One of the biggest challenges on the GMAT^® is the battle against the timer. But the first thing that test-takers have to realize that it is not a test where a section cannot be completed in 75 minutes.

Even the seemingly time-taking problems or calculations always have a straightforward solution provided you pause to think about the best way of going about solving a problem rather than jump into it.

Usually the speed-breakers are Data Sufficiency questions involving inequalities. The first reaction of test-takers is to randomly plug numbers and see how things will pan out. Since they have not taken the time to first define the problem well, plugging numbers tends to become a trial and error process in the hope of hitting upon the answer rather than a process of testing the inequality!

Over the next few Quantitative posts we will look at methods to effectively tackle Data Sufficiency problems involving inequalities.

One of the first methods of decreasing the solving time required on these problems is to test the converse of the inequality — Can the converse be true?

Let us look at what this means using the GMAT question below.

DIRECTIONS: Each data sufficiency problem below consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you are to select option

A if statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked;
B if statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked;
C if BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient;
D if EACH statement ALONE is sufficient to answer the question asked;
E if statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If n is an integer between 10 and 99, is n< 80?
(1) Sum of the two digits of n is a prime number
(2) Each of the two digits of n is a prime number

The question is asking you if n is less than 80. What is the converse of this inequality?

Can n be greater than 80?

Instead of checking whether n < 80, we can check whether n can be greater than 80; if it cannot be then it has to be less than 80. The big advantage with this is that it narrows down the range of numbers you have to test for – instead of 1 to 80 you are just checking for 81 to 90 to begin with.

Statement 1, says that the sum of the two digits of n is a prime number. Whenever you have check for a condition it is best to do so systematically:

• In this case since you are checking for numbers between 81 and 99, the sum of whose digits are prime, you know that the sum has to be a prime greater than 8.
• The first prime greater than 8 you can check for is 11 (it is always best to find a template for the possibilities, rather than blindly plugging numbers, a grossly over-rated strategy of there ever was one) . So straightaway if one digit is 8 the other has to be 3 — 83. So n can be greater than 80.
• Since the digits can be reversed to get 38, n can be less than 80 as well.

Hence, (1) alone is not sufficient since it can be both greater and less than 80 as per this statement and hence you cannot answer conclusively whether n is greater than 80 or not.

Statement 2, straightaway shows that n cannot be greater than 80 since there cannot be number between 81 and 99 with both digits prime since 8 and 9 are not prime. Hence n has to be less than 80 and (2) alone is sufficient.

We were able to reach the answer so easily because we re-defined the question, else we would need first need to check for values less than 80 and then for values greater than 80.

This method of testing can be applied to inequalities involving two variables as well. For example, if the question asks is r > b?, you can test whether r can be less than b. You will find that with one statement it will be revealed that r cannot be less than b given the condition so r has to be greater than b.

Asking if the converse can be true is a very useful strategy to simplify problems involving inequalities based Data Sufficiency questions and save precious time on the Quantitative Section of the GMAT^®. Practice the same with 700-800 level Data Sufficiency questions to get a better grip on how to can be apply it effectively.