Sometimes the process of teaching helps the teacher as much as the student. I was in Hyderabad the last weekend, conducting a GMAT Boot Camp for IMS Hyderabad students and during the course of the session, I discovered a better way to solve an old problem.

Towards the end of a long and grueling day of solving  700-800 level problems we came to this Data Sufficiency problem as part of the segment where we focus specifically on the toughest type of Data Sufficiency problems – those involving inequalities. 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 choose 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 mv < pv < 0, then is v > 0?

(1) m < p
(2) m < 0

The standard way of approaching this problem is by plugging in numbers: we test the conditions for positive, negative, fractions and see if we can draw any conclusive inference about v.

I gave the problem to the participants to solve and as I was slowly pacing across the room, I was thinking that to take them through testing the conditions for positive, negative and fractions might be both time-consuming and exhausting.

One of the things that I love about the GMAT is that on most problems, not ALL, one can find a smarter way out. So I decided to explore an easier way, a more elegant solution.

I looked at statement (2) first and thought out verbally – m is negative, I need to know something about v, the question stem says mv is negative, so there is no way v can be negative since if m and v are both negative mv will become positive, so v has to be positive (I ignored pv < 0  because the statement did not speak about p and I needed to determine something about v). So (2) alone is sufficient.

I then moved to statement (1), which I would normally have solved by taking different cases: m & v – both positive, m & v – both negative, m & v – one positive, one negative. Since I wanted to find a better method I thought I will proceed the same way as I did with statement (2).

The statement said  m < p and the question stem said mv < pv. So even after multiplying the first inequality m < p with v, the sign remains the same, mv is still less than pv. This can only happen if v is positive!

Whenever you multiply or divide any number with a negative number its sign changes, a negative number becomes a positive number and vice-versa. So if the number is part of an inequality and it gets multiplied with or divided by a negative number the sign of the inequality has to change. So if the sign of the inequality is not changing after being multiplied by v on both sides, then v has to be positive.

It again went on to prove what I strongly believe about problem solving — if we truly engage & evaluate the problem at hand and do not start solving it in the auto-pilot mode, we will find the best solutions.

When I came back to Chennai I discussed this problem with a student, he said something that confirmed what I have always maintained about the GMAT Quant — the problems posed will never test a concept or formula that is not covered in the Math Review section of the OG; even if there are formulas not mentioned in the Math Review that can be used to solve GMAT Quant problems, it does not mean that those formulas are required to solve them.

The student told me that this concept — multiplying or dividing an inequality by a negative number changes the sign of the inequality — is clearly stated in the inequalities section of the Math Review. I looked it up and it sure is mentioned.

The GMAT can be taken by someone as young as 13 years old anywhere across the world. This itself shows that the Quant concepts tested will not be advanced concepts by any stretch of imagination. What they are testing is whether you can reason in a Quantitative context!