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A market research company surveyed users of the toothpaste

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A market research company surveyed users of the toothpaste

by BTGmoderatorLU » Tue Mar 19, 2019 3:02 pm

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Source: Veritas Prep

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.

The OA is C
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Source: — Data Sufficiency |
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by swerve » Wed Mar 20, 2019 9:27 am

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The question is asking for the ratio \(Z\)

\(Z = \frac{y-\text{only}}{x-\text{only} + y-\text{only} + \text{both}}\)

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 \(3:2\)"

\(\Rightarrow \frac{y-\text{ only} + \text{ both }}{x-\text{ only} + \text{ both}} = \frac{3}{2}\)

\(\Rightarrow \text{ both } = 2y - 3x\). Insufficient.

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

\(\Rightarrow x-\text{ only} = \text{both}\). Insufficient.

Combining both statements

\(1+2 \Rightarrow 2y - 3x = x = \text{ both} \)

\(\Rightarrow y = 2x = \text{both}\)

\(\Rightarrow Z = \frac{2x}{x + 2x + x} = \frac{1}{2}\)

C is sufficient.
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