Official Guide Explanation:
Problem Solving #134




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Solution and Metadata

Question: 134
Page: 79
Difficulty: 6 (Moderately Difficult)
Category 1: Algebra > Inequalities > Negatives
Category 2: Algebra > Simplifying Algebraic Expressions >

Explanation: To focus in on the values of c and d, simplify the inequalities by first subtracting one from each part:

0 - 1 < 1 - ((c)/(d)) - 1 < 1 - 1

-1< - ((c)/(d)) < 0

Now, divide each part by -1, remembering to reverse the inequality signs:

1 > ((c)/(d)) > 0

If ((c)/(d)) is between 0 and 1, c and d must have the same sign; if they are both positive, c < d, if they are both negative, c > d. Since the question tells us that d is positive, c must be positive as well. Armed with that knowledge, evaluate the statements:

I.    c must be positive..

II.    Yes, ((c)/(d)) is less than 1.

III.    This isn't clear: we don't know anything about the size of c and d, just their relationship with each other.

I and II must be true, so choice (C) is correct. Note that the statements were very easy to work through after doing all of the analysis beforehand. Roman numeral questions can be very similar to Data Sufficiency questions that way.

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