Great post gmatfeelizzle and we'll gladly help you out with your query. So let's get started :
We are looking for the value of x + y.
Now in value-based DS question, you should expect to get one and only one value per question.
If, when solving the question, you stumble on two distinct values, then the answer will most likely be E.
Statement 1 :\(x*y = 24\)
This statement alone is insufficient because we can think of a multitude of possibilities for both x and y that yield the result given (24) but when plugged into x + y, it will yield different results. For instance :
x = 4 and y = 6 =>\(x + y = 10\)
x = 3 and y = 8 =>\(x + y = 11\)
And so on. You should also think about negative integers as well (since the DS question stem doesn't seem to indicate any signum constraint on the variables we're manipulating).
Statement 2 : \(\frac{x}{y} =\frac{8}{3}\)
Again the statement alone is insufficient for the same reasons as statement 1. We can think of many values for x and y that yield 8/3 but when plugged into x + y, it will yield different results.
Let's combine (1) and (2).
From 2 we get : \(x = \frac{8}{3}*y\)
Substituting in 1 yields : \(x*y = 24\)=> \(\frac{8}{3}*y*y = 24\) => \(y^2 = 24*\frac{3}{8}\) => \(y^2 = 9\)
At this point, you are tempted to say that\(y = 3\), but since
we have no signum constraints on the variables in hand, y can be 3 just as it can be -3.
Remember that even powers hide the base's sign !!!So if \(y = 3\), then \(x = \frac{8}{3}*y = 8\). Yielding : \(x + y = 11\)
If \(y = -3\), however, then \(x = \frac{8}{3}*y = -8\). Yielding : \(x + y = -11\)
So even when combining the two statements,
you get different results !! And that clashes with the basic outcome of a value-based DS question
: if the question is solvable, then the value is unique ! Otherwise, the answer will almost always be E.Which is actually the case in this question.
The answer was deliberately lengthy so as to answer your query about why "C" isn't the right answer.
Hope that helped.