The concept of abstraction involves taking things from specific values to general ideas. On the GMAT, abstraction is one of the simplest ways to turn an easy problem into a difficult one. A simple example would be to ask someone what “5 times 6” would be, and then to expand that to “x times y” or “odd number times even number.” Abstraction helps by giving broad strokes to concepts, but it also requires a deeper understanding of the underlying principles. (This is the same principle as abstract art… apparently).
The GMAT is known for employing abstraction to make simple questions harder to grasp. Sometimes, a concrete problem using specific numbers can be very difficult, but the difficulty lies in the execution of the solution. An abstract problem, however, introduces an entirely different level of complexion, where even understanding the question at hand isn’t obvious (think of a Georgia O’Keefe painting). Once you’ve figured out what the problem is asking, then you can go about solving it. But until then you’re scratching your head wondering what the next step could be.
There is a lot of value in understanding the abstract, overarching theme of a question. After all, instead of saying that 2 + 2 gives you an even number, and 2 + 4 gives you an even number, and 2 + 6 gives you an even number, you can summarize that the sum of any two even numbers will be even. Once you understand this principle, it makes all future questions on this topic easier to solve. However, if you happen to see something on test day that you’re unfamiliar with, you might be better off concentrating on the question at hand than the unbreakable rule that guarantees the consistency of the answer.
As such, digging into why problems work is important during the time you prepare for the GMAT, so that problems seem easier on test day. Let’s explore one such relatively simple problem, made difficult by the abstract phrasing of the question:
If the operation ∆ is one of the four arithmetic operations addition, subtraction, multiplication, and division, is (6 ∆ 2) ∆ 4 = 6 ∆ (2 ∆ 4)?
- 3 ∆ 2 > 3
- 3 ∆ 1 = 3
A) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.
B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.
C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.
D) Each statement alone is sufficient to answer the question.
E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.
Data sufficiency questions tend to be somewhat abstract on their own because they are asking whether something is sufficient or not. There aren’t specific values you are being asked to evaluate, but rather the entire spectrum of possibilities. To make things even more abstract, the question is asking about some equation ∆ (which looks isosceles to me), which could represent any of the four basic operations. This question is very abstract, and contains a pitfall or two if you’re not careful.
Before even looking at the statements, let’s revisit the equation in the question:
(6 ∆ 2) ∆ 4 = 6 ∆ (2 ∆ 4)
This equation is actually asking about the commutative property of operations, because the numbers are all the same, but the order of operations is different. Replace all the ∆ operations by +, and we quickly see that the answer is 12 on both sides. You may already know that addition and multiplication are commutative, whereas subtraction and division are not (and this holds for all problems, so it’s a great shortcut). However, we may as well demonstrate it to ourselves here:
(6 + 2) + 4 = 6 + (2 + 4) –> 8 + 4 = 6 + 6 –> 12 = 12. This holds, meaning the operation is commutative.
(6 x 2) x 4 = 6 x (2 x 4) –> 12 x 4 = 6 x 8 –> 48 = 48. This holds, meaning the operation is commutative.
(6 – 2) – 4 = 6 – (2 – 4) –> 4 – 4 = 6 – (-2) –> 0 = 8. This doesn’t hold, meaning the operation is not commutative.
(6 ÷ 2) ÷ 4 = 6 ÷ (2 ÷ 4) –> 3 ÷ 4 = 6 ÷ ½ –> ¾ = 12. This doesn’t hold, meaning the operation is not commutative.
This means that we will have sufficient data if a statement can narrow down the choices to any one operation or to either multiplication & addition or division & subtraction. The data will be insufficient if we cannot narrow down the operations or have at least one commutative operation (x or +) and a non-commutative operation (- or ÷) as possibilities.
Next, we must look through the statements and see what information we can glean. For simplicity’s sake, I’m going to begin by evaluating statement 2. This is because the equation will yield less abstraction than the inequality of statement 1. If the ∆ equation can satisfy this equation, it’s a possible answer. If it cannot, we can remove it from the list of potential equations.
Statement 2 says that 3 ∆ 1 = 3. We can replace this by the four basic equations and see which ones hold:
3 + 1 = 3 –> This should give 4. Doesn’t hold. Eliminate addition.
3 – 1 = 3 –> This should give 2. Doesn’t hold. Eliminate subtraction.
3 x 1 = 3 –> This should give 3. Holds. Keep multiplication.
3 ÷ 1 = 3 –> This should give 3. Holds. Keep division.
You may be able to quickly ascertain that addition and subtraction do not hold for this equation, so only multiplication and division could work. Since we have two operations that could work, one of which is commutative and one of which is not, we can definitely say that this statement is insufficient.
Moving on to statement 1, we approach it in the same way and see if the operations can hold (i.e. the answer is greater than 3):
3 + 2 > 3 –> This gives 5. Holds. Keep addition.
3 – 2 > 3 –> This gives 1. Doesn’t hold. Eliminate subtraction.
3 x 2 > 3 –> This gives 6. Holds. Keep multiplication.
3 ÷ 2 > 3 –> This gives 1.5. Doesn’t hold. Eliminate division.
For this statement alone, we see that addition and multiplication both work, but the other two equations don’t. This means that we don’t know exactly which operation this ∆ represents, but either way it will give the same answer to the question given. The two operations left standing (last operation standing?) both yield the same answer to the statement, which means we don’t need to narrow down the choices or put the statements together. A common pitfall on this question is to put the statements together, because then only multiplication can work for both statements. However, that’s a trap, as you don’t need statement 2 at all. The correct answer is A, because statement 1 is sufficient on its own to answer the question posed.
For abstract problems, it’s easy to get lost in the generalization of the problem. What happens whenever I add two even numbers together? The magnitude of the scope is almost overwhelming, and as such the best strategy is to turn it concrete using simple examples. If no numbers are provided, try picking small, useful numbers like 2, 3 and 10. If the numbers are given but other variables, such as the operations, are left blank, then just go through all the possibilities until the rule becomes clear. The best way to overcome abstraction is to make it concrete.
Ron Awad is a GMAT instructor for Veritas Prep based in Montreal, bringing you weekly advice for success on your exam. After graduating from McGill and receiving his MBA from Concordia, Ron started teaching GMAT prep and his Veritas Prep students have given him rave reviews ever since.