While this problem isn't necessarily a difficult question, it highlights a great trick that the GMAT sometimes employs to deliberately confuse some students. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full "GMAT Jujitsu" for this question:
First, how the GMAT uses the "
%" symbol here is an example of a common problem structure:
temporarily-defined, abstract functions. Such functions often borrow symbols that would normally be seen in entirely different contexts. This is a deliberate, psychological attempt on the part of the GMAT to trap novice test takers. You expect the symbol to do one thing, but the GMAT defines it as something different. Now, we are actually accustomed to using abstract symbols to represent mathematical functions with a defined order of operations (after all, we know that "
a÷b" means "
take the value in the 'a' spot and divide it by the value in the 'b' spot.") Temporarily defining other characters to perform other mathematical operators is just the next level of this same idea. Any symbol can be temporarily defined as something else. For example, I have seen
%,
^,
&,
#,
@,
§,
◊,
@, and various characters of the alphabet used.
In the context of this question, the symbol "
%" doesn't mean "percent." The question
defines "
%" as a mathematical operator that takes two inputs: \(a\) and \(b\). For this question, for any given \(a\) and \(b\), \(a\)
%\(b\) is defined as \((a + b)(a - b)\). Once we understand the basic rules of the function, then the mathematics of the question become very obvious.
There are two main ways you can do this question. First, just
Do the Math, plugging in the values the problem gives you and then solving for \(x\):
\(a\)%\(b=(a + b)(a - b) = a^2 -b^2\)
(This is called the "difference of squares.")\(9 = 5^2 - x^2\)
\(x^2 = 25-9 = 16\)
\(x = +4\) or \(-4\)
(Don't forget the negative solution for \(x^2\))The answer is A.The second way to solve this problem is to
Back Solve, looking for the answer choice that would work for the equation
\((5 + x)(5 - x) = 9\). Since the answer choices are all integers, this means that as soon as we can tell that either
\((5+x)\) or
\((5-x)\) ISN'T a multiple of \(9\), we know that such an answer CAN'T be right.
A) \((5-4)(5+4) = 1(9) = 9\)
(This works, and so we don't even need to go further! The answer must be A.)B) \((5-3)(5+3) = 2(8) \neq 9\)
C) \((5+2)(5-2) = 7(3) \neq 9\)
D) \((5+3)(5-3) = 8(2) \neq 9\)
E) \((5+6)(5-6) = 11(-1) \neq 9\)
And
that is how you think like the GMAT.
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Aaron PondVeritas Prep Teacher of the YearHit "+1 Kudos" if this post helped you understand the GMAT better.
Visit me at https://www.veritasprep.com/gmat/aaron-pond/ if you would like to learn even more "GMAT Jujitsu"!