Many business schools teach using the famous “case method”, in which you will analyze the real-world situation of a specific business at a time of crisis/transition/decision in order to gain practical knowledge of business theory as applied to an actual situation. The theory behind the case method is that, by analyzing how, for example, Kodak needed to transition from a conventional (film) to a new (digital) business model, you will gain large-scale comprehension of a business principle in general, and not just an intimate understanding of one business. With this experience, you can then apply your theoretical-and-practical understanding of a vast array of business principles to whatever situations will arise in your future role as a manager.
The GMAT gives you similar opportunities to glean information from a specific case and extrapolate it to another, perhaps more complicated situation. In fact, while many questions may seem to require you to have memorized a variety of specific tricks, formulas, and rules, the GMAT will reward you for being able to derive these rules from specific cases, and may even punish you for memorizing-without-understanding. Consider the question:
What is the sum of the even integers between 300 and 400, inclusive?
There are a few “rules” that can help you solve this question efficiently:
- For evenly-spaced sets (like a set of consecutive even integers), the mean and median of the set will be the same. In this case, the middle number, 350, will be the average of all the values in the set.
- To find the number of values in an inclusive set, take the range of (usable) values, then add one. (The counterpart to this is that, for exclusive sets, you subtract one).
- So here, knowing that we can only use the even numbers – every second number will count – we’d take the range (100) divide by 2 (to eliminate the non-useful odd numbers), and then add one (because it’s an inclusive set) to note that there are 51 terms with an average of 350. Accordingly, the answer will be 350*51, or 17,850.
Now, that seems like a lot of memorization needed for a fairly unique question type. Furthermore, memorization can be tough to implement – you will likely remember that for inclusive/exclusive sets, you add one in one case and subtract one in the other, but it may be tough when you are under pressure to remember exactly which is which. So keep in mind that you can use small cases in which you can prove rules like the above to prove your point, then extrapolate it to the question at hand – like your own personal GMAT “case method”:
- The range 300 to 400 is pretty vast, but once you recognize that it is an evenly spaced set of consecutive even integers, you can recognize that it will react similarly to any other set of similar numbers. If you take a more manageable set of consecutive even integers, like 2 through 10, you can experiment to see if a pattern exits. In that case, there are 5 values: 2, 4, 6, 8, and 10. Playing with those values, you’ll find that the ends (2 and 10) add to 12, and the next values inward (4 and 8 ) do the same, but that 6 won’t have a pair. The sum, then is, 12+12+6, or 30, a multiple of 6. Looking for patterns in these numbers, you may well find that the average value is the same as the middle value, or at least that you can find pairs to add to the same thing (12) unless there is an odd-man-out middle value, in which case it will be half the value of each pair (the same logic, just without mathematical terminology like “mean” and “median”). If you extrapolate this pattern to a larger set of consecutive even integers like 300-400, you can determine that they’ll have an average value of 350, or that each pair (other than the middle number) will add to 700.
- You are probably at least aware that there is a rule for inclusive and exclusive sets, but it comes up so infrequently that you may not have it down cold when the time comes to use it. That’s okay! It’s more important to know that a rule exists than to know the specifics of the rule! If you know that a rule exists for inclusive/exclusive sets, you can just prove it to yourself using a set like 1, 2, and 3. The range of that set is 3-1 = 2, but you can clearly see that if you include all numbers, there are 3 total. Accordingly, the rule for inclusive sets is to add one to the range. Similarly, if you excluded the ends of the range (1 and 3) there is only one value left, so you would have to subtract one from the range.
Many a GMAT prep student has read an explanation to a question like the one above and thought to himself “sure, that’s great if you remember the rule, but there are so many rules to remember”. When you recognize, however, that you can pretty quickly prove to yourself any rule that you know (or even suspect) exists, you can use small-number case methods to do for you what your memory just may not be able to.
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