In a survey on three products – A, B, and C – 50% of those surveyed

Question

In a survey on three products – A, B, and C – 50% of those surveyed liked product A, 30% liked product B, 20% liked product C, and 85% liked at least one of the three products. If 5% of those surveyed like all three products, then what percentage of those surveyed liked more than one of the products?

A. 5
B. 10
C. 15
D. 20
E. 25

A. 5
B. 10
C. 15
D. 20
E. 25

pushpitkc · Accepted Answer

Formula used 
P(Total) = P(A) + P(B) + P(C) - P(exactly 2) - 2*P(all 3)

Substituting values
85 = 50 + 30 + 20 - P(exactly 2) - 2*5
P(exactly 2) = 90 - 85 = 5

Percentage or people who liked more than one product = p(exactly 2) + p(all 3) = 5+5 = 10(Option B)

khushbumodi · Answer

Sorry but I am a bit confused here!

In the formula that you used, isn't that supposed to ADD and not subtract the number for All Three?
You subtracted twice that number instead of adding it once. Can you please explain how did you come up with that formula?

Thanks

Sent from my SM-N9200 using  GMAT Club Forum mobile app

ScottTargetTestPrep · Answer

The problem really asks for the percentage of people who liked 2 or 3 products.

We can create the following equation:

Total percentage of people = percentage who like product A + percentage who like product B + percentage who like product C - (percentage who like 2 products) - 2(percentage who like 3 products) + percentage who like neither product

Let’s represent the percentage who like 2 products as D and percentage who like neither product as N. Then:

100 = 50 + 30 + 20 - D - 2(5) + N

100 = 90 - D + N

We are also given that 85% of the people surveyed liked at least one of the three products. Thus, 100 - 85 = 15 percent of the people liked none of the three products. So we have:

100 = 90 - D + 15

D = 5

Thus, 5 + 5 = 10 percent of the people like more than one product.

Answer: B

pushpitkc · Answer

P(Total) = P(A) + P(B) + P(C) - P(exactly 2) - 2*P(all 3) is the formula whose clarification you had asked for

The clarification is as follows :
 P(A) = P(Only A) + P(A U B) + P(A U C) + P(A U B U C)
P(B) = P(Only B) + P(B U C) + P(B U A) + P(A U B U C)
P(C) = P(Only C) + P(C U A) + P(C U B) + P(A U B U C)

P(exactly 2) = P(A U B) + P(A U C) + P(B U C)

P(all 3) = P(A U B U C)

Substituting these values in the formula,
P(Total) = P(A) + P(B) + P(C) - P(exactly 2) - 2*P(all 3)

P(Total) = P(Only A) +  P(A U B) + P(A U C) + P(A U B U C)  + P(Only B) +  P(B U C)  + P(B U A) +  P(A U B U C)  + P(Only C) + P(C U A) + P(C U B) + P(A U B U C) -  P(A U B) - P(A U C) - P(B U C) - 2* P(A U B U C)

P(Total) = P(Only A) + P(Only B) + P(Only C) + P(B U A) + P(C U A) + P(C U B) + P(A U B U C)

Dreamer0215 · Answer

Hi Scott,

can you please tell me why you have not used the formula ?

Total = A+B+C - (sum of 2 group overlaps) + all three + neither

By this, we will get 25 as the answer

ScottTargetTestPrep · Answer

Solution:
 
As a matter of fact, you can also solve this question using that formula and again get the correct answer of 10. I am not sure how you got 25 as the answer, but here’s how you solve it:

We have Total = 100, A = 50, B = 30, C = 20, all three = 5, and neither = 100 - 85 = 15. Substituting these values in your formula, we get:

100 = 50 + 30 + 20 - (sum of 2 group overlaps) + 5 + 15

sum of 2 group overlaps = 20

Now, we need to understand what “sum of 2 group overlaps” is representing. This is actually (the percentage who like A and B) + (the percentage who like A and C) + (the percentage who like B and C). Notice that the group “all three” is counted three times in the above calculation. If we subtract twice the percentage of those who like all three products, we will get the percentage who like two or three products. Thus, the percentage of people who like more than one product is once again found to be 20 - 2 * 5 = 10 percent.

Answer: 10

KarishmaB · Answer

Ignoring percentages, 
50 + 30 + 20 = 100 gives us 100 instances but only 85 people liked at least one of the products so we have 15 extra instances. 
5 people liked all three products so this accounts for 10 instances (2*5 extra instances were counted for these 5 people). 
We still have 5 more instances to account for and these must be the people who like 2 products. 
So 5 + 5 = 10 people like more than one product.

Answer (B)

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