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Math Expert V
Joined: 02 Sep 2009
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16 00:00

Difficulty:   75% (hard)

Question Stats: 68% (02:48) correct 33% (02:51) wrong based on 80 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

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Math Expert V
Joined: 02 Sep 2009
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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 $$100-75=25$$.

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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 V
Joined: 02 Sep 2009
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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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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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Intern  B
Joined: 20 Aug 2016
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Isn't it should be A+B-2(both) as both part is common to both A and B?
Math Expert V
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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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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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2
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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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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0.20A=0.25B
B=4/5 A

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

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Joined: 09 Dec 2013
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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

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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Joined: 31 May 2017
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Hi Bunuel, Can you show how this can be solved using the 2*2 matrix?
Math Expert V
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Ujaswin wrote:
Hi Bunuel, Can you show how this can be solved using the 2*2 matrix?

Check here: https://gmatclub.com/forum/m15-184048.html#p1563789
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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. Not able to understand the equations.
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I think this is a high-quality question and I agree with explanation.
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