How to test cases for specific questions

Question

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 greate rthan 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.

AvinashT2303 · Answer

Cases can be made based on the option given in question

1. It says that SQ root of X is an integer which means it can be 1, 4, 9.....Since it has multiple option/cases, we cannot consider this option

now 2.

Its says SQ Root of X-24 is also an integer. Now add a number in 24 which makes the TOTAL a perfect quare. You will have to do some mental calc on this or you can write down each square like 2, 4, 9, 16.... and add 24 and see if the total is also a perfect SQ.

1+ 24 will be 25, SQ of 25 is 5
1 + 25 will be 49, SQ of 49 is 7

Again two different answers. If we add 1&2 again both has two options, 1 and 5, and therefore, both options are insuf and answer should be E.

If you liked it, gv me a kudos

Posted from my mobile device

DavidTutorexamPAL · Answer

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!

Bunuel · Answer

First question: https://gmatclub.com/forum/if-x-is-a-po ... 20428.html Second question: https://gmatclub.com/forum/for-any-posi ... 07706.html Number plugging: How to Do Math on the GMAT Without Actually Doing Math The Power of Estimation for GMAT Quant How to Plug in Numbers on GMAT Math Questions Number Sense for the GMAT Can You Use a Calculator on the GMAT? Why Approximate? GMAT Math Strategies — Estimation, Rounding and other Shortcuts The 4 Math Strategies Everyone Must Master, Part 1 (1. Test Cases and 2. Choose Smart Numbers.) The 4 Math Strategies Everyone Must Master, part 2 (3. Work Backwards and 4. Estimate) Intelligent Guessing on GMAT How to Avoid Tedious Calculations on the Quantitative Section of the GMAT GMAT Tip of the Week: No Calculator? No Problem. The Importance of Sorting Answer Choices on the GMAT For more check Ultimate GMAT Quantitative Megathread Hope it helps.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

How to test cases for specific questions

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards