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

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18 Sep 2003, 15:38

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In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% 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) 5

B) 10

C) 15

D) 20

E) 25

I found the answer and explanation on Score 800 test, but cant understand it at all. Would any of u be kind enough to explain the answer.

Out of 100%, only 85% liked at least one product.
i.e 85 = Sum of who liked only 1 product(Sum1) - sum of who like 2 products(Sum2) + sum of who liked all 3(Sum3).

I can't make Venn diagram here. But that'll help.

so eqn would be

85 = 50+30+20 - (Sum2)+5 --> Sum2 = 20.

Now, to find % who liked more than one product would be Sum2 + Sum3
= 20 +5 = 25. Hence E.

Pls. let me know if my initial eqn is correct. That's the key here.

The eqn I used is this:
Number of people who liked at least 1 product = Prod1+Prod2 + Prod3 - Prod1&2 - Prod2&3 - Prod1&3 - Prod1&2&3

The numbers of people liking prod 1 include those liking 1&2, 1&3, and 1,2&3. The same is true for the people liking product 2 and product 3. That's why you need to subtract out the people liking multiple products.

So (using a basis of 100 for easy numbers):
85=50+30+20-(1&2)-(2&3)-(1&3)-5
85=95-(Number of People Who Liked 2 Products)
Number of People who liked exactly 2 products = 10

From question stem, number of people who liked all three products =5
Number of people who liked more than 1 product = 10+5=15

Number of people who liked at least 1 product = Prod1+Prod2 + Prod3 - Prod1&2 - Prod2&3 - Prod1&3 - 2*Prod1&2&3

Infact, if u draw vein diagram u will see that we have counted the intersection of 1,2 & 3 three times, thats why we substract 2 times. As for intersection 1&2, 2&3, 3&1 we count it two times and subtract only 1 time to get the correct number.

praet/stolyar correct me if i am wrong.
Answer wud be B.

Number of people who liked at least 1 product = Prod1+Prod2 + Prod3 - Prod1&2 - Prod2&3 - Prod1&3 - 2*Prod1&2&3

Infact, if u draw vein diagram u will see that we have counted the intersection of 1,2 & 3 three times, thats why we substract 2 times. As for intersection 1&2, 2&3, 3&1 we count it two times and subtract only 1 time to get the correct number.

praet/stolyar correct me if i am wrong. Answer wud be B.

Hi Vicky,

Your answer seems correct. However, there is a correction required in the formula you mentioned.

Number of people who liked at least 1 product = Prod1+Prod2 + Prod3 - Prod1&2 - Prod2&3 - Prod1&3 + Prod1&2&3

If you pick up few numbers then you will be able to verify this.

Formula for two elements a and b
Union of a and b aUb = a + b - a^b

To derive the formula for 3 elements, consider aUb as one unit and expand the union of a,b and c using the formula for two elements
aUbUc = (aUb) + c - (aUb)^c
= a + b - a^b + c -(a+b-a^b)^c
= a + b - a^b + c- (a^c + b^c-a^b^c)
= a + b - a^b + c-a^c-b^c+a^b^c
= a + b + c - (a^b+a^c+b^c) + a^b^c

In the current question, the values given are

aUbUc= 85, a=50,b=30,c=20, a^b^c = 5, What is (a^b+a^c+b^c) + a^b^c?
Applying the formula
(a^b+a^c+b^c) = a+b+c -aUbUc+ a^b^c = 50+30+20-85+5 = 20

Formula for two elements a and b Union of a and b aUb = a + b - a^b

To derive the formula for 3 elements, consider aUb as one unit and expand the union of a,b and c using the formula for two elements aUbUc = (aUb) + c - (aUb)^c = a + b - a^b + c -(a+b-a^b)^c = a + b - a^b + c- (a^c + b^c-a^b^c) = a + b - a^b + c-a^c-b^c+a^b^c = a + b + c - (a^b+a^c+b^c) + a^b^c

In the current question, the values given are

aUbUc= 85, a=50,b=30,c=20, a^b^c = 5, What is (a^b+a^c+b^c) + a^b^c? Applying the formula (a^b+a^c+b^c) = a+b+c -aUbUc+ a^b^c = 50+30+20-85+5 = 20

(a^b+a^c+b^c) + a^b^c = 20 + 5 = 25

Hi pawargmat,

Your derivation of the formula is correct. But what the question is asking is NOT = (a^b+a^c+b^c) + a^b^c

What the questio is asking is = (a^b+a^c+b^c) - 2(a^b^c)

This can become clear if you use the ven diagram.

If you draw the ven diagram, you will realize that the term "a^b+a^c+b^c" itself includes the term "a^b^c" THRICE. So we need to take out 2(a^b^c) from "a^b+a^c+b^c" in order to arrive at the answer.