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# M15-16

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Math Expert
Joined: 02 Sep 2009
Posts: 43828

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15 Sep 2014, 23:55
Expert's post
15
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Difficulty:

95% (hard)

Question Stats:

58% (02:39) correct 43% (02:44) wrong based on 120 sessions

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Of 200 surveyed students, 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$. If each student read at least one of the books, what is the difference between the number of students who read only book $$A$$ and the number of students who read only book $$B$$?

A. 20
B. 25
C. 30
D. 35
E. 40
[Reveal] Spoiler: OA

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15 Sep 2014, 23:55
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Expert's post
Official Solution:

Of 200 surveyed students, 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$. If each student read at least one of the books, what is the difference between the number of students who read only book $$A$$ and the number of students who read only book $$B$$?

A. 20
B. 25
C. 30
D. 35
E. 40

Say the number of students who read book $$A$$ is $$A$$ and the number of students who read book $$B$$ is $$B$$.

Given that 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$, so the number of students who read both books is $$0.2A=0.25B$$, and therefore $$A=1.25B$$.

Since each student read at least one of the books then $$\{Total\}=\{A\}+\{B\}-\{Both\}$$, hence $$200=1.25B+B-0.25B$$, which gives $$B=100$$, $$A=1.25B=125$$ and $$\{Both\}=0.25B=25$$.

The number of students who read only book $$A$$ is $$\{A\}-\{Both\}=125-25=100$$;

The number of students who read only book $$B$$ is $$\{B\}-\{Both\}=100-25=75$$;

The difference is $$100-75=25$$.

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Joined: 24 Jun 2015
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09 Jul 2015, 07:40
Bunuel wrote:
Official Solution:

Of 200 surveyed students, 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$. If each student read at least one of the books, what is the difference between the number of students who read only book $$A$$ and the number of students who read only book $$B$$?

A. 20
B. 25
C. 30
D. 35
E. 40

Say the number of students who read book $$A$$ is $$A$$ and the number of students who read book $$B$$ is $$B$$.

Given that 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$, so the number of students who read both books is $$0.2A=0.25B$$, or: $$A=1.25B$$.

Since each student read at least one of the books then $$\{Total\}=\{A\}+\{B\}-\{Both\}$$, hence $$200=1.25B+B-0.25B$$, which gives $$B=100$$, $$A=1.25B=125$$ and $$\{Both\}=0.25B=25$$.

The number of students who read only book $$A$$ is $$\{A\}-\{Both\}=125-25=100$$;

The number of students who read only book $$B$$ is $$\{B\}-\{Both\}=100-25=75$$;

The difference is $$100-75=25$$.

Hi Bunuel,

I got this answer right but it took me almost 5 minutes.... I dont know If I am too slow but I think I did it in a normal pace.... How can this problem be solved faster? Usually I am good with overlapping sets using double matrix, but this problem has different restrictions that makes it a harder... Could you help me?

Thanks a lot.

Regards

Luis Navarro
Looking for 700
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19 Jul 2015, 08:03
1
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The third statement concludes with "or: A=1.25B". "Or" in math isn't quite the same as "or" in english. Indeed, "Or" represents an alternative to the expression, but it's not logical with the noun it modifies. That is, "or: A=1.25B" isn't the number of those who read A and B anymore, but the ratio of A and B. I know what you meant, but the logic could be enhanced. Perhaps you could add " .20A = .25B, and therefore, A=1.25B." Otherwise the meaning isn't as clear as it could be.

Thanks,
Math Expert
Joined: 02 Sep 2009
Posts: 43828

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20 Jul 2015, 04:04
mejia401 wrote:
The third statement concludes with "or: A=1.25B". "Or" in math isn't quite the same as "or" in english. Indeed, "Or" represents an alternative to the expression, but it's not logical with the noun it modifies. That is, "or: A=1.25B" isn't the number of those who read A and B anymore, but the ratio of A and B. I know what you meant, but the logic could be enhanced. Perhaps you could add " .20A = .25B, and therefore, A=1.25B." Otherwise the meaning isn't as clear as it could be.

