Among 250 viewers interviewed who watch at least one of the three TV

Question

Among 250 viewers interviewed who watch at least one of the three TV channels namely A, B &C. 116 watch A, 127 watch C, while 107 watch B. If 50 watch exactly two channels. How many watch exactly one channel?

a) 185
b) 180
c) 175
d)190
e) 195

source - pearson

(A) 185
(B) 180
(C) 175
(D) 190
(E) 195

source - pearson

ScottTargetTestPrep · Accepted Answer

We can use the equation:

Total = watched A + watched B + watched C - watched exactly two - 2(watched all 3) + watched none

250 = 116 + 107 + 127 - 50 - 2T + 0

250 = 300 - 2T

2T = 50

T = 25

So we have:

250 = watched exactly 1 + watched exactly 2 + watched all 3 + watched none

250 = n + 50 + 25 + 0

175 = n

Answer: C

KarishmaB · Answer

250 = n(Exactly 1 channel) + n(Exactly 2 channels) + n(Exactly 3 channels)
250 = n(Exactly 1 channel) + 50 + n(Exactly 3 channels)

Let's find the value of n(Exactly 3 channels) = x

250 = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C) 
Note that each of n(A and B) is the sum of 'number of people watching exactly two channels A and B' and 'number of people watching all three channels'. 
250 = 116 + 127 + 107 - n(Exactly 2 channels) - 3x + x
250 = 116 + 127 + 107 - 50 - 2x
x = 25

250 = n(Exactly 1 channel) + 50 + 25
n(Exactly 1 channel) = 175

Answer (C)

Johnjojop · Answer

Hi Veritasprep,
Thanks for this great explanation. But I m little confused about this statement: 250 = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C) 
I drew the venn diagram and I wasnt able to come up to this formulae. Please help!
Thanks again!

yezz · Answer

Since none = 0 , therefore at least 1 = total = 250 .

and since A+B+C = 350 thus (A and B)+ (B and C)+(C and A) - (A and B and C) = 100

but (A and B)+ (B and C)+(C and A) = Exactly 2 sets + 3 (A and B and C)

Thus(A and B)+ (B and C)+(C and A) - (A and B and C) = 100 =Exactly 2 sets + 2(A and B and C) and since Exactly 2 sets = 50 thus all three (A and B and C) = 25

Exactly 1 = Total - Exactly 2 - All 3 = 250-50-25 = 175

stanoevskas · Answer

Could someone please clarify the 3x+x part at the end of the equation as a substitute for what should be "neither" - n(A and B and C) in the equation ? Or am I looking at this from the wrong perspective ?

tapabrata · Answer

For a 3 set overlapping venn diagram, 
n(A and B) = n (people who exactly watch A and B but not C) + n(A and B and C)
Similarly
n(B and C) = n (people who exactly watch B and C but not A) + n(A and B and C)
n(C and A) = n (people who exactly watch C and A but not B) + n(A and B and C)

n(A and B)+n(B and C)+n(C and A) = n(people who watch exactly 2 channels) + 3x .. equation 1

250 = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(C and A) + n(A and B and C)

Substituting equation 1 in the above equation we get

250 = 116 + 127 + 107 - n(people who watch Exactly 2 channels) - 3x + x

Hope it is clear

guialain · Answer

Tough most GMAT takers must be familiar to the concept of 3 overlapping sets , I think the difficulty of this question's diffculty is average.
I would say Level 600-650 question.

manishtank1988 · Answer

My 2 cents:
I solved this question as follows:

Personally, i would to say that the only formulas i remember are
T = A + B + C - [exactly 2 sets] - 2*[all 3 sets] + None &
T = A + B + C - [among 2 sets] + [all 3 sets] + None.
If you can't solve with these 2 formulas than the best option will be dividing 3 overlapping sets into various sections as a, b, c, d, e, f and g as you might already have seen in various examples solved here.
(If not refer - https://gmatclub.com/forum/three-table- ... 05637.html)
Honestly, after 2 tries i have understood one thing that GMAT does NOT give time to anaylse question and see which formula to use. So remembering lot of formulas isn't a way to prepare for GMAT. Also, question where you can apply it directly will hardly be ever asked.

Attachments

a+b+c.jpg [ 600.44 KiB | Viewed 13631 times ]

AkshdeepS · Answer

Easiest and the best way from all above. Kudos as I was about to giving up on this question.

Thanks

QZ

ShubhiChauhan · Answer

250 = 116 + 127 + 107 - n(Exactly 2 channels) - 3x + x 
How is this coming? -3X+x which formula is this?

even if we put value in the above equation:
we should get only - 
250 = 116 + 127 + 107 - 3x + x ( why n(Exactly 2 channels) is also subtracted?

I am getting a little confused here

MHIKER · Answer

Help me If I do wrong!

TOTAL=A+B+C- EXACTLY 2- 2 
250 = 116+127+107-50-2X

X= 25

NOW, TOTAL = EXACTLY ONE + EXACTLY 2 + ALL THREE
 250 = EXACTLY ONE +50+25
 EXACNTLY ONE = 175

ANS. C

bumpbot · Answer

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Among 250 viewers interviewed who watch at least one of the three TV

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