100 = x + x - 20 + 23 - 23

x = 60,

So, Product A = 60%, Product B = 40%, Both = 23%, Neither = 23%

23% of the total no. of people should be an integer. So, A,B & C are out.

60% of D and 40% of D are both integers. So, D satisfies all conditions.

So, answer is D.

Even if it does not affect the result and the reasoning, in this case, I believe that the sentence is not completely clear because it should clarify if:

- x contains people who likes ONLY A, or not;
- (x - 20) contains people who likes only B, or not.

The result in this case is not affected (D in both cases), but the computation of x and x-20 is.

Patrizio

x percent of respondents liked product A means that x% contains those who also liked product B. No further clarification is needed here.

Otherwise it would be written: x percent of respondents liked ONLY product A.

Hope it's clear.
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AUB = 100 – 23 = 77 % = A + B – A and B
77 = x + x – 20 – 23
X = 60
A – 40 B – 60 Both – 23
Now the only key is that number of people cannot be fraction. All the percentages are integers for option D. Hence answer is D.
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What if the value of x is 23.5% then 100 would also not suffice. So you need to calculate all the values to make sure that you don't get an integer.
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true. agreed. oversight from my end.
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200 also satisfies the condition, but why do you prefer D. Since, 23%, 40% and 60% of 200 is some integer.

Hi,
could you please clarify the formula? why do you subtract 23?
thanks

sorry, still don't get it is there a formula you are referring to when finding a minimum?
thanks

Hi LinaNY, there's no formula for finding minimum. The formula used above is:

Total = Set X + Set Y - Both + Neither
100 = x + (x-20) - 23 + 23

Since 23 is a prime number, the total cannot be less than 100 and keep it as an integer. Your only two options are 100 and 200. Which one is the minimum? The smaller one. 200 works as well, you'd have 120 people who like X, 80 who like Y, 46 who like both and 46 who like neither. But you can get smaller than this, much like 2/4 can be reduced to 1/2.
This yields 60 people who like X, 40 who like Y, 23 who like both and 23 who like neither. Must be D if you want the minimum.

Hope this helps
-Ron
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Thank you very much, Ron. It totally makes sense now.

A company conducted a survey about its two brands, A and B. x percent of respondents liked product A, (x – 20) percent liked product B, 23 percent liked both products, and 23 percent liked neither product. What is the minimum number of people surveyed by the company?

1)46
2)80
3)90
4)100
5)200
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Merging topics. Please refer to the discussion above.
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Total = A + B - (Both A and B) + Neither

Solving in terms of percentages, let the Total = 100

Like B = x - 20

Like A = x

100 = (x) + (x - 20) - (23) + (23)

120 = 2x

X = 60%


Like A only = 60% - 23% = 37%

Like B only = 40% - 23% = 17%


(D) is the first number given that, when you multiply by these given percentages, the result is a Whole Number (integer constraint because we are surveying people)

Minimum is 100 people surveyed

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