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## Official Guide Explanation:

Data Sufficiency #87

**Background**

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

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**Solution and Metadata**

**Question****: 87**

Page: 280

Difficulty: **6** (Moderately Difficult)

Category 1: Arithmetic > Ratio and Proportion >

Category 2: Word Problems > Rate Problems > Speed

**Explanation:** Rate equals distance over time:

r = ((d)/(t))

so t = ((d)/(r)). Thus, the question boils down to this:

((d_{1})/(r_{1})) > ((d_{2})/(r_{2}))?

Statement (1) is insufficient. It allows us to substitute one of the variables, since d_{1} = d_{2} + 30:

((d_{2} + 30)/(r_{1})) > ((d_{2})/(r_{2}))?

While we know that the first distance is greater, we don't know anything about the rates.

Statement (2) is also insufficient. Again, it allows us to substitute one variable, but this time we don't know anything about distance:

((d_{1})/(r_{2} + 30)) > ((d_{2})/(r_{2}))?

Taken together, the statements are still insufficient. We can substitute both statements into the original inequality:

((d_{2} + 30)/(r_{2} + 30)) > ((d_{2})/(r_{2}))?

An easier way to see the relationship is to cross - multiply. We can do that because distances and rates must be positive:

(d_{2} + 30)r_{2} > (r_{2} + 30)d_{2} ?

d_{2}r_{2} + 30r_{2} > d_{2}r_{2} + 30d_{2} ?

30r_{2} > 30d_{2} ?

r_{2} > d_{2} ?

What matters is the relationship between the second rate and distance. We don't know that, so we can't answer the question. Choice (E) is correct.

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