Bunuel wrote:
If the operation € is defined for all x and y by the equation x € y = 2*x*y, then 3 € (4 € 5) =
(A) 80
(B) 120
(C) 160
(D) 240
(E) 360
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:The exam is using the € symbol to stand in for another ad hoc equation, but the fact that your brain has to process this extra information is enough to throw some students out of their comfort zone. Added to this, the question does not ask for a single execution of this operation, but rather the resolution of a nested € equation. These foreign symbols may seem daunting, but remember there’s nothing here that wouldn’t be trivial without the bloated wording.
Let’s break this question down into its component parts. The symbol € is being defined for x and y as 2*x*y, which basically means take the two numbers together and multiply them. Once you’ve finished that, double the result, and you’re done. So if I ask for 5 € 10, I’d take 5*10, which is 50, and then double it. The answer would be 100. It’s relatively simple once you translate the equation into something meaningful, so we’re set up to execute a € equation on any two variables.
Of course the equation doesn’t give us only two variables, it gives us three. It’s logical to assume that the order of operation will matter here (hint: it actually doesn’t in this case), so we should start with the nested arguments before expanding outwardly. Within the bracket is 4 € 5, which would mean we multiply 4 by 5 and then double it, yielding a total of 20 * 2, or 40. The equation now reads 3 € 40, which means we again multiply together and then double, leaving a total of 120 * 2, or 240. Answer choice D is 240, so we have reached the correct answer.
Why did I mention that the order doesn’t matter? Because this specific example uses only multiplication, which is a commutative equation, or in other words: a x b = b x a. This isn’t always the case (think division), so it’s a good habit to always execute operations in the correct order. You may remember the mnemonic PEMDAS, which reminds you that the order of operations is {Parentheses, Exponents, Multiplication, Division, Addition, Subtraction}. In this instance the results would have been the same but that’s one more trap the GMAT test makers have at their disposal.
Another potential solution involves eliminating answer choices that cannot possibly work. If we look at the arguments provided, we have 3, 4 and 5, all of which need to be multiplied together. That product yields 60, which means that the correct answer choice must be a multiple of 60. Answer choices A and C can both be eliminated based on knowing that much. Perhaps from there you can recognize that this number needs to be doubled twice, leading you once again to answer choice D. However, this type of question is not particularly easy to backsolve unless you understand what is going on with the symbols.
In conclusion, people usually fail to correctly answer these questions because they get caught up in the abstract notation. The GMAT is a test about how you think, and the goal of many questions is simply to see if you can successfully navigate unfamiliar terminology. The same question, without the layering mechanism of the € sign would be significantly easier. Similarly, adding in another argument, such as squaring the parentheses, would appear to make this question significantly higher. In both cases, the questions should be solved in the same way, understanding the result of the symbol and methodically applying it to each argument. With some preparation, you can use your ease with these questions as a sign that you’re going to do well on test day.