There have been a lot of posts here that focus a lot on the raw algebra, but there are not a lot of posts on quantitative reasoning. Of course, as you are studying for the GMAT, I do not want to undersell the importance of being able to use algebra. If you can't see another way to think about a Quantitative question, there is nothing wrong with some algebraic gymnastics. But the GMAT calls it "
Quantitative Reasoning" for a reason. Let's look at this question from that perspective...
First of all, there is been a fair amount of discussion in this section about not needing ANY information to solve for
\(u^2 + v^2\). But this assumes from the beginning that
\(u\) and
\(v\) must be integers because
\(11 = u^2 - v^2\). However, just because
\(u^2 - v^2\) is equal to an integer value, doesn't mean
\(u\) and
\(v\) must both be integers. For example, if
\(u=100\), then
\(v\) has a non-integer solution:
\(v = \sqrt{89}\). This is a perfectly valid solution if there are no other additional constraints. Therefore, the math, as it stands, is not sufficient by itself. You should know that Data Sufficiency questions are
NEVER sufficiently limiting without at least one of the two supplementary statements. That is not how Data Sufficiency problems work.
So, let's begin with our critical thinking takedown of this question. The problem tells us that
\(11 = u^2 - v^2\) and
\(y=2uv\), while asking us to solve for
\(u^2 + v^2\). (Don't let those other variables get in the way.
\(x\) and
\(z\) are really just placeholders.) Let's analyze each statement to see if it is sufficiently limiting to get to a single answer for
\(u^2 + v^2\).
Statement #1
Here is a simple algebraic approach to this statement (just to show it)...If
\(60=2uv\), then
\(uv=30\) and
\(v=\frac{30}{u}\). Therefore,
\(11= u^2-v^2=u^2-(\frac{30}{u})^2=u^2-(\frac{30^2}{u^2})\).
If we multiply both sides by
\(u^2\) to cancel the denominator and bring everything to one side, we have:
\(u^4 - 11u^2 - 30^2 = 0\)This is similar in shape to a quadratic that we can factor. It turns out to be:
\((u^2 - 6^2)(u^2 + 5^2) = 0\)Since
\((u^2 + 5^2) = 0\) only has imaginary solutions, we can see that
\(u\) =
\(6\) or
\(-6\). Plugging these values into the original equation gives us two pairs of
\((u,v)\) solutions:
\((6,5)\) and
\((-6, -5)\). And yet, because the problem is asking us for
\(u^2 + v^2\), both sets of solutions result in the same solution:
\(36 + 25 = 61\).
Statement #1 is sufficient.Here is the "critical thinking" approach...However, it is possible to avoid all of that algebra. Statement #1 tells us that
\(y=60\). Therefore,
\(60=2uv\) and
\(uv=30\). This is massively limiting. Think about it... the hypothetical solution above (
\((u,v) = (100, \sqrt{89})\))
cannot be an option if
\(uv=30\). Whenever you square two different numbers, the difference between these numbers will be magnified. The only exception to this is if both numbers are fractions less than 1.
\((u,v)=(30,1)\) is too extreme, since
\(30^2 - 1^2 = 899\). The original difference between
\(u\) and
\(v\) must be very small in order for
\(u^2 - v^2\) to be equal to only
\(11\). Plus, because the problem asks us for
\(u^2 + v^2\), we don't have to worry about negative values. No matter what, the negative options for
\(u\) and
\(v\) would be cancelled out for purposes of this problem. We must be looking for a single solution where
\(u\) and
\(v\) are close together. Logically speaking, if we don't care about negatives there can be only one possibility where
\(u^2-v^2 = 11\). While this does turn out to be an integer solution, (
\(6^2 - 5^2=36-25=11\)), we don't even need to know that
\(u\) and
\(v\) must be integers for this to work.
Statement #1 is definitely sufficiently limiting.Statement #2
This statement is a lot easier to think about, and the algebra isn't that bad. If we know that
\(u=6\), then:
\(11 = u^2-v^2= 6^2 - v^2\)Solving for v gives us:
\(v^2 = 25\) and
\(v=\pm 5\). We now know both
\(u\) and
\(v\). Therefore, we can determine
\(u^2+v^2\).
Statement #2 is also sufficient.With both statements being independently sufficient, the correct answer to this question is "D".
Now, for those of you that are studying for the GMAT, let's step back a little bit to identify the strategies that got us to the answer. Thinking at the "strategy" level will allow you to solve numerous questions on the GMAT, not just this one. First, recognize that even exponents (most commonly "squares") can simultaneously hide and/or eliminate negative options. Second, many quantitative questions aren't about the algebra at all. If you find yourself slogging through a massive amount of math, there might be a quicker way to think about the problem. After all, the GMAT calls this section "
Quantitative Reasoning" for a reason. Of course, if you are an algebra wiz and you can finish off a question algebraically without it costing you too much time, go for it (especially on the test, when test anxiety might cause you to miss the more holistic solutions.) But while you are studying, be on the constant lookout for quicker, holistic ways of thinking about the problems that you can add to your strategic arsenal. That is what I call "
GMAT Jujitsu."
Attachments
File comment: Shows the conceptual reason for the "critical thinking" approach to Statement #1
Number Line Visualization.png [ 8.93 KiB | Viewed 37378 times ]