Bunuel wrote:

Is ab < 12?

(1) a < 3 and b < 4

(2) 1/3 < a < 2/3 and b^2< 169

Kudos for a correct solution.

OFFICIAL SOLUTION:This data sufficiency problem asks you to draw a conclusion about an inequality. Data sufficiency questions often deal with inequalities because their solutions are often ambiguous. For this problem, you need to determine whether the product of a and b is less than 12.

Statement (1) tells you that a is less than 3 and b is less than 4. If you didn’t think about this statement thoroughly, you may have been tempted to say that it sufficiently determines that ab is less than 12 because 3 × 4 is 12 and the values are less than those.

Consider all possibilities for a and b. If both a and b represent negative numbers that are less than –2 and –3, their product would actually be equal to or greater than 12.

For instance, if a = –3 and b = –4, their product would be 12, which equals 12 and therefore isn’t less than 12. Values for a and b of –9 and –10, respectively, would produce a product of 90, which is far more than 12. Statement (1) isn’t sufficient to determine whether ab < 12, so eliminate A and D.

Consider statement (2). Start with the inequality it gives you for b2. Solve the inequality by taking the square root of both sides, and make sure you consider both positive and negative possibilities:

b^2 < 169

b < 13 or b > –13

The other information in the statement tells you that a is greater than 1⁄3 of b but less than 2⁄3 of b. So when you multiply a and b, a will be at most 2⁄3 of b (which is 13). 2⁄3 × 13 is 8.67, so the product of a and b will certainly be less than 12 for all positive values of b. You don’t need to worry about the negative values of b, because a negative multiplied by a positive is always a negative, which means the product is less than 12. Statement (2) give you enough information to answer the question, so you can eliminate C and E.

Your answer is B.

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