Official Guide Explanation:
Data Sufficiency #120

Background

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Solution and Metadata

Question: 120
Page: 283
Difficulty: 7 (Very Difficult)
Category 1: Arithmetic > Percents > Percent Change

Explanation: This question can involve quite a bit of algebra. Jump right in. If we call the 1997 rent r, we know that the 1998 rent is x percent more:

1998: r + (((x)/100))r

In other words: it's the 1997 rent plus x percent (x divided by 100) of the 1997 rent.

The algebra gets messier when we subtract y percent from the 1998 rent to get the 1999 rent:

[r + (((x)/100))r] - (((y)/100))[r + (((x)/100))r]

(r + (((x)/100))r)(1 - ((y)/100))

We want to know whether that expression (the 1999 rent) is greater than the 1997 rent, r:

(r + (((x)/100))r)(1 - ((y)/100)) > r ?

Before jumping into the statements, recognize that we can divide r from both sides:

(r)(1 + ((x)/100))(1 - ((y)/100)) > r ?

(1 + ((x)/100))(1 - ((y)/100)) > 1 ?

Given all those 1's, let see if we can get any further:

1 + ((x)/100) - ((y)/100) - ((xy)/(10,000)) > 1 ?

((x)/100) - ((y)/100) - ((xy)/(10,000)) > 0 ?

x - y - ((xy)/100) > 0 ?

x - y > ((xy)/100) ?

That's a lot of work, and it takes a lot of time, but glance at the statements and you can recognize just how much work it can save you.

Statement (1) is insufficient. Given that x > y, the left side of the inequality will always be positive, but it could be very small. For instance, if x = 20 and y = 19, the left side is 1, and the right side is ((20(19))/100), or about 4. By constrast, if there's a large difference between x and y, the left side will be larger.

Statement (2) is sufficient. This is where our work pays off. It answers the question directly (if we've simplified the question, anyway). ((xy)/100) is less than x - y, so the answer to the question is "yes." Choice (B) is correct.

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