Of the 100 best-selling children's books in 2018, 60 were illustrated,

Question

Of the 100 best-selling children's books in 2018, 60 were illustrated, 39 had fewer than 100 pages, 43 featured talking animals as book characters, and 9 had none of these qualities. If 33 of the books had at least two of these qualities, how many of the books had all three of the qualities?

(A)9
(B)12
(C)15
(D)18
(E)24

(A)9
(B)12
(C)15
(D)18
(E)24

Solution 
 
• Let us draw a Venn-Diagram • In the above Venn – diagram,  
o Number of books which are illustrated  = I + x + p + y = 60 .....Eq.(i) o Number of books which have fewer than 100 pages  = F+ y + p + z = 39 .....Eq.(ii) o Number of books which have taking animals  = T +x +p + z = 43 .......Eq.(iii)  o Number of books with at least two qualities  = x + y + z + p = 33 ........Eq.(iv) o Number of books with all three qualities  = p  
  
We need to find the value of p.
 
• Now,  I +F+T+x+y+z+p + 9 = 100   
  ⟹I + F + T + 33 + 9 = 100     ⟹I+F+T =58 .............Eq.(v)  
• Adding Eq.(i), Eq.(ii), and Eq.(iii), we get, 
  
o  I+F+T+2x + 2y +2z + 3p = 60+39 +43 
  ⟹58 + 2x + 2y+ 2z + 3p = 142   
  ⟹ 2(x+y+z+p) +p =84 
  ⟹2*33 + p = 84   
  ⟹ p = 84-66 =18

Thus, the correct answer is  Option D.

ScottTargetTestPrep · Accepted Answer

Solution:

We can use the formula:

Total = Quality A + Quality B + Quality C - Exactly Two Qualities - 2 * All Three Qualities + None

If we let All Three Qualities = x, then Exactly Two Qualities = 33 - x (notice that the problem says that at least two qualities is 33).

100 = 60 + 39 + 43 - (33 - x) - 2x + 9

100 = 142 - 33 + x - 2x + 9

100 = 118 - x

x = 18

Answer: D

NandishSS · Answer

GMATWhizTeam  Venn-Diagram is missing!!!

GMATWhizTeam · Answer

Hey NandishSS We have uploaded the image along with the solution. Can you try and refresh the page once? You should be able to see it. It may be taking some time to upload on the page. Regards,

GMATWhiz Team

Sri07 · Answer

My answer for this one is 9

100 = A + B + C - (a+x+b+x+c+x) +x + None
100 = 60 +39+43 -(a+b+c+3x)+x+none
= 142-a-b-c-3x+x+9
=142+9-a-b-c-2x
a+b+c=33
2x= 142-33+9
2x=18
x=9

Can you help please

themightygaby · Answer

Your assessment of "Exactly Two Qualities" is incorrect. The problem tells us 33 books have  at least  two of these qualities. That means a portion of those 33 books have all 3 qualities. Let's call this portion x. It follows that 33-x is the expression for "Exactly Two Qualities."

The reason your answer is 1/2 of the correct answer is because you duplicated the number of Quality A, Quality B, and Quality C in your expression for "Exactly. Two Qualities"

MathRevolution · Answer

Total - neither = Type1 + Type 2 + Type 3 - Exactly Two - (2 * All three)

Exactly Two = 33 - All Three

=> 100 - 9 = 60 + 39 + 43 - 33 + All three - 2 All three

=> All Three = 109 - 91 = 18

Answer D

ScottTargetTestPrep · Answer

From what I understand, you are letting using a, b and c denote the elements that belong to exactly two of the sets and letting x denote the elements that belong to all three sets. Then later in your solution, you are substituting a + b + c = 33; however, notice that in the question we are told that "If 33 of the books had at least two of these qualities...", which means that the number of elements that belong to exactly two sets + the elements that belong to all three sets is 33. In your notation, that would mean a + b + c + x = 33. If you resolve the question by substituting a + b + c = 33 - x, you should get the correct answer.

GianKR · Answer

Dear  KarishmaB  can you please help me understand where I am going wrong in the following?

Applying the 3 overlapping sets formula, I got the following:

# of unique elements= # of illustrated books + # of books with < than 100 pages + # of books featuring talking animals - # of books having at least two qualities + # of books having all three qualities + # of books having none of the qualities.

100= 60+39+43 - 33 + # of books having all three qualities + 9
Hence, 
# of books having all three qualities= 100-9- (60+39+43) +33 = -18

Where am I going wrong that I am getting -18 instead of 18? Is my understanding of the formula wrong?

Thank you for your usual support!

KarishmaB · Answer

Look here at the diagram first: https://anaprep.com/sets-statistics-thr ... ping-sets/ The '33 of the books had at least two of these qualities' means 33 books = (d + e + f + k) region We cannot use this formula: Total = A + B + C - (A and B + B and C + C and A) + (A and B and C) + Neither Why? Because '(A and B + B and C + C and A)' is not given to be 33. A and B = d + k region B and C = e + k region C and A = f + k region When you are given the overlap of A and B, B and C and C and A, then we can use this formula. But in 33, k is not counted thrice. It simply gives us (d+e+f+k) region. Look at the information you have as per the diagram: 60 were illustrated = a + d + f + k region 39 had fewer than 100 pages = b + d + e + k region 43 featured talking animals as book characters = c + e + f + k region So when I say 60+39+43, it is (a + d + f + k) + (b + d + e + k) + (c + e + f + k). When I remove 33 from this, I remove (d+e+f+k) and am left with a + b + c + d + e + f + 2k So 60 + 39 + 43 - 33 = (a + b + c + d + e + f + k) + k = (100 - 9) + k k = 18

HarshavardhanR · Answer

- Exactly one quality - I
- Exactly two qualities- II
- Exactly three qualities - III
- None of the qualities - N

I + II + III + N = total = 100

- II + III = books with at least two qualities = 33
- N = 9

So,

I + 33 + 9 = 100. So,

I = 58.

Visualize the venn diagram here.  When all the 3 circles are added (60+39+43), in effect, the I regions get counted once, the II regions get counted twice and the III region gets counted thrice.
  
So,

I + 2II + 3III = 60 + 39 + 43 = 142

i.e.,

I + 2(II + III) + III = 142

i.e.,

58 + 2 (33) + III = 142

This gives us:

III = 142 - 124 =    18 . Choice D .

---
Harsha

Of the 100 best-selling children's books in 2018, 60 were illustrated,

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