## Number Properties - Algebra - Connected Concepts

GMAT Quant they say is hard. Though we know that it tests you on high school level Math only. Then why do most people find it hard? Because the questions are framed in some seriously innovative ways. For example, I came across the below question on GMAT Club. It is from Manhattan GMAT (as per the tag). Another tag claimed that it was a number properties question though all that came to my mind was a concept I discuss in my inequalities module in Algebra.

Before I discuss the question, let me re-iterate that I am a big proponent of holistic approaches. I like to look at the big picture and solve the question conceptually. So until and and unless I really really need to, I do not put my pen to paper. More often than not, I don't even keep a paper pen with me and work straight on my laptop giving solutions to problems. It works because I am able to connect concepts (some of these connections are mentioned in the modules)

Let me give you the question now.

Question: For non-zero integers a, b, c and d, is ab/cd positive?

Statement 1: ad + bc = 0

Statement 2: abcd = -4

Solution: Recall what we discuss in our inequalities module: the sign of the answer doesn't change whether you multiply or divide factors. It is something we discuss when working with inequalities with factors. We say that

(x+a)(x+b)(x+c) < 0 has the same solution as (x+a)(x+b) / (x+c) < 0, as (x+a) / (x+b)(x+c) < 0 and as 1 / (x+a)(x+b)(x+c) < 0 etc because it doesn't matter whether the factors are getting multiplied or divided. The sign of ab will be the same as the sign of a/b or b/a. If a and b are both positive, all ab, a/b and b/a will be positive. If a is negative and b is positive, all ab, a/b and b/a will be negative. If both a and b are negative, all ab, a/b and b/a will be positive. Extending the same logic here, we know  that sign of abcd, ab/cd, ad/bc etc will be the same.

Let's consider statement 2 first.

Statement 2: abcd = -4

When all a, b, c and d are multiplied together, we get a negative result. Then when two are multiplied and other two divide them i.e. with ab/cd, the result will be negative too.

This statement is sufficient alone.

Statement 1: ad + bc = 0

This means ad = - bc
So ad and bc have opposite signs (e.g. if bc = 10, ad = -10).

Then a/d and b/c will have opposite signs too since sign of ad is same as sign of a/d and sign of bc is same as sign of b/c.

Then ab/cd = (a/d) * (b/c) will be negative since one of them is positive and the other negative.

This statement is sufficient alone.