M15-16

Question

Of the 200 surveyed students, 20% of the students who read book A also read book B, and 25% of the students who read book B also read book A. If each student read at least one of the two books, what is the positive 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

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

Bunuel · Accepted Answer

Official Solution:

Of the 200 surveyed students, 20% of the students who read book A also read book B, and 25% of the students who read book B also read book A. If each student read at least one of the two books, what is the positive 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

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

Let the number of students who read book A be A and the number of students who read book B be B. 
 
We know that 20% of those who read book A also read book B, hence the number of students who read both books is 0.2A. We also know that 25% of those who read book B also read book A, hence the number of students who read both books is 0.25B. Therefore, 0.2A = 0.25B, which gives A = 1.25B.
 
Since we're given that each student read at least one of the books, then {None} = 0 and we have {Total} = {A} + {B} - {Both}. Substituting A = 1.25B and Total = 200, we get 200 = 1.25B + B - 0.25B, which gives B = 100, therefore, A = 1.25B = 125, and Both = 0.25B = 25.
 
Thus, the number of students who read only book A is A - Both = 125 - 25 = 100, and the number of students who read only book B is B - Both = 100 - 25 = 75. The positive difference between these two numbers is 100 - 75 = 25.

Answer: B

SharmaGyan · Answer

If someone is done reading those long explanations.

0.2A = 0.25B = {Both - Say X}

0.2A=X , A = 5X
0.25B=X, B = 4X

Total = A+B - Both = (4x+5x) - X = 8X 
8X=200
X=25

(At this stage, draw a Venn Diagram and you would be 30 seconds away from the answer.)

Difference = (5X-X) - (4X-X) = 4X-3X = X =25

Mo2men · Answer

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

AkshdeepS · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Purnank · Answer

Simplest way i could think of assuming 100x as students who read book A,
20% of that 100x who also reads book B. There for n(A∩B) = 20x and these are those sudents 25% who read book B also read book A. Therefore the total of n(B) = 4*20x = 80x.
n(AUB) = n(A) + n(B) - n(A∩B) = 100x + 80x - 20x = 160x
However, each student read at least one of the two books.
Therefore n(AUB) = 160x = 200 
x = 5/4. 
A only = n(A) - n(A∩B) = 100x-20x = 80x
B only = n(B) - n(A∩B) = 80x-20x = 60x
Therefore, Difference between the number of students who read only book A and the number of students who read only book B.
80x - 60x = 20x = 20*5 / 4 = 25. B.

Scholar94 · Answer

I think this is a  high-quality  question and I agree with explanation.

Mayzone · Answer

How did 0.2A=0.25B, become A=1.25B ?

Bunuel · Answer

0.2A = 0.25B

A = 0.25B/0.2

A = 1.25B (because 0.25/0.2 = 1.25)

Hope it's clear.

Mayzone · Answer

Yes, it is thanks!

Msk666 · Answer

Since we're given that each student read at least one of the books, then {None} = 0 and we have {Total} = {A} + {B} - {Both}.

I get the none part but why is the total equation constructed the way it is? why are we subtracting both

Bunuel · Answer

That's because when you sum A and B, the group of students who read both books is included in A and also in B, effectively counting them twice. To avoid double-counting, we subtract that group once from the sum A + B, ensuring they are counted only once.

This is the fundamental formula for two overlapping sets, and it's crucial to understand it before attempting related questions.

Overlapping Sets

Theory: 
 
  Overlapping Sets Made Easy!  
  How to draw a Venn Diagram for problems  
  ADVANCED OVERLAPPING SETS PROBLEMS  
  Formulae for 3 overlapping sets

Questions: 
 
  The E-GMAT Sets Triad: 3 Exciting Sets Questions!  
  The Word “Or” in GMAT Math  
  DS Questions  
  PS Questions

For more: 
 
  ALL YOU NEED FOR QUANT ! ! !  
  Ultimate GMAT Quantitative Megathread

Hope it helps.

abishekcmouli · Answer

I like the solution - it’s helpful.

AJB1991 · Answer

I like the solution - it’s helpful.

bumpbot · Answer

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