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# In a consumer survey, 85 percent of those surveyed liked at

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In a consumer survey, 85 percent of those surveyed liked at [#permalink]

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24 Nov 2005, 17:24
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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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24 Nov 2005, 18:19
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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24 Nov 2005, 19:00
I agree with 15%. Good variation of this type of question.
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24 Nov 2005, 19:02
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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24 Nov 2005, 19:10
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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24 Nov 2005, 19:21
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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24 Nov 2005, 19:31
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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24 Nov 2005, 19:53
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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24 Nov 2005, 20:25
Great explanation qsr, HongHu. thanks.
OA is 10%
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25 Nov 2005, 03:28
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#139938

Please 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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25 Nov 2005, 08:39
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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26 Nov 2005, 03:10
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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04 Dec 2005, 04:20
Shouldn't do you be adding P1andP2andP3 to 10% giving an answer of 15%?
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04 Dec 2005, 08:02
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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20 May 2008, 12:53
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.
Re: PS: Set   [#permalink] 20 May 2008, 12:53
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