What is the value of (2t + t - x)/(t - x)?

Okay, I am going to be blunt here. Many explanations of Quantitative questions focus blindly on the math, but remember: the GMAT is a critical-thinking test. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full “GMAT Jujitsu” for this question:

First, let’s dissect the structure of this question. The two answer choices together comprise a perfect example of what I call a “C Trap” in my classes. It is part of human nature to want to make decisions using all the available information. The Testmaker knows this. If you encounter a problem where it is painfully obvious that the two statements together are sufficient, be careful. It should be very easy to see that if we were to use both statements together, we could solve this problem. We would have two equations and two variables. We could easily plug in Statement #2 into the Statement #1 and solve for \(t\), then we could plug in our value for \(t\) into Statement #2 and solve for \(x\). Knowing both \(t\) and \(x\) would obviously give us the value for the expression in the question stem. But if it is that obvious that the two statements together work, what is this problem even doing on the GMAT? It seems too easy. When you encounter a problem structured like this, look closer for additional leverage.

Always look for large chunks of equations you can cancel or simplify all at once. Many questions creatively combine or eliminate large chunks so you can solve the question without needing to solve for each variable. In my classes, I call this strategy “Chunky-quations.” If you know how to dissect a question, these common “chunks” show up all over the place. This particular question asks us to solve for an expression – in other words, we don’t even have a full equation. In order to eliminate variables to get to a numerical solution for an expression, our options are very limited. One method is to substitute, thereby eliminating variables we don’t want to see. Another strategy is to “Divide and Conquer”, finding and cancelling chunks common to both the numerator and denominator of fractions. This can also eliminate variables.

Let’s begin by focusing on the easier of the two statements. (I call this strategy “Low-Hanging Fruit” in my classes. Looking at the easier statement first may help you to evaluate the problem and may even give you clues on how to evaluate the harder statement.) The question stem gives us the expression \(\frac{2t+t -x}{t -x}\). It has a common chunk “\(t-x\)” in both the numerator and denominator, but the extra “\(2t\)” remains. If we were to substitute Statement #2 into this expression, we could either eliminate the “\(t-x\)” chunk, leaving the \(2t\), or we could solve Statement #2 for one of the two variables, \(t\) or \(x\), and then substitute, eliminating that variable. But this would still leave the other variable. There is no way to eliminate both variables with Statement #2 alone. Eliminate it.

We could technically also solve Statement #1 for one of the two variables, \(t\) or \(x\), and then substitute, but that would cause the same problem. Don't do math for kicks and giggles. Do math because it helps you to get the answer to the question. We can easily rearrange Statement #1 so it reads “\(2t = 3(t-x)\)”. This contains the “\(t-x\)” chunk we were looking for. Thus, we can substitute “\(3(t-x)\)” for the “\(2t\)” in the original expression. This leaves:

\(\frac{2t+t -x}{t -x}=\frac{3(t-x)+(t -x)}{t -x}=\frac{4(t-x)}{t -x}\)

We have a common chunk in the top and bottom of the denominator, allow us to cancel both chunks, leaving only a numerical value remaining. Statement #1 is sufficient, and the answer is “A”.

Now, let’s look back at this problem through the lens of strategy. This question can teach us patterns seen throughout the GMAT. First, this problem highlights a common trap of the test where it baits you into picking an “obvious” answer without looking closer at the underlying structure. Be careful of obvious answers. There is almost always more going on. Next, you can see how we can use the structure of the problem to think about possible solutions. If you need to solve for an expression (instead of a full equation), you need to look for common “chunks", allowing you to eliminate entire pieces of the equation simultaneously. With expressions involving fractions, one of those ways of eliminating chunks is to find, factor, or create chunks that you can “Divide and Conquer.” And that is how you think like the GMAT.