Official Guide Explanation:
Problem Solving #14

Background

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Solution and Metadata

Question: 14
Page: 63
Difficulty: 4 (Moderately Easy)
Category 1: Arithmetic > Fractions >

Explanation: There are a number of handy ways to compare fractions. The typical way is to find a common denominator, which is easiest on this problem for choice (D). To compare (2/3) and (13/27), convert (2/3) to ((2(9))/(3(9))) = (18/27). (2/3) is bigger, so eliminate (D).

More powerful methods rely on comparing numbers with consecutive denominators and/or numerators. For instance, to compare (2/3) to (33/50), convert (2/3) to something nearby, like ((2(17))/(3(17))) = (34/51). If you add one to the numerator and denominator of a fraction, it gets bigger, so (34/51) > (33/50). Eliminate (A).

You can analyze (B) in a similar way. Convert (2/3) to ((2(4))/(3(4))) = (8/12). If two fractions have the same numerator, the one with the smaller denominator is the bigger number, so (8/11) > (8/12), so (B) is the choice that is larger than (2/3).

For the sake of completeness, let's compare (C) and (E) to (2/3), as well. (2/3) = (4/6), and (4/6) > (3/5), so (C) is smaller than (2/3). Similarly, (2/3) = (6/9), and (6/9) > (5/8), so (E) is smaller, as well.

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