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Math Expert V
Joined: 02 Sep 2009
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M28-55  [#permalink]

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10 00:00

Difficulty:   85% (hard)

Question Stats: 47% (01:22) correct 53% (01:44) wrong based on 179 sessions

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Data set A consist of 10 terms, each of which is a reciprocal of a prime number, is the median of the data set less than $$\frac{1}{5}$$?

(1) Reciprocal of the median is a prime number.

(2) The product of any two terms of the set is a terminating decimal.

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Math Expert V
Joined: 02 Sep 2009
Posts: 57238
Re M28-55  [#permalink]

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1
4
Official Solution:

(1) Reciprocal of the median is a prime number. If all the terms equal $$\frac{1}{2}$$, then the $$median=\frac{1}{2}$$ and the answer is NO but if all the terms equal $$\frac{1}{7}$$, then the $$median=\frac{1}{7}$$ and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Answer: B
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Re: M28-55  [#permalink]

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Hello Bunuel. Can you please explain why do you have 7/20 and not 1/10...1/2*1/5 = 1/10..why do you have to add 1/5 and 1/2 and then divide it by 2? I did not get the rational behind it....
Math Expert V
Joined: 02 Sep 2009
Posts: 57238
Re: M28-55  [#permalink]

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p2bhokie wrote:
Hello Bunuel. Can you please explain why do you have 7/20 and not 1/10...1/2*1/5 = 1/10..why do you have to add 1/5 and 1/2 and then divide it by 2? I did not get the rational behind it....

Set A consist of 10 terms. The median of a set with even number of terms is the average of two middle terms, when arranged in ascending/descending order.

If two middle terms are 1/5, then the median is simply 1/5.
If two middle terms are 1/2, then the median is simply 1/2.
If two middle terms are 1/5 and 1/2, then the median is $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$.

Hope it's clear.
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Re: M28-55  [#permalink]

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Hi Bunuel,
i could not understand the 2nd statement

(2) The product of any two terms of the set is a terminating decimal

so what is the set looks like? Is it five 1/2 and five 1/5?

Thanks for your help.

Bunuel wrote:
Official Solution:

(1) Reciprocal of the median is a prime number. If all the terms equal $$\frac{1}{2}$$, then the $$median=\frac{1}{2}$$ and the answer is NO but if all the terms equal $$\frac{1}{7}$$, then the $$median=\frac{1}{7}$$ and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Answer: B

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Kindly press +1 Kudos if my post helped you in any way Math Expert V
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Re: M28-55  [#permalink]

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sunita123 wrote:
Hi Bunuel,
i could not understand the 2nd statement

(2) The product of any two terms of the set is a terminating decimal

so what is the set looks like? Is it five 1/2 and five 1/5?

Thanks for your help.

Bunuel wrote:
Official Solution:

(1) Reciprocal of the median is a prime number. If all the terms equal $$\frac{1}{2}$$, then the $$median=\frac{1}{2}$$ and the answer is NO but if all the terms equal $$\frac{1}{7}$$, then the $$median=\frac{1}{7}$$ and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Answer: B

The set could be any combination of 1/2's and 1//5:
{1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2}
{1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5}
{1/5, 1/5, 1/5, 1/5, 1/5, 1/2, 1/2, 1/2, 1/2, 1/2}
....
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Re: M28-55  [#permalink]

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so for example if set is {1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2}
then it does not satisfy this condition (2) The product of any two terms of the set is a terminating decimal

sorry it does. Thanks Bunuel

Bunuel wrote:
sunita123 wrote:
Hi Bunuel,
i could not understand the 2nd statement

(2) The product of any two terms of the set is a terminating decimal

so what is the set looks like? Is it five 1/2 and five 1/5?

Thanks for your help.

Bunuel wrote:
Official Solution:

(1) Reciprocal of the median is a prime number. If all the terms equal $$\frac{1}{2}$$, then the $$median=\frac{1}{2}$$ and the answer is NO but if all the terms equal $$\frac{1}{7}$$, then the $$median=\frac{1}{7}$$ and the answer is YES. Not sufficient.

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Answer: B

The set could be any combination of 1/2's and 1//5:
{1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2}
{1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5, 1/5}
{1/5, 1/5, 1/5, 1/5, 1/5, 1/2, 1/2, 1/2, 1/2, 1/2}
....

