A market research company surveyed users of the toothpaste

[BTGmoderatorDC]

A market research company surveyed users of the toothpaste industry's two most popular brands, Brand X and Brand Y. If each person contacted reported that they used at least one of the two brands, what percent of respondents reported that they only use Brand Y?

(1) Among the respondents, the ratio of those who reported that they use Brand Y to those who reported that they use Brand X was 3:2.
(2) Among the respondents who reported that they use Brand X, one half also use Brand Y.

OA C

Source: Veritas Prep

[ceilidh.erickson]

We're given that everyone in the survey used at least one of Brand X and Brand Y. We can think of this as:
(Brand X users) + (Brand Y users) - (people who used both) = Total
or, since we're asked for a percentage:
X + Y - B = 100

If we want to know the percentage who used Brand Y, we'll need information about the proportion who used Brand X, and the proportion who used both.

(1) Among the respondents, the ratio of those who reported that they use Brand Y to those who reported that they use Brand X was 3:2.

This would be sufficient if the categories were mutually exclusive, but they're not: we have a category for "both." This only gives us a relationship between 2 of our variables:
Y/X = 3/2 --> 2Y = 3X
But since we have 3 variables and only 2 equations, this won't be enough.

You can also try testing cases to prove insufficiency:
Case 1: 60% could use Brand Y, 40% could use Brand X, and 0% use both. 60 + 40 - 0 = 100
Case 2: 90% could use Brand Y, 60% could use Brand X, and 50% use both. 90 + 60 - 50 = 100

These are both consistent with the statement information, but they give us different answers to the question. Insufficient.


(2) Among the respondents who reported that they use Brand X, one half also use Brand Y.

Again, this only gives us a relationship between 2 of our variables:
B = (1/2)X
But since we have 3 variables and only 2 equations, this won't be enough.

Alternatively, you could test cases again:
Case 1: If 50% of respondents use Brand X, and half of those (25% of the total) use both, then 75% of respondents use Brand Y. 50 + 75 - 25 = 100
Case 2: If 20% of respondents use Brand X, and half of those (10% of the total) use both, then 90% of respondents use Brand Y. 20 + 90 - 10 = 100

These are both consistent with the statement information, but they give us different answers to the question. Insufficient.

(1) & (2) Together:

Each statement gives us a relationship between 2 of our variables:
(1) Y/X = 3/2 --> 2Y = 3X
(2) B = (1/2)X

Since we have the given equation X + Y - B = 100, we now have 3 equations for 3 variables, and we can solve. At this point, DON'T actually solve! I also wouldn't recommend testing cases, since you might have to wrack your brain to come up with a value that fits both statements (that should be your indication that there's probably only 1 case that will fit.

The answer is C.

Just to prove that we get an answer (but again, do NOT do this work on test day if you already see that you'll have sufficient information!):

2Y = 3X --> X = (2/3)Y
B = (1/2)X --> B = (1/2)(2/3)Y --> B = (1/3)Y
X + Y - B = 100 -->
(2/3)Y + Y - (1/3)Y = 100
(4/3)Y = 100
Y = 100(3/4)
Y = 75