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A group of 49 consumers were offered a chance to subscribe to three ma

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New post 05 Nov 2017, 01:04
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A group of 49 consumers were offered a chance to subscribe to three magazines: A, B, and C. Thirty-eight of the consumers subscribed to at least one of the magazines. How many of the 49 consumers subscribed to exactly two of the magazines?

(1) Twelve of the 49 consumers subscribed to all three of the magazines.
(2) Twenty of the 49 consumers subscribed to magazine A.
[Reveal] Spoiler: OA

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Re: A group of 49 consumers were offered a chance to subscribe to three ma [#permalink]

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New post 05 Nov 2017, 04:43
Bunuel wrote:
A group of 49 consumers were offered a chance to subscribe to three magazines: A, B, and C. Thirty-eight of the consumers subscribed to at least one of the magazines. How many of the 49 consumers subscribed to exactly two of the magazines?

(1) Twelve of the 49 consumers subscribed to all three of the magazines.
(2) Twenty of the 49 consumers subscribed to magazine A.


The answer should be E as follows
Total=49
Let
subscribers to A=x
subscribers to B=y
subscribers to C=z
subscribers to exactly 2 magazines = m
subscribers to all three magazines = n
subscriber to no magazines = p = 49-38 = 11 [Given in question stem]

Formula is Total = subscribers to A + subscribers to B + subscribers to C - (subscribers to exactly 2 magazines) - 2(subscribers to all three magazines) + subscriber to no magazines
Total = x + y +z - m - 2*n +p
49 = x + y +z - m - 2*n + 11 ----------------------------------------------------------- Equation 1
We need to find out the value of m

(1) Twelve of the 49 consumers subscribed to all three of the magazines.
n=12
Putting the this value in the Equation 1, we get

49 = x + y +z - m - 2*12 + 11
49 = x + y +z - m - 34 + 11 ------------------------------------------------------------- Equation 2
Since we do not know the value of x,y,z we can not find the value of m
INSUFFICIENT

(2) Twenty of the 49 consumers subscribed to magazine A.
x=20
Putting this value in equation 1, we get
49 = x + y +z - m - 2*n + 11
49 = 20 + y +z - m - 2*n + 11
Since we do not know the value of y,z,n we can not find the value of m
INSUFFICIENT

Combining both, we get
49 = 20 + y +z - m - 24 + 11
Since we still do not know the value of y, z or (y+z), we can not find the value of m
INSIFFICIENT

Hence, E is the answer.
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Re: A group of 49 consumers were offered a chance to subscribe to three ma [#permalink]

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New post 06 Nov 2017, 17:15
Jabjagotabhisavera why are we not using the other formula, shouldn't we use this formula Total= A+B+C -(sum of 2 group overlaps)+ (All three) + Neither

Though the answer will still be 'E'

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A group of 49 consumers were offered a chance to subscribe to three ma [#permalink]

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New post 06 Nov 2017, 23:57
Amirfunc wrote:
Jabjagotabhisavera why are we not using the other formula, shouldn't we use this formula Total= A+B+C -(sum of 2 group overlaps)+ (All three) + Neither

Though the answer will still be 'E'


Hi Amirfunc,

We can use any formula we wish, but we need to understand the the meaning of the the term we are using and also need to understand how the formula is derived basically.

Attachment:
Statistics.jpg
Statistics.jpg [ 30.03 KiB | Viewed 214 times ]


Refer to the figure above, we can say that.

Total = a+b+c+d+e+f+g+Neither (N) ------------------------------------------------------Equation 1
The question is asking us to find out the value of exactly 2 group overlapping, so basically the value of d+e+f.
So, sum of exactly 2 groups overlapping is d+e+f ------------------------------------------Equation 2
Notice that A=a+d+g+f => a = A-d-g-f -----------------------------------------------------Equation 3
Similarly B=b+d+g+e => b = B-d-g-e -----------------------------------------------------Equation 4
and C=c+e+g+f => c = C-e-g-f --------------------------------------------------------------Equation 5
Putting the value of equation 2,3 and 4 in equation 1, we get
Total = (A-d-g-f) + (B-d-g-e) + (C-e-g-f) +d +e + f + g + N = A+B+C -2*(d+e+f)+(d+e+f) - 2*g +N = A+B+C -(d+e+f) - 2*g +N
Using the definition from equation 2 in the above formula, we can say
Total = A+B+C -(d+e+f) - 2*g +N -----------------------------------------------------------Equation 6
Total = A+B+C -(Sum of exactly 2 groups overlapping) - 2*intersection of all three +N ------------Equation 7

Now coming back to the formula you refer to, first we need to understand what is the meaning of different terms used here.

Total= A+B+C -(sum of 2 group overlaps)+ (All three) + Neither ---------------------Equation 7
sum of 2 group overlaps = A∩B + B∩C + C∩A = d+g + e+g + f+g
Using the formula from Equation 6
Total = A+B+C -(d+e+f) - 2*g +N
Add and subtract 3*g in the above equation, we get
Total = A+B+C -(d+e+f) - 2*g +N +3*g -3*g = A+B+C -(d+e+f + 3*g) - 2*g +3*g +N = A+B+C -([d+g]+ [e+g]+ [f+g]) - g +N = A+B+C -(A∩B + B∩C + C∩A) - A∩B∩C +N

Total = A+B+C -(A∩B + B∩C + C∩A) + A∩B∩C +N
Total = A+B+C -(sum of 2 group overlaps) + intersection of all three +N

So, summery is
1. Formula I used and the formula you refer to can be derived using the basic concept as long as we are a using the correct meaning of the different terms used here.
i. sum of 2 group overlaps = A∩B + B∩C + C∩A = d+g + e+g + f+g
ii. Sum of exactly 2 groups overlapping = d+e+f

Hope, I was able to explain how both the formula are derived from the basic concept and why I used a particular formula [As question is asking about the value of "Sum of exactly 2 groups overlapping"] to solve this question.
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A group of 49 consumers were offered a chance to subscribe to three ma   [#permalink] 06 Nov 2017, 23:57
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