In a consumer survey, 85 percent of those surveyed liked at least one of three products. 50 percent of those surveyed 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 percent of the survey participants liked more than one of the three products?

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

(B) Overall, 85% of the surveyed people like at least one product, but when we add up the percent of people who like each product individually, we get a sum that is more than 85%: 50% + 30% + 20% = 100%. These two figures differ because the people that like all three products are counted three times in the 100% figure, and the people that like exactly two products are counted twice.

To correct for the triple-counting, we can subtract 2 times the number of people that like all three products, so these people are now counted just once. We can also correct for the double-counting by subtracting the number of people that were double counted, so they are now counted just once. The equation will look like this: 85% = 100% – 2 × (like all three) – (like exactly 2). 85% = 100% – 2 × (5%) – (like exactly 2) 85% = 90% – (like exactly 2) 5% = the number of people that like exactly 2 products.

Since 5% like all three products, and 5% like exactly 2 products, 10% like more than one product. The correct answer is choice (B). -------------

Why shouldn't the number of people who like all 3 be subtracted 3 times since this would logically make sense?

There is nothing in the question to suggest there are only 3 outcomes. Product a , b and c. It is possible that 15% did not like any of the 3 products. This would then yield A as the answer.

There is nothing in the question to suggest there are only 3 outcomes. Product a , b anc c.

The question does NOT suggest there are only 3 outcomes. It suggests there are 8 possible outcomes: - did NOT like any - liked Product 1 only - liked Product 2 only - liked Product 3 only - liked Products 1 and 2 only - liked Products 1 and 3 only - liked Products 2 and 3 only - liked ALL the Products

questioner wrote:

It is possible that 15% did not like any of the 3 products. This would then yield A as the answer.

It is not only possible, but the question IMPLIES that 15% did NOT like any of the products, because 85% liked at least one. (100% - 85% = 15%). So this fact does NOT change the reasoning and the correct answer is B, NOT A.

Probably, you got confused with the figure 100% = 50% + 30% + 20%. Note, that this figure "100%" does NOT refer to all surveyed people, but it refers to 85% (that liked at least one of the product) + some extra counted groups.

Take a look at Venn diagrams above to clearly understand what groups and how many times are counted in 100% = 50% + 30% + 20%. First of all, think what groups these 20%, 30%, 50%, 85% consist of. Think of those groups as of areas. When they overlap, the overlapping parts are counted twice or more times (depending on how many areas overlap).

50% = liked Product 1 only + liked Products 1 and 2 only + liked Products 1 and 3 only + liked all the Products 30% = liked Product 2 only + liked Products 1 and 2 only + liked Products 2 and 3 only + liked all the Products 20% = liked Product 3 only + liked Products 1 and 3 only + liked Products 2 and 3 only + liked all the Products 85% = liked Product 1 only + liked Product 2 only + liked Product 3 only + liked Products 1 and 2 only + liked Products 1 and 3 only + liked Products 2 and 3 only + liked all the Products

Also note, that LIKED MORE THAN ONE = liked Products 1 and 2 only + liked Products 1 and 3 only + liked Products 3 and 4 only + liked all the Products

When you get a good understanding of how different groups overlap, you will be able to deal with larger groups instead of 8 original small disjoint ones. For example, to solve this problem, you can deal with "liked at least one", "liked two only", "liked all three".

To practice, think what groups and how many times are counted in 80% = 50% + 30% or 70% = 50% + 30%.

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