Drawing Borders


When attending a multiple choice time limited test, one is expected to operate with the provided info fairly quickly. Drawing conclusions based on the recently acquired data in timely manner is something that all those GRE, SAT, GMAT etc. tests want to check you ability at.

This is why examinees are not allowed to use a calculator or a computer of some kind (which is, in fact, not necessary for the calculations required – but that is another topic) and the formulae list is fairly limited (but quite sufficient, however one should really remember those basic formulas in order to save much time at the exam).

Although time is not EVERYTHING in the exam (a common mistake committed by both the students and instructors during the preparation period), it is important. There is a possibility that we will find ourselves with less than a minute left – and yet there are still several questions unsolved.

As those are multichoice questions, we can – and should – try and put all the answers in the answer scorecard. There will be people who choose a number and put it as an answer for all the questions they did not succeed to solve. Not a bad tactics, giving you an average of 25% score in those questions. Yet, there is a way to improve our “guessing statistics” – and sometimes even solve the question with almost no time spent! The technique is called “intelligent guessing”. Remember, in the standardized test we don’t have to find the EXACT answer – we need to find the RIGHT one. This can be done by eliminating the wrong results – sometimes requiring no computation whatsoever. Example:

The field of the size of 1800 square feet was divided between three brothers. The elder brother got half of the field, the second brother got two thirds of the remainder and the youngest brother inherited the rest. What is the total area that the young brother now possesses, if before this division process he already owned several fields totaling 500 sq. ft.?

1) 500 sq. ft
2) 800 sq. ft
3) 1000 sq. ft
4) 1400 sq. ft

Let’s solve this one by elimination and approximation. First of all, let’s draw the upper and lower “borders” for the possible answer. If the youngest brother already possessed 500 sq. ft, now he should own MORE than that. So, answer number (1) is wrong. Now let’s check the upper limit. It is obvious, that the younger brother inherited less than the whole field of 1800 sq. ft – thus his total cannot be over 1800+500=2300 sq. ft. Well, all the answer suit that criterion.

But, it is also clear that the younger brother inherited less than HALF of the field (as the elder brother got half and the remainder was divided between the two other brothers). Wait a second! It as the “middle” brother got MORE of the remainder (2/3), the youngest, actually, inherited less than QUARTER of the total field. So his total cannot be over 500+1800/4 = 950 sq. ft. So, we will now look for answer that is greater than 500 and smaller than 950. Fortunately, there is only ONE such an answer – (2) 800 sq. ft. I am not saying this will happen in the exam (although it might) – but you could definitely eliminate several wrong answers by this technique, making your chances of “guessing” correctly higher.

Here is another problem:
An experienced programmer can finish a project in 6 days. A novice programmer can complete it in 12 days. If they work on the project together, how much time will they require for completing it?

1) 15 days
2) 9 days
3) 5 days
4) 4 days

Let’s “draw the limits” once again. Of course, “15 days” is a wrong answer. Since an experienced programmer can complete the work in 6 days, when he is aided by the novice he will need less time than that. By the way, this eliminates the answer “9 days” as well. Let’s look further. IF we had two experienced programmers, they would work twice the speed of one experienced programmer, completing the project in 6/2 = 3 days.

IF we had two NOVICE programmers, they would work twice the speed of one novice programmer, completing the project in 12/2 = 6 days. But in reality we have ONE NOVICE and ONE EXPERIENCED programmers. So the answer should be somewhere in between the 3 and 6. Unfortunately, there are two answers that match the criterion – both (3) and (4). Ok, let’s take a guess – at least is it one of two answers, instead of four. We have 50% chance of guessing right! Well, my guess will be 4 days. And this would be the right answer – remember, the “power” of the faster worker is actually “higher” than that of the “slower” one.

So, generally, when we have the border of 3 and 6 we can usually say that the answer should be slightly below the average (which is 4.5 in our case).

Now, let’s solve this one “for real”: the working rate of the experienced programmer is 1/6; the working rate of the novice programmer is 1/12. Their total working rate is 16+1/12 = ¼. Thus, they will complete the project in 1/(1/4) = 4 days.


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