Of 200 surveyed students, 20% of those who read book A

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

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

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

TirthankarP · Answer

It depends on what information you have with you. If you know the values of TOTAL, ONLY A and ONLY B and you need to calculate the value of BOTH, then you can use the formula: Total= only A+ only B + both But if you don't know the values of ONLY A and ONLY B (like in this case where we are able to deduce the values of A and B but not of ONLY A and ONLY B), the we should use the following formula: Total= A+ B - BOTH. So use of formula varies depending on the information available. I hope I have addressed your query :-D

TirthankarP · Answer

Total students = A + B - BOTH

where, A = students who read book A
 B = students who read book B

BOTH = 0.2A (20% of those who read book A also read book B)
BOTH = 0.25B (25% of those who read book B also read book A)

So, 0.2A = 0.25B
  A = 1.25B

Replacing A with 1.25B and BOTH with 0.25B in the original equation, we get
Total = A + B - BOTH
200 = 1.25B + B - 0.25B
200 = 2B
  B=100

So,   BOTH = 0.25B = 0.25 * 100 = 25

Again, Total = A + B -BOTH
 200 = A + 100 - 25
   A = 125

We have to find   ONLY A - ONLY B  
 ONLY A = A - BOTH = 125 - 25 = 100
ONLY B = B - BOTH = 100 - 25 = 75

So  ONLY A - ONLY B = 100 - 75 = 25   (Correct Answer : B)

LinaNY · Answer

Bunuel,

Could you please give an example of a problem where we would use Total= only A+ only B + both. Why aren't we using this formula in this problem?
thanks

Bunuel · Answer

PS: a-polling-company-found-that-of-300-households-surveyed-102377.html DS: a-data-company-recently-conducted-a-survey-to-determine-wh-154001.html Theory on Overlapping Sets: advanced-overlapping-sets-problems-144260.html how-to-draw-a-venn-diagram-for-problems-98036.html All DS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=45 All PS Overlapping Sets Problems to practice: search.php?search_id=tag&tag_id=65 Hope this helps.

ASHISHBAHL · Answer

Y+.8X=200
.2X=.25Y

X=125 Y=100

We have to find 
 
(x-.2x)-(y-.25y) 
 
100-75==>25 Ans

KarishmaB · Answer

Note that "20% of those who read book A also read book B" is the same group as "25% of those who read book B also read book A".

20% of A = 25% of B
A/B = 5/4
On the ratio scale, we have 5 students in A and we have 4 in B with an overlap of 1 (counted twice). So 8 corresponds to 200 students which gives us a multiplier of 25.
On the ratio scale, 4 students of A read only A and 3 students of B read only B so difference between them is 1. 
Since multiplier is 25, 1 corresponds to 25 and the actual value of the difference is 25.

Answer (B)

Rishm · Answer

I solved it like this - 20x/100*200=25y/100*200 (as they would be the same considering overlapping sets). The ratio comes down to 4x:5y and if u notice that the multiple of them would be 40x:50y where difference is 10, 80x:100y where difference would be 20 but we increase it to 120x:150y it crosses 200 students hence difference between 20-30. Hence B is the only answer.

Of 200 surveyed students, 20% of those who read book A

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