toshpoint0 wrote:
Hi there -
I've been told to practice to test cases for specific Data Sufficiency questions. However, I have no idea how to set up the problems with multiple variables. Here are some examples.
OG2017 - DS327
If x is a positive integer, what is the value of Square root of (x + 24) - Square root of (x)?
1. Square root of (x) is an integer
2. Square root of (x + 24) is an integer
OG2017 - DS365
For any positive integer x, the 2-height of x is defined to be the greatest nonnegative integer n such that 2^n is a factor of x. If k and m are positive integers, is the 2-height of k greater than the 2-height of m?
1. k > m
2. k/m is an integer
I am hoping the experts here can help me figure this out whole "testing cases" strategy.
Testing cases / picking numbers is one of the more important skills you need to learn to master the GMAT.
There isn't a very large difference between multiple variables and single variables; only that at 3+ variables it is usually easier to just solve the question.
There 2 main ways to pick numbers:
Option 1: Flat out guess - this requires very little thought; just 'plug and chug'.
1. Pick a random number, see how it works.
2. Try to challenge the result of (1) by picking a different number.
For example, let's look at your first question:
In (1) you could guess x =1 and see that sqrt(x+24) - sqrt(x) = 5 - 1 = 4
Then you would challenge this by trying to find a different x that gives a different result, for example x = 4.
For (2) it's a bit harder to guess: you would need to say, 'let's guess that sqrt(x+24) = 5 and then solve this to get x = 1'
Then you could challenge this with 'let's guess that sqrt(x+24) = 6 and solve to get x=12.
Since you've succesfully challenged both results, neither is sufficient
Testing the combination by guesswork essentially amounts to checking values of x that fulfill (1):
x = 1 works in both equations; then next square is x = 4 which doesn't work; then x=9,16 which don't work and finally x=25 which does.
So you would mark (E).
Option 2: Try 'sophisticated guessing'; this requires some thought but can prevent missing cases.
1. Pick a 'likely number', for example something at the edge of your range or a known 'probelmatic number' like 0 or a negative
2. Try to infer a trend based on the result of (1). Use this trend to see if you can challenge the result.
For example, in your second question this would look as follows:
(1) The question asks about powers of 2 so let's guess that k = 2m. In numbers, if m = 1 then k = 2. In this case the 2-height of m is 0 and that of k is 1. If we choose a different factor of 2, say 4 then m=1 and k=4 and the 2-height of m is 0 and k is 2. The 'trend' in this case is that multiplying the factor by a power of 2 increases the 2-height. So we'll challenge this by trying to multiply by something that isn't a power of 2, say 3. So m = 1, k = 3 and they both have the same 2-height, 0.
(2) We'll do the same as above - say m=k=1 giving them the same 2-height. Since the question asks about powers of 2 we'll try a factor of 2, say m=1,k=2 and get different 2-heights.
Combined: By now we should have realized that factors of 2 and of 3 give different results. So we can test m = 1, k=2 and m=1,k=3 which fulfill both criteria but give different results. Therefore we mark (E)
In summary, the bottom line for testing numbers is to pick a number and then try to challenge your result with another number. You can do this with some thinking or just by flat-out guessing.
Hope that helps!
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