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Kindly press +1 Kudos if my post helped you in any way Intern  Joined: 16 Jul 2014
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Re: M28-55  [#permalink]

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Bunuel wrote:
Official Solution:

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Hi Bunuel,

When you say that the second statement implies that the set must consist of 1/2, 1/5, or a mix of both, what exactly implies this? Are 1/2 and 1/5 the only terminating reciprocals of prime numbers and that this is just a rule we should memorize? Or is there something else that gives this away?
Math Expert V
Joined: 02 Sep 2009
Posts: 57238
Re: M28-55  [#permalink]

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2
1
OutOfTheHills wrote:
Bunuel wrote:
Official Solution:

(2) The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of $$\frac{1}{2}$$ or/and $$\frac{1}{5}$$. Thus the median could be $$\frac{1}{2}$$, $$\frac{1}{5}$$ or $$\frac{\frac{1}{5}+\frac{1}{2}}{2}=\frac{7}{20}$$. None of the possible values is less than $$\frac{1}{5}$$. Sufficient.

Hi Bunuel,

When you say that the second statement implies that the set must consist of 1/2, 1/5, or a mix of both, what exactly implies this? Are 1/2 and 1/5 the only terminating reciprocals of prime numbers and that this is just a rule we should memorize? Or is there something else that gives this away?

The stem says that set A consist of 10 terms, each of which is a reciprocal of a prime number. Therefore the elements of A could be 1/2, 1/3, 1/5, 1/7, 1/11, 1/13, ...

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 the 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.

Therefore, for the product of any two terms of the set to be terminating decimal, it must consists of only of 1/2's, 1/5's, or both.

Check Terminating and Recurring Decimals Problems in our Special Questions Directory.

Hope it helps.
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Re: M28-55  [#permalink]

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Bunuel wrote:
OutOfTheHills wrote:
Bunuel wrote:
Official Solution:

Check Terminating and Recurring Decimals Problems in our Special Questions Directory.

Hope it helps.

I was still a bit confused at first, but the directory of problems is a huge, huge help. For anyone with similar problem is me, this is just a rule you should remember.
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Re: M28-55  [#permalink]

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I think this is a high-quality question and I agree with explanation.
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Joined: 25 Dec 2014
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Re: M28-55  [#permalink]

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@Bunnel

For statement 2, the median is either 1/2 or 1/5, there are two different answers.
either less or equal to 1/5. How can we take B to be sufficient?

The question asks if median is less than 1/5. It's either yes with median=1/2 or no with median=1/5.

This makes the answer E, as even after combining the two options, we wouldn't get a definite answer.

Please advice.
Math Expert V
Joined: 02 Sep 2009
Posts: 57238
Re: M28-55  [#permalink]

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KaushalPandey wrote:
@Bunnel

For statement 2, the median is either 1/2 or 1/5, there are two different answers.
either less or equal to 1/5. How can we take B to be sufficient?

The question asks if median is less than 1/5. It's either yes with median=1/2 or no with median=1/5.

This makes the answer E, as even after combining the two options, we wouldn't get a definite answer.

Please advice.

How is 1/2 less than 1/5? 1/2 > 1/5.
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Re: M28-55  [#permalink]

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what if the set consists of 1/2^6-s and 1/5^6-s? In that case, does statement 2 suffice?
Math Expert V
Joined: 02 Sep 2009
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Re: M28-55  [#permalink]

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NaeemHasan wrote:
what if the set consists of 1/2^6-s and 1/5^6-s? In that case, does statement 2 suffice?

We are told that "Set A consist of 10 terms, each of which is a reciprocal of a prime number". Are 1/2^6 and 1/5^6 reciprocals of prime numbers?
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Re: M28-55  [#permalink]

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I don't agree with the explanation. set A could be (1/5,...10 times)
median= 1/5
this being one possibility
Math Expert V
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Posts: 57238
Re: M28-55  [#permalink]

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suraj_patil wrote:
I don't agree with the explanation. set A could be (1/5,...10 times)
median= 1/5
this being one possibility

You should read solution more carefully.

If the set is {1/5, ..., 1/5}, then the median is 1/5 and we have a NO answer to the question (the median is NOT less than 1/5), as in all other possible cases.
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Re: M28-55  [#permalink]

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I think this is a high-quality question and I don't agree with the explanation. The very definition of set says that it is a collection of distinct objects. Hence, each term must be unique. We can not assume, all terms to be same.

Help me understand better.
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Neither does anybody else" Re: M28-55   [#permalink] 01 Jan 2019, 11:17
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