Is ab < 12? (1) a < 3 and b < 4 (2) 1/3 < a < 2/3 and b^2< 169

Picking values is the most accurate way to deal with DS involving inequalities...

Statement 1

a = 2
b = 3
Answer to the question is Yes
a = -6
b = -7
Answer to the question is No

hence statement 1 is Insufficient

Statement 2

0.34 < a < 0.67
b < +- 13

take a = 0.65 (close to the max value)
take b = 12.9
Somewhere around 8.5
(you can consider integer values for a quick reasoning)
Answer to the question is Yes
As a > 0
Consider b to be negative
It will anyways be less than 12

hence Statement 2 is sufficient.

Correct answer B

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.

This question tests you on the max/min concept of inequalities.

Let us first discuss the when and how to use the max/min concept of inequalities:

When to use the Max/Min Concept of Inequalities:

Whenever you encounter a question with two finite ranges (x and y in this case) and the question asks us to find the sum (x+y), difference (x-y) and product (xy) of the two ranges, then this concept needs to be used

How to use the Max/Min Concept of Inequalities:

1. Place the two [b]finite ranges one below the other
2. Make sure the inequality signs are the same. If they are not the same then we make them the same by flipping one finite ranges inequality sign. This can be done by reversing the inequality or multiplying throughout by -1
3. Perform the mathematical operation only between the extreme values of the finite ranges.[/b]

Now lets look at the question.

Is ab < 12?

Statement 1 : a < 3 and b < 4

Here we have been given two infinite ranges. If a < 3 and b < 4 then ab can take any value on the negative scale and any value on the positive scale. Insufficient.

Statement 2 : 1/3 < a < 2/3 and b^2 < 169

If b^2 < 169 ------> -13 < b < 13

Now here we have two finite ranges and the question asks us for the product ab.

1/3 < a < 2/3
-13 < b < 13

Placing the ranges one below the other and multiplying we get 4 values

1/3 * -13 ----> -1/39
2/3 * 13 -----> 2/39
1/3 * 13 -----> 1/39
2/3 * -13 ----> -2/39

So the range of ab will be -2/39 < ab < 2/39. Here ab will always be less than 12.

Answer : B

AdityaCrackVerbal There's a typo in your solution. The answer must be B according to your solution(you have written the answer is C).

Thanks. Made the change.

Statement 1 - Assume a = 2, b = 3 .... ab = 6 YES
Assume a = -4, b = -5 .... ab = 20 NO ........ Insuff

Statement 2 - -13<b<13
a max ~ 2/3 ..... ab ~ 26/3 which is less than 12 hence Suff

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