation if the denominator is not in the form of 2^n 5^m then the fraction will be terminal decimal. If you look at the denominator of other answer choices they are also not in the above form
1. 189 = 3^3 *7^1
2. 196 = 2^2 * 7^2
3. 225 = 3^2 * 5^2
4. 144 = 2^4 * 3^2

So how the last answer choice is correct still not clear based on your explanation?

As per solution:

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 $$2*5^2$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

A. $$\frac{10}{189}=\frac{10}{3^3*7}$$ --> denominator has primes other than 2 and 5 in its prime factorization, hence it's repeated decimal;

B. $$\frac{15}{196}=\frac{15}{2^2*7^2}$$ --> denominator has primes other than 2 and 5 in its prime factorization, hence it's repeated decimal;

C. $$\frac{16}{225}=\frac{16}{3^2*5^2}$$ --> denominator has primes other than 2 and 5 in its prime factorization, hence it's repeated decimal;

D. $$\frac{25}{144}=\frac{25}{2^4*3^2}$$ --> denominator has primes other than 2 and 5 in its prime factorization, hence it's repeated decimal.

E. $$\frac{39}{128}=\frac{39}{2^7}$$, denominator has only prime factor 2 in its prime factorization, hence this fraction will be terminating decimal. All other fractions' denominator have primes other than 2 and 5 in its prime factorization, hence they WILL BE repeated decimals:

Hope it's clear.

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08 Jan 2010, 14:54
Excellent explanation buddy!!!!
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Re: Which of the following fractions  [#permalink]

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12 Jan 2010, 03:35
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Bunuel:

numbers with terminating decimals basically should have 5 or 2 or both in its denominators, right? So any numerator with denominator 125 or 8 would be a terminating decimal?

Thanks.
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Re: Which of the following fractions  [#permalink]

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12 Sep 2010, 09:41
Nice formula for checking for 2 and 5 factors for denominator. I was dividing all numbers....
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Re: Which of the following fractions  [#permalink]

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13 Sep 2010, 02:20
Terminating... means has to have 2s or 5s excusively in the denominator
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Re: Which of the following fractions  [#permalink]

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13 Sep 2010, 20:22
great explanation on terminating decimals!
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Re: Which of the following fractions  [#permalink]

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14 Sep 2010, 00:43
Hi Bunuel
As per your explanation if the denominator is not in the form of 2^n 5^m then the fraction will be terminal decimal. If you look at the denominator of other answer choices they are also not in the above form
1. 189 = 3^3 *7^1
2. 196 = 2^2 * 7^2
3. 225 = 3^2 * 5^2
4. 144 = 2^4 * 3^2

So how the last answer choice is correct still not clear based on your explanation?
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Re: Which of the following fractions has a decimal equivalent  [#permalink]

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10 Dec 2013, 18:23
First thought: Terminatieng and non terminating
My concept after reading the question: Terinating – non repeating, Non terminating – Repeating numbers after decimal
My Strategy : 1.All numbers are squares or cubes 2. Simply these 3. Then divide
I have learned – Denominator having 2^m5^n are terminating numbers
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Re: Which of the following fractions has a decimal equivalent  [#permalink]

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11 Dec 2013, 10:40
kanusha wrote:
First thought: Terminatieng and non terminating
My concept after reading the question: Terinating – non repeating, Non terminating – Repeating numbers after decimal
My Strategy : 1.All numbers are squares or cubes 2. Simply these 3. Then divide
I have learned – Denominator having 2^m5^n are terminating numbers

Dear kanusha,
I am responding to your private message.

First of all, you may find this blog helpful.
http://magoosh.com/gmat/2012/gmat-math- ... -decimals/

What you say in the last line is correct and is key to understanding this problem. My only caution would be: use proper mathematical grouping symbols. You are not thinking like a mathematician when you write
2^m5^n
That is precisely the way it is written by someone who isn't thinking carefully about the mathematical symbols. What you meant is:
(2^m)(5^n)
Those parenthesis are not garnish, not extra decorative elements --- they are absolute essential pieces of mathematical equipment, and you are setting yourself up for mistake if you casually ignore their tremendous importance. See:
http://magoosh.com/gmat/2013/gmat-quant ... g-symbols/

In this problem, all of the denominators happen to be squares and other powers.
289 = 17^2
196 = 14^2
225 = 15^2
144 = 12^2
128 = 2^7
I think it's good to know the perfect squares up to 20^2 = 400. It's also good to know the first eight powers of 2. It just saves time, and helps to deepen number sense.

Nevertheless, the fact that most of these are squares is not particularly relevant. All you have to do is find the prime factorization of the denominator. As soon as you find a prime factor other than 2 or 5, then you know the decimal would be repeating & non-terminating.

If the denominator an odd number not ending in a 5, then it can't be divisible by 2 or 5: it must have other prime factors and must lead to a repeating & non-terminating decimal. If the denominator is divisible by 3, a very easy check, then it lead to a repeating & non-terminating decimal.

The easy way to handle these, even without knowing they are perfect squares ----
(A) 289 --- an odd number, so not divisible by 2, and clearly not divisible by 5, so it must have other prime factors. No good.
(B) 196 --- divide by 2 = 98 --- divide by 4 = 49 --- other odd factors. No good.
(C) 225 --- 2 + 2 + 5 = 9, which is divisible by 3, so that means 225 is divisible by 3. No good.
(D) 144--- 1 + 4 + 4 = 9, which is divisible by 3, so that means 144 is divisible by 3. No good.
(E) only one left

Does all this make sense?
Mike
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27 Feb 2014, 11:44
uwengdori wrote:
Which of the following has a decimal equivalent that is a terminating decimal?

10/189

15/196

16/225

25/144

39/128

To be honest, I don't even understand what the question is asking for. Help is appreciated.

Dear uwengdori,
I'm happy to respond. You may find some help in the other posts in this merged thread, but I will be happy to explain it as well.

First of all, I would highly suggest reading this post, which will clarify a great deal:
http://magoosh.com/gmat/2012/gmat-math- ... -decimals/

So, as that blog explains, if the denominator of a fraction has no prime factors other than 2 and 5, the fraction will terminate instead of repeat.

The next step is to recognize that 128 is a power of 2. It's highly worthwhile to have the first ten powers of 2 memorized:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024

Since 128 is a power of 2, it has only factors of 2, no other prime factors. This means, any fraction with 128 in the denominator will be a terminating decimal.

If you have any questions after you read that blog post, please let me know.

Mike
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Re: Which of the following fractions has a decimal equivalent  [#permalink]

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06 Jul 2014, 21:12
I look for terminating decimals..I look for even numbers...

Anything halved is always terminating ...E is all 2's...So E wins
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Re: Which of the following fractions has a decimal equivalent  [#permalink]

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30 Apr 2016, 02:20
just look options

128=2^7

10 sec question but i did in 30

E
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Re: Which of the following fractions has a decimal equivalent  [#permalink]

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29 Jul 2019, 13:28
Thank you, Bunuel. Very helpful. Thousand Kudos to you..
Re: Which of the following fractions has a decimal equivalent   [#permalink] 29 Jul 2019, 13:28