Of the 45 housholds in a certain neighborhood, 28 subscribe to Newspap

Question

Of the 45 households in a certain neighborhood, 28 subscribe to Newspaper Q, 17 subscribe to Newspaper R, 12 subscribe to Newspaper S, 7 subscribe to both Q and R, 8 subscribe to both Q and S, and 9 subscribe to both R and S. The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers. If 39 of the households subscribe to at least one of the three newspapers, how many households subscribe to only one of the newspapers?

A) 15
B) 21
C) 27
D) 33
E) 46

A) 15
B) 21
C) 27
D) 33
E) 46

MartyMurray · Accepted Answer

Of the 45 households in a certain neighborhood, 28 subscribe to Newspaper Q, 17 subscribe to Newspaper R, 12 subscribe to Newspaper S, 7 subscribe to both Q and R, 8 subscribe to both Q and S, and 9 subscribe to both R and S. The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers. If 39 of the households subscribe to at least one of the three newspapers, how many households subscribe to only one of the newspapers?

A) 15
B) 21
C) 27
D) 33
E) 46

We don't need to draw a Venn diagram to answer this question. If we understand the math, we can answer the question by simply doing the math on the sheet or in our heads.

However, for the purpose of illustrating what's going on, I'll use a Venn diagram.

We can start with three circles to represent the sets of households that subscribe to the three newspapers.

https://i.postimg.cc/L4fTz1fs/Triple-sets-1.png

Then, we work our way from the center outward, starting with the households that subscribe to all three newspapers.

The passage says, "The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers." So, we know that the number that subscribe to all three is the following:

Households That Subscribe to None = All Households - Households That Subscribe to at Least One = 45 - 39 = 6.

https://i.postimg.cc/kGxv909S/Triple-sets-2.png

Then, to find the number that subscribe to two but not all three, we subtract 6 from each number of households that subscribe to two.

Q and R only = 7 - 6 = 1

Q and S only = 8 - 6 = 2

R and S only = 9 - 6 = 3

https://i.postimg.cc/nr8GyF8Z/triple-sets-3.png

Finally, to find the number that subscribe to one newspaper only, we subtract the numbers already in each circle from the total number that subscribe to each newspaper.

Q only = 28 - (2 + 6 + 1) = 19

R only = 17 - (1 + 6 + 3) = 7

S only = 12 - (2 + 6 + 3) = 1

https://i.postimg.cc/B6PC78VM/triple-sets-4.png

Then, the total number of households that subscribe to only one newspaper is Q only + R only + S only = 19 + 7 + 1 = 27

A) 15
B) 21
C) 27
D) 33
E) 46

Correct answer:  C

KarishmaB · Answer

First notice that we are given that out of 45, 39 households  subscribe to at least one of the three newspapers. This means 6 households subscribe to none. This also means 6 susbcribe to all three (since "The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers")

Those that subscribe to 2 or more newspapers = 7 + 8 + 9 - 2*6 = 12 
Why do we subtract 6 twice? Because it is counted each time in 7, 8 and 9. We need to count it only once.

Since 39 households  subscribe to at least one of the three newspapers, if we subtract from this the number that subscribe to at least 2 or more, we will get the number that subscribe to only 1 newspaper. 
Those that subscribe to only 1 newspaper = 39 - 12 = 27

Answer (C)

coucoudanger · Answer

7 subscribe to both Q and R : x+y=7
8 subscribe to both Q and S : y+z=8
9 subscribe to both R and S : w+y=9

If 39 of the households subscribe to at least one of the three newspapers then 6 households subscribe to none of them
The number of households who subscribe to all three newspapers is equal to the number of households who subscribe to none of the three newspapers
y=6
x=1 ; z=2 ; w= 3
households who subscribe to only Q : 28-x-y-z =19
households who subscribe to only R : 17-w-y-x =7
households who subscribe to only S : 12-z-y-w =1

Number of households that subscribe to only one of the newspapers is 19+7+1=27

rakman123 · Answer

JeffTargetTestPrep  could you please advise - thank you

JoeAa · Answer

Got a question here: how could 12 subscribe to Newspaper S, and 8 subscribe to both Q and S, and 9 subscribe to both R and S. Shouldn't 8+9

sanebeyondone · Answer

I think wording of this problem is a bit unclear

For example,
7 subscribe to both Q and R

Does this mean 
7 subscribe to ONLY both Q and R (exclusive)
OR
7 subscribe to NOT only both Q and R and to something else. (nonexclusive)

Though somehow I know it right the way for example,
28 subscribe to Newspaper Q
That means nonexclusive

I guess I need to remove from my memory that the word 'both' is nonexclusive then.....

Posted from my mobile device

Bunuel · Answer

The statement '7 subscribe to both Q and R' means exactly that – 7 households subscribe to both, without implying exclusivity. For instance, when the stem states '28 subscribe to Newspaper Q', it's understood that these 28 households are subscribers of Q, yet some may also be subscribed to other newspapers. In a similar vein, '7 subscribe to both Q and R' suggests that these 7 households subscribe to Q and R, but it does not exclude the possibility of them also being subscribed to S.

Furthermore, as this is an official question, the wording is as good as it gets.

NEYR0N · Answer

KarishmaB  ,

"Those that subscribe to 2 or more newspapers = 7 + 8 + 9 - 2*6 = 12 
Why do we subtract 6 twice? Because it is counted each time in 7, 8 and 9. We need to count it only once. " -- are we treating three sets as if they were two sets ?

thanks,

NEYR0N · Answer

Attached my solution using the 2 formulas from gmat club. If an expert can evaluate this approach that could be great.

KarishmaB · Answer

I am not sure what you mean by that. How are we treating 3 sets as 2? Look at the different regions of overlapping sets here: https://anaprep.com/sets-statistics-thr ... ping-sets/ d+k = 7 f + k = 8 e + k = 9 So k is counted 3 times in 7+8+9 but we should count it only once. That is why we subtract it out twice.

Mohak01 · Answer

KarishmaB

Can you help me understand where I went wrong with my approach.

Assume household with 3 newspaper = no newspaper household = x, 
Then
Total households - households with no newspaper = ( Q+R+S) - (QR+RS+SQ) +2(QRS)

45-x=(28+27+12) -(7+8+9) +2x
45-x=33+2x
3x=12, hence x=4

Here is where I got stuck as to why the additional info. in last sentence is given.

Thanks in advance.

KarishmaB · Answer

The error is the formula you have used.

Total - None = A + B + C - (AB + BC + CA) + ABC
Use it and you will get the right answer. We add the triple overlap region only once back because it gets added thrice (in A, B and C) and gets subtracted thrice (in AB, BC and CA). But we must count it once so we add it back.

I sincerely advise all my learners to ignore the various formulae and focus on a couple of basic ones and Venn diagram. It is easy to mess up if you are not connecting your formulae to the Venn diagram.

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Of the 45 housholds in a certain neighborhood, 28 subscribe to Newspap

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