Let me show you what I mean.

Here’s a classic GMAT problem.

Seed mixture K is 30 percent ryegrass and 70 percent wheat by weight; seed mixture N is 15 percent ryegrass and 85 percent barley. If a mixture of K and N contains 20 percent ryegrass, what percent of the weight of the mixture is N?

(A) 10%     (B) 33.3%     (C) 40%    (D) 50%    (E) 66.6%

Wow ok. So there are a lot of things going on; all kinds of grasses in all kinds of percentages. First of all, we don’t care at about bluegrass or whatever fescue is. If we combine the mixtures to arrive at 20 percent ryegrass, we can just keep track of how we mix them according simply to the ryegrass; whatever else is in the mixture will follow accordingly and work out as it will. We just need to see how a mixture with 30 percent ryegrass and a mixture with 15 percent ryegrass combine to land at 20 percent. With me? If not, read again (this is where tutoring is helpful).

Now, the straightforward academic explanation of how to solve this mathematically is long, exhaustive, involves all sorts of 1 over r fractions and equations and common denominator-finding and whatnot and that’s all very good if you have all day (but you don’t). You need to get this problem done in 2 minutes (depending on how your timing is going, which we’ll discuss) and I’ve already wasted 30 seconds talking your ear off just now.

So ask yourself, what if I poured these 2 mixtures in equal quantities? Where would the ryegrass percentage land? It would of course land right in the middle. Equal parts of a 30 percent solution and a 15 percent solution will land us right in the middle, at 22.5 percent. However, in this problem that isn’t where we land; we land at 20 percent. Think to yourself: if we mixed them evenly, the result would be a 22.5 percent solution and here we have 20, so which must there be more of? Which of the two mixtures, 15 or 30 percent, is the result closer to?

The 15 percent solution! 20 is closer to 15 than it is to 30, so there is more of the 15 percent solution, mixture N. Even if you had no idea where to go from here, you can eliminate answers d and e. There has to be less of mixture K.

Now, let’s return to thinking about that average mixture (because when you mix liquids of different salience, you’re arriving at an average salience). Because 20 is twice as close to 15 as it is to 30 (5 away versus 10 away), we can reliably infer that there will be twice as much of the 15 percent mixture. That means it will be 2/3 Mixture N (the 15 percent mixture) and 1/3 Mixture K (30 percent). Our answer is E.

However much this explanation clicked with you, were I able to explain it in different ways and help you master this kind of thinking, you could do this otherwise difficult GMAT problem in a matter of seconds and move on with your life.