In this video from the new GMAT tutorial series produced by PrepAdviser and examPAL we are going to focus on algebra by referring to both arithmetic and the use of variables, as both go hand in hand in the test.
Because there are so many subtopics, if you are going to study just one thing for the GMAT – algebra should be it. In fact, algebra is even more important than just being the most common subject on the test: the material that is included in algebra is the basis for all other quantitative subtopics. For example, we need to know powers and roots in order to use the Pythagorean theorem in geometry, and we have to have a keen grasp of ratio in order to tackle many word problems.
Algebra, more than any other topic, is something we are probably already quite familiar with from school. But while it’s obviously good to have some prior acquaintance with the material, this experience also has a downside. In school, algebra questions are usually solved with one strategy, and one strategy only: simplifying. But here, that impulse can spell trouble.
For example, take a look at the following equation:
X squared minus (y plus z) squared, divided by z plus y plus x.
So, if we want to simplify, we start by opening up the parentheses, which gives us x^2 minus all of that, whose negative we then need to take and we get that, and… quickly we find ourselves in a mess! We have actually drifted farther away from the answer choices.
For questions such as this, we have to use cognitive flexibility, which means looking at the question’s characteristics and thinking what could be the most efficient way to solve it. In the GMAT, “efficient” means “fast and correct”. In this case, the question has only variables, and thus we can take the alternative approach of simply choosing any numbers we want that make the equation work, and solving the question with them!
For example, we could say that x=1, y=2, and z=3 and, placing these values in the expression, we are going to end up with some specific answer that works with those specific values. In this case, when we simplify, we get a negative 6. Then, we are going to plug these values into the answer choices, hoping that one of them works and four do not. So, going through the answer choices and replacing x, y, and z, with 1, 2, and 3 – we get different results: 0, 0, -6, 4, and 2.
So, the only answer which works is (C). Do we understand why it worked? Not exactly. But that doesn’t matter! It’s enough for us to be able to eliminate the four other ones, since we found one example at least for which they definitely do not work. And so, we can confidently pick answer choice (C).
As we can see, this was much quicker and easier than simplification.
The important point, though, isn’t just that we can use numbers instead of variables. The point is that the GMAT is all about variation, and we always need to be flexible about how we choose the fastest way to solve each question.
For example, in the following question, a precise simplification is, once again, possible, but not at all easy.
For which of the following values of m is (m^2+ m+ 3) a prime number?
In this case, since we are asked about “which of the following values” does something, being cognitively flexible means simply trying the answers out and seeing which works.
In A, we get 2 squared plus 2 plus 3, which equals 9 – not a prime number.
B gives us 3 squared plus 3 plus, which is 5 – not a prime number either.
C is 4 squared plus s 4 plus 3, which is 23 – that is a prime number!
No need to check (D) and (E) – that’s our answer!
Not very hard at all! And we don’t even have to check all the answers this time – only one answer can be correct.
Sometimes, we don’t really have to solve at all, but only to think about the question logically.
Negative 99 plus Negative 98 plus Negative 97 plus Negative 96, and on and on right up to 97 plus 98 plus 99 plus 100, equals…?
Here once again, simplification is technically possible – but not likely to be rewarding. But let’s just look at the question logically, and look for patterns.
When summing up negatives and positives, we would be happy to find for every negative – a positive number that cancels it out. That is, two numbers that when added together, give us zero. We can do that for -99 and 99. We can also do it for -98 and 98, for -97 and 97, and on and on and on as we approach zero. Eventually, the only thing we will be left with is 100 – which is answer choice (B).
To summarize, algebra is a wide-ranging, absolutely essential GMAT topic. To handle it successfully, we have to both study a lot of material, and practice being flexible when it comes to picking the fastest solution strategy for each question.