Great question - the more-theoretical number properties items allow the GMAT to add quite a bit of difficulty, as does the Data Sufficiency format, so the combination can be tricky, and waste a lot of your time. A few thoughts on that:
-Plugging in numbers is a perfectly viable strategy, but as you mentioned, it can be inefficient if not done properly. Here's what I'd suggest for plugging in numbers (and for training yourself to not have to plug them in as often):
1) Your goal when plugging in numbers should always be to "prove insufficiency". That is, once you've gotten an answer of "yes" to the overall question, your only goal should be to get "no" with your next set of numbers.
2) Keep in mind that the GMAT loves to test the "fringes" of its parameters, so pick numbers that push the boundaries of what is possible in each case as you seek to prove that the statement is not sufficient.
3) As you're picking those numbers that push the boundaries in order to prove insufficiency, think about the types of numbers you're picking. Often if you're doing that, you'll see the 'catch' in the question.
Let's use your first example as our example:
The numbers x and y are not integers. The value of x is closest to which integer?
(1) 4 is the integer that is closest to x+y
(2) 1 is the integer that is closest to x-y
Neither statement one nor statement two is remotely sufficient, as the numbers could take on a huge range: -1,000,000.5 and 1,000,004.6, or 4.1 and 0.2 in the first case. So let's start by discussing the combination of statements 1 and 2.
Again, we want to pick numbers that test the boundaries. If x + y is closest to 4, then it gives us the range of 3.5000001 to 4.49999999, or essentially from 3.5 to 4.5. If x - y is closest to one, it gives us the range of 0.5000001 to 1.49999999.
If we pick the lower estimates for the two statements, we could have:
x + y = 3.500001
x - y = 0.500001
2x = 4.0000002
x = 2.0000001, so it's closest to 2.
Now, we want to find a different value for x so that it's closer to 3, or another integer. Well, using higher numbers should get us there:
x + y = 4.4999999
and if we want x to be as high as possible, we'd want y to be as small as possible, so the difference between the two would be largest, so we'll use the largest possible difference:
x - y = 1.4999999
2x = 5.999999999
x = 2.9999999 or closest to 3.
Therefore, the statements are insufficient.
Now, in doing the math, you'll notice I took some liberties with the decimals, because they didn't matter much - I didn't need to be precise, as long as I stayed within the constraints. Similarly, we probably didn't even need to do all of that math. Once we realized that the sum gave us a range of almost a full unit (3.5 to 4.5, exclusive) and the difference gave us the same range (0.5 to 1.5 exclusive), then we should have seen a path to get a range of x values that would cross over a .5 threshold to bump us from closer to the lower number to closer to the higher one. If we see that, we don't even do the math, and it's really the setup that lets us know that the correct answer is E.
What I like about the idea of "trying to prove insufficiency" is that it forces you to think about those limits that the GMAT often approaches with its ranges of potential values. As you're thinking about that, you can quite often see the rule and not have to finish the math; however, even if you do end up needing to do the math, you're at least being efficient, as you're trying to push the limits of those numbers to achieve an insufficient answer.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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