Consumer Survey

IMO B



Let total people surveyed = 100
85 people like at least one of the product. 5 of all liked all three.
So, all (blue) + all (red) = 80 (please refer the diagram)

Also,

p1+p1p2+p1p3+5=50 -->1

p2+p2p3+p2p1+5=30 -->2

p3+p3p1+p3p2+5=20 -->3

Adding 1,2,3 equations, we get

(p1+p2+p3)+ 2*(p1p2+p2p3+p3p1) + 15 = 100
(p1+p2+p3)+ 2*(p1p2+p2p3+p3p1)=85
[(p1+p2+p3)+(p1p2+p2p3+p3p1)] + (p1p2+p2p3+p3p1) = 85

Adding 5 on both sides
[(p1+p2+p3)+(p1p2+p2p3+p3p1)]+5+(p1p2+p2p3+p3p1)=90 ---> 4

we know that [(p1+p2+p3)+(p1p2+p2p3+p3p1)]+5 = all(blue) +all (red)+ all green =85
So,
equation 4 can be written as

85+(p1p2+p2p3+p3p1) =90
(p1p2+p2p3+p3p1)=5 (all red)

Now asked:
The survey participants liked more than one of the three products: either two products or three products

Also means all(red)+ all(green) = 5+5 = 10

So B

pankajks2010 wrote:In a consumer survey, 85 percent of those surveyed liked at least one of three products. 50% of those asked liked product 1, 30% like product 2, and 20% liked product 3. If 5% of the people in the survey liked all the three of the products, what percentage of the survey participants liked more than one of the three products:

a) 5%
b) 10%
c) 15%
d) 20%
e) 25%

[spoiler]OA: B[/spoiler]

courtsey: 800score.com

Here is the formula for 3 overlapping groups in which sometimes 2 of the groups overlap and sometimes all 3 groups overlap:

T = G1 + G2 + G3 - (those in 2 of the groups) - 2*(those in all 3 groups)

The big idea with overlapping group problems is to subtract the overlap.
When we add together everyone who likes product 1, everyone who likes product 2, and everyone who likes product 3, those who like exactly 2 of the products get counted twice, so they need to subtracted from the total once.
Those who like all 3 products get counted 3 times, so they need to be subtracted from the total twice.

In the problem above:
T = 85
G1+G2+G3 = Product 1 + Product 2 + Product 3 = 50+30+20
Those who like exactly 2 of the products = x
Those who like all 3 products = 5

Plugging into the formula, we get:

85 = 50 + 30 + 20 - x - 2*5
x = 5 consumers who like exactly 2 of the products.

Since 5 consumers like exactly 2 of the products and 5 like all 3 products, the number who like at least 2 of the products = 5+5 = 10.

The correct answer is B.