Thanks,

Edited as suggested. Thank you!
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19 Aug 2015, 07:07
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Need help as to how the calculation for both which is 0.25B has been derived?
Math Expert
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19 Aug 2015, 09:05
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Expert's post
schak2rhyme wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Need help as to how the calculation for both which is 0.25B has been derived?

From $$\{Total\}=\{A\}+\{B\}-\{Both\}$$ we get that $$B=100$$. Next, the stem says that "25% of those who read book $$B$$ also read book $$A$$.", thus 25% of those who read book $$B$$ read both book A and book B. Therefore, $$\{Both\}=0.25B=0.25100=25$$.

Hope it's clear.
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20 Aug 2015, 06:24
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Bunuel wrote:
Of 200 surveyed students, 20% of those who read book $$A$$ also read book $$B$$ and 25% of those who read book $$B$$ also read book $$A$$. If each student read at least one of the books, what is the difference between the number of students who read only book $$A$$ and the number of students who read only book $$B$$?

A. 20
B. 25
C. 30
D. 35
E. 40

After constructing the matrix as attached:
From here: 0.2X=0.25Y......> 4x=5Y or the same X=1.25Y
Difference between reading only A and only B= 0.8X - 0.75Y, simplify and from above equation to 0.2(4X) - 0.75Y= 0.2*5Y -0.75 Y = Y-0.75Y= 0.25Y.......(1)
From the matrix: Y+0.8X=200…..> Y + 0.2 (4X)+Y=200……> 2Y=200…> Y=100.

Applying in (1)… Difference=0.25Y=25
>> !!!

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Joined: 10 May 2015
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06 Sep 2015, 19:55
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. I actually thought the question implied that the no of people reading both A and B is 0.2 A + 0.25 B. Can it not be the case upon reading the question? Please throw some light on this.
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31 Oct 2015, 20:27
0.2A = 0.25B, this can be inferred from the question.
this can be rewritten as 0.8A = 1B, or total who reads A = 1.25B and those who read only A = B
now we have B-0.75B = 0.25B
the only left to find out is B.
we have 0.25B both, 0.75B both, and B - those who read only A. We have 2B=200 or B=100. The difference thus must be 100-75 = 25.
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24 Sep 2016, 02:47
Isn't it should be A+B-2(both) as both part is common to both A and B?
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25 Sep 2016, 22:37
Saps wrote:
Isn't it should be A+B-2(both) as both part is common to both A and B?

What should be A + B - 2(both) ? If it's the number of students who read only book A or only book B, then yes.
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06 Dec 2016, 01:43
Twenty percent of A is common with Twenty five percent of B. Rest of A =80% ,Rest of B =75%
If we consider the common part as 1 part of A and 1 part of B, 4 parts of A and 3 parts of B are left. therefore the total number of parts is 8 and the difference between only A and only B is 1 part which is equal to 200/8 =25
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18 Jan 2017, 05:13
1
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The ratio of A to B is 5 to 4.
We are given that each student chooses at least one book. The intersection of A & B is 'x' (assuming 5X choose A and 4X choose B).
There are a total of 200 students. The number of people reading just A is 4x. The number of people reading just B is 3X. X read both A & B.

8X = 200 --> X=25. 4X-3X=X=25.
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20 Jan 2017, 09:46
I think this is a high-quality question and the explanation is clear enough.
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Joined: 29 Aug 2012
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02 May 2017, 05:43
very simple ! m=ones who just read A , n=ones who just read B
0.2(m+x)=x ... m=4x
0.25(n+x)=x ... n=3x
200=m+x+n=8x ... x=25
m-n=x=25
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02 May 2017, 15:14

0.20A=0.25B
B=4/5 A

200 =0.8A+0.6A+0.2A
A= 125

Re: M15-16   [#permalink] 02 May 2017, 15:14
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# M15-16

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