This is a classic question type that demonstrates a pattern common to many questions on the GMAT. Several of the approaches in this forum focus blindly on the math, but remember: the GMAT is a critical-thinking test. The tactics I will show you here will be useful for numerous questions, not just this one. For those of you preparing for the GMAT, my solution is going to walk through not just what the
answer is, but how to strategically
think about it. As a result, I will probably include some steps that I would normally just do in my head if it were the actual test, but I want to be as thorough as possible so you can see each step of the process. Ready? Here is the full "GMAT Jujitsu" for this question:
First, we need to make sure we understand the difference between the phrases "
directly proportional" and "
inversely proportional." Both of these terms describe relative relationships between two variables (or, potentially, relationships between entire chunks of equations.) For purposes of this discussion we will relate
\(x\) to
\(y\).
If \(x\) and \(y\) are "
directly proportional", it means that the relationship between \(x\) and \(y\) can be represented by the equation
\(x = Ky\), where \(K\) is a constant called the "coefficient of proportionality" or the "constant of variation." For example, the circumference of a circle is directly proportional to the circle's diameter (\(C = \pi d\)), with \(\pi\) being the "coefficient of proportionality." Some people oversimplify this rule by saying, "when \(x\) increases, so does \(y\)." However, that isn't always accurate. It is possible for \(K\) to be negative, meaning that as \(x\) increases, \(y\) would actually
decrease. (Picture a line drawn in coordinate space with a
negative slope and you can visualize this quickly. Negative linear slopes are still
directly proportional.)
If \(x\) and \(y\) are "
inversely proportional", it means that the relationship between \(x\) and \(y\) can be represented by
\(xy = K\), with \(K\) still serving as the "coefficient of proportionality." Notice that in this case, in order for the product \(xy\) to always equal the constant, \(K\), as \(x\) increased, \(y\) would have to decrease proportionally, in effect "cancelling out" the change in \(x\). Another way to write this relationship would be
\(x = K(\frac{1}{y})\). With the equation in this form, you should be able to see why they call it "
inversely proportional" instead of "
negatively proportional." \(x\) is proportional to the
inverse of \(y\) (in other words,
\(\frac{1}{y}\)), not the negative of \(y\). This relationship isn't linear, but is actually curved in coordinate space.
Now that we have the basics, let's take a look at this specific question. It states, "The variable \(x\) is inversely proportional to the square of the variable \(y\)." This means that:
\(x(y^2)=K\). Alternately, we can also visualize it as
\(x = K(\frac{1}{y^2})\).
The problem then states that we will be manipulating \(y\) by dividing it by \(3a\). This will make \(y\) (and thus \(y^2\)) proportionally smaller. (Of course, we don't know what "\(a\)" is, but this is a good way to visualize what is happening.) Since \(x\) and \(y^2\) are inversely proportional, this means that anything we do to \(y^2\), we must adjust \(x\) in the
proportionately opposite way to cancel out the change and keep \(x(y^2)\) equal to the constant, \(K\).
Thus, when we divide \(y\) by \(3a\), we change \(y^2\) to:
\((\frac{y}{3a})^2=\frac{y^2}{9a^2}\)
The "new" \(y^2\) is now divided by \(9a^2\). In order to reverse this change out, this means that \(x\) would need to be multiplied by something so that \(x(y^2)\) still equals the constant, \(K\). For purposes of visualization, I will call that something "?". Here is what it would look like mathematically:
\((x*?)*(\frac{y^2}{9a^2}) = x(y^2)\)
Solving for "\(?\)" allows us to cancel out \(x\) and \(y^2\) from both sides of the equation, and moves the \(9a^2\) in the denominator over to the other side.
\(?=9a^2\)
Thus, the factor by which we must multiply \(x\) by to maintain the "inversely proportional" relationship is \(9a^2\). The answer is "
E".
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Aaron PondVeritas Prep Teacher of the Year / Professionally mentoring GMAT students since 2006 Visit me at https://www.veritasprep.com/gmat/aaron-pond/ if you would like to learn even more "GMAT Jujitsu"!