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M15-16

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M15-16  [#permalink]

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New post 16 Sep 2014, 00:55
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A
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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

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Re M15-16  [#permalink]

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New post 16 Sep 2014, 00:55
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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 \(100-75=25\).


Answer: B
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Re: M15-16  [#permalink]

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New post 09 Jul 2015, 08:40
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 \(100-75=25\).


Answer: B


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.

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Re: M15-16  [#permalink]

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New post 19 Jul 2015, 09:03
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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.

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Re: M15-16  [#permalink]

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New post 20 Jul 2015, 05:04
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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Re M15-16  [#permalink]

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New post 19 Aug 2015, 08:07
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?
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Re: M15-16  [#permalink]

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New post 19 Aug 2015, 10:05
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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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Re: M15-16  [#permalink]

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New post 20 Aug 2015, 07:24
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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
Answer is B
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Re M15-16  [#permalink]

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New post 06 Sep 2015, 20:55
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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Re: M15-16  [#permalink]

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New post 31 Oct 2015, 21:27
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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Re: M15-16  [#permalink]

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New post 24 Sep 2016, 03:47
Isn't it should be A+B-2(both) as both part is common to both A and B?
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New post 06 Dec 2016, 02:43
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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Re: M15-16  [#permalink]

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New post 18 Jan 2017, 06:13
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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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Re: M15-16  [#permalink]

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New post 20 Jan 2017, 10:46
I think this is a high-quality question and the explanation is clear enough.
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Re: M15-16  [#permalink]

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New post 02 May 2017, 06:43
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very simple 8-) ! 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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Re: M15-16  [#permalink]

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New post 02 May 2017, 16:14
20% who read A also reads B.....read both(A+B)=0.20A
25% who read B also reads A.....read both(A+B)=0.25B

0.20A=0.25B
B=4/5 A

read only A= read A - read both(A+B)=A-0.2A=0.8A
read only B= read B - read both(A+B)=B-0.25B=0.75B=0.75*4/5A=0.6A [B=4/5 A]

Total= read only A + read only B + read both(A+B)
200 =0.8A+0.6A+0.2A
A= 125

read only A =0.8A=0.8*125=100
read only B=0.6A=0.6*125=75

read only A-read only B=100-75=25

Answer: B
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M15-16  [#permalink]

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New post 07 Apr 2018, 13:35
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



My technique may not be the most accurate but this was logical to me in understanding while I was going through the complex solutions above
20% of A is 25% of B.
Think this way, out of the 200 students there are those who read A and those who read B and those who read both,
Lets name them Group X who read book A and Group Y who read book B. Since there are only 2 books we are talking about.
In both the groups there are students who also like to read the other groups book apart from their own group.
WHEN I SAY A & B I AM REFERRING TO THE BOOKS A & B
Group X
80% only A,20% A+B
Group Y
75% only B, 25% A+B
The number that is highlighted should be same for both groups.
Lets take :

X=100
80% of X is 80 ; 20% is 20 which is also 25% of Y so,
25% of Y is 20 ; 75% is 60 which makes 80
So total is 180 but we want 200

X=120
80% of X is 96 ; 20% is 24 which makes Y=80 because total should be 200
25% of Y is 20 ; This does not match because look at the highlighted part above, it has to be the same number of students

X=125
80% of X is 100 ; 20% is 25 ( now ask 25 is 25% of what?) 100 right
75% of Y is 75 ; 25% is 25 (its a match)

Therefore 100 who read only A -75 who read only B =25
P.S. This is just meant for understanding since it takes a lot of time while doing it in actual.
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