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In a consumer survey, 85 percent of those surveyed liked at [#permalink]
 24 Nov 2005, 18:24
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In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?
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Let the three products be A, B, and C.
Let the number of people surveyed be 100.
85 people liked at least one of the three products.
A+B+C-(A&B)-(B&C)-(C&A)-3(A&B&C)
= 50 + 30 + 20 - (A&B + B&C + C&A) - 15
= 85 - (A&B + B&C + C&A)
= 85
Therefore, (A&B + B&C + C&A) = 0
15/100 = 15%
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I agree with 15%. Good variation of this type of question.
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gamjatang wrote: Let the three products be A, B, and C.
Let the number of people surveyed be 100.
85 people liked at least one of the three products.
A+B+C-(A&B)-(B&C)-(C&A)-3(A&B&C)
= 50 + 30 + 20 - (A&B + B&C + C&A) - 15
= 85 - (A&B + B&C + C&A)
= 85
Therefore, (A&B + B&C + C&A) = 0
15/100 = 15%
I doubt if bold portion of formula correct in this case.
A&B includes A&B&C
so it will ne deducted thrice already :
- (A&B + B&C + C&A)
am i missing something?
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A+B+C-[(A&B)+(B&C)+(C&A)]+(A&B&C) = 85
100-[(A&B)+(B&C)+(C&A)]+5=85
(A&B)+(B&C)+(C&A) = 20 or 20%
Is this correct?
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gsr wrote: A+B+C-[(A&B)+(B&C)+(C&A)]+(A&B&C) = 85
100-[(A&B)+(B&C)+(C&A)]+5=85
(A&B)+(B&C)+(C&A) = 20 or 20%
Is this correct?
I solved the same way, but OA is not 20% 
Looks like:
(A&B)+(B&C)+(C&A) above includes A&B&C thrice...?
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duttsit wrote: Looks like:
(A&B)+(B&C)+(C&A) above includes A&B&C thrice...?
True 
(A&B)+(B&C)+(C&A) - 2*(A&B&C) = 20 - 10 = 10% ?
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duttsit wrote: In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?
A=0.5
B=0.3
C=0.2
AUBUC=.85
AandBandC=.05
AUBUC=A+B+C-(people who liked at least two) - AandBandC
.85=1-X-.05
x=1-.85-.05=.1
10%
(I tried to summerize all related formulas into one post, please check to see if they are right.)
http://www.gmatclub.com/phpbb/viewtopic ... 938#139938
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Great explanation qsr, HongHu. thanks.
OA is 10%
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HongHu wrote: duttsit wrote: In a consumer survey, 85 percent of those surveyed liked at least one of three products: 1, 2, and 3. 50 percent of those asked liked product 1, 30 percent liked product 2, and 20 percent liked product 3. If 5 percent of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products? A=0.5 B=0.3 C=0.2 AUBUC=.85 AandBandC=.05 AUBUC=A+B+C-(people who liked at least two) - AandBandC .85=1-X-.05 x=1-.85-.05=.1 10% (I tried to summerize all related formulas into one post, please check to see if they are right.) http://www.gmatclub.com/phpbb/viewtopic ... 938#139938Please correct me if I am wrong, but as per what I have understood out of sets,
n(A U B U C) = n(A) + n(B) + n(C) - n(A U' B) - n(B U' C) - n(A U' C) + n(A U' B U' C)
ie. it should be
AUBUC=A+B+C-(people who liked at least two) "+" AandBandC
So it has to be 20% isn't it.
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You get different results when you count people together or when you count them seperately. For example, say there are three items A, B, and C. If 3 people like A and B, 3 people like B and C, and 3 people like A and C. The number of people who like at least two items are not necessarily 9. Perhaps there are only 7 people who like at least two items, because 1 person likes all the three of them. (2 A and B only, 2 B and C only, 2 A and C only, and 1 like all of them.) This is why the two formulas are different. When you count them seperately and add them up, you have twice overcounted the center when they intersept each other.
In other words, the original formula is this:
N(AUBUC) = N(A) + N(B) + N(C) - N(A n B) - N(A n C) - N(C n B) + N(A n B n C)
When you subsititute N(A n B) + N(A n C) + N(C n B) = N(at least two) + 2N(A n B n C)
You will get
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - 2N(A n B n C) + N(A n B n C)
Which means
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - N(A n B n C)
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HongHu wrote: You get different results when you count people together or when you count them seperately. For example, say there are three items A, B, and C. If 3 people like A and B, 3 people like B and C, and 3 people like A and C. The number of people who like at least two items are not necessarily 9. Perhaps there are only 7 people who like at least two items, because 1 person likes all the three of them. (2 A and B only, 2 B and C only, 2 A and C only, and 1 like all of them.) This is why the two formulas are different. When you count them seperately and add them up, you have twice overcounted the center when they intersept each other.
In other words, the original formula is this:
N(AUBUC) = N(A) + N(B) + N(C) - N(A n B) - N(A n C) - N(C n B) + N(A n B n C)
When you subsititute N(A n B) + N(A n C) + N(C n B) = N(at least two) + 2N(A n B n C)
You will get
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - 2N(A n B n C) + N(A n B n C)
Which means
N(AUBUC) = N(A) + N(B) + N(C) - N(at least two) - N(A n B n C)
Wow.. I finally get this 
Thanks HongHU
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Shouldn't do you be adding P1andP2andP3 to 10% giving an answer of 15%?
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rianah100 wrote: Shouldn't do you be adding P1andP2andP3 to 10% giving an answer of 15%?
that 10% is resulted with the inclusion of that percentage figure for (P1 and P2 and P3).
the required % = % liked 2 products + % liked 3 or all products
= % of people liked products (1 & 2 +2 & 3+ 1 & 3) + % of people liked all 3 (P1 and P2 and P3)
= 5%+5%
= 10%
note: % of people liked products (1 & 2 +2 & 3+ 1 & 3) is calculated as under:
total % of people surveyed = % of people liked product 1 + % of people liked product 2 + % of people liked product 3 - % of people liked products (1 & 2) - % of people liked products (2 & 3) - % of people liked products (1 & 3) - 2 [% of people liked products (1, 2, & 3)]
85%= 50+30+20 - p(1 & 2) - p(1 & 2) - p(2 & 3) - p(1 & 3) - 2 (5%)
p(1 & 2) + p(1 & 2) + p(2 & 3) + p(1 & 3) = 100-85-10 = 5%
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AUBUC - N = A + B + C - X - 2Y
N = Liked none/neither
X = Liked excatly two
Y = Liked all three.
100 - 15 = 50 + 30 + 20 - X -2*5
85 = 100 - 10 -X
X = 5
So, 5% of the people like exactly two and 5% like exactly/all 3.
So 10% like more than one.
B.
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