Math in Critical Reasoning Questions
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Stacey is a GMAT Instructor living in Montreal. Click here to read more articles from Manhattan GMAT and to learn more about Manhattan GMAT's classes. |
We have a guest-author this week: Chris Ryan, Manhattan GMAT’s intrepid Director of Instructor and Product Development. If you’ve used any Manhattan GMAT material in your prep, Chris is the man to thank: he may have written it, proofed it, managed it – somehow, he has touched everything that Manhattan GMAT publishes. This week, Chris is giving us some pointers on how to deal with math (math?) in Critical Reasoning questions. (There’s math on Critical Reasoning?!?) Without further ado, here’s Chris:
A very common flaw on the GMAT, and in life, is the confusion of absolute numbers and percentages. For example, which is larger, one-third of x or one-half of y? Without any information to compare x and y, we cannot answer this question. It is true that one-half is larger than one-third when applied to the same quantity, but when applied to quantities of different sizes, one-third could be much larger than one-half. For example, one-third of the population of New York City is a greater quantity than one-half the population of Boise, Idaho.
How does this play out on the GMAT? Consider the following argument:
At any given time, approximately fifteen percent of all homes in Florida are on the market. In Texas, however, only seven percent of all homes are on the market at any given time. Therefore, one will have a wider selection of homes to choose from if one looks for a home in Florida rather than in Texas.
This argument falsely assumes that the number of homes for sale in Florida is greater than the number of homes for sale in Texas, based on the fact that a larger proportion of homes in Florida are for sale. Imagine, however, that there are only 100 homes in Florida, yielding an available housing stock of 15 homes (because 15% of 100 = 15). And imagine that there are 1000 homes in Texas, yielding an available housing stock of 70 homes (because 7% of 1,000 = 70). In this particular case, the conclusion of the argument would not hold true. We can’t actually conclude anything definitive about the real numbers when we’re only giving information about percentages.
(Bonus Question: At least what percentage of the number of homes in Texas would the number of homes in Florida have to be in order for the argument to hold true? Answer found at end of article.)
The relationship between number and percent can also go the other way. Consider the following argument:
More people in California own air conditioners than do people in Illinois, Indiana, and Ohio combined. Therefore, Californians are clearly more concerned with their physical comfort than are people in those other three states.
This argument falsely assumes that the percentage of people who own air conditioners is higher in California than it is in Illinois, Indiana, and Ohio together, based on the fact that the number of people who own air conditioners is greater in California. Imagine, for example, that the population of California were 10,000,000, of whom 1,000,000 owned air conditioners – representing 10%. Imagine as well that the combined population of Illinois, Indiana, and Ohio were 1,000,000, of whom 900,000 owned air conditioners. (SK note: If you know the real populations of these states, don’t use them – use numbers that make your life easy!)
Now, it would indeed be true that more people owned air conditioners in California, but it would represent only 10% of the population, whereas 90% of the population of the other states owned air conditioners. In these circumstances, it would be difficult to maintain that Californians care more about their physical comfort (and we haven’t even discussed the different climates in these states!). When dealing with arguments that involve comparisons of quantities and/or percents, be sure you determine whether the comparison is valid. Most of the time, there’s a trap in there somewhere.
(Answer to Bonus Question: In order for the argument to be valid, it would have to be true that the figure representing 15% of homes in Florida is greater than the figure representing 7% of homes in Texas. We can represent this as an equation: 0.15F > .07T, where F = total homes in Florida and T = total homes in Texas.
If we isolate F, we get:
0.15F > .07T
> 
F > 0.47T
Therefore, the total number of homes in Florida has to be greater than 47% of the total number of homes in Texas.)
Major take-aways from Chris’s article:
- If you see a CR premise that gives real numbers and then the conclusion discusses a proportion / percentage, or vice versa, be skeptical.
- Test the given information to see whether you really can conclude what’s claimed in the argument. You can even test real numbers to see how things work – just make sure that you follow any constraints given by the problem (just as you would on any math problem!).
- If there’s a flaw in the comparison between proportion / percentage and real numbers, then the correct answer is likely going to hinge on that fact somehow. The author is assuming that the conclusion part (e.g., real number) does actually follow from the premise (e.g., proportion) – but now you know that it isn’t necessarily the case that a particular proportion tells us something about real numbers (or vice versa).
8 comments
suman on February 1st, 2010 at 8:26 am
Really nice !!
Akalpita on February 1st, 2010 at 3:19 pm
That makes total sense!! Thank you for very good article.
prinit on February 1st, 2010 at 4:59 pm
Thanks for bringing this article from the Creator's shelf. Great Piece of information.
rahul on May 22nd, 2010 at 5:04 am
can u help me with this problem stacy
Ronald: According to my analysis of the national economy, housing prices should not increase during the next six months unless interest rates drop significantly.
Mark: I disagree. One year ago, when interest rates last fell significantly, housing prices did not increase at all.
It can be inferred from the conversation above that Mark has interpreted Ronald’s statement to mean that
(A) housing prices will rise only if interest rates fall
(B) if interest rates fall, housing prices must rise
(C) interest rates and housing prices tend to rise and fall together
(D) interest rates are the only significant economic factor affecting housing prices
(E) interest rates are likely to fall significantly in the next six months
Stacey Koprince on May 22nd, 2010 at 7:47 am
Yes! But not here.
Discussions about separate problems need to be posted in the forums.
Post this problem in the forums and MAKE SURE to cite the author of the problem. Also make sure that you tell me what you think about this problem - what do you think you understand, what answer choices did you eliminate, and what questions do you have?
Then send the link to me via PM. I will follow the link and we can discuss!
p.s. If you don't cite the author or tell me what you think about the problem, then I will just ask you again in the post before I answer, so save yourself some time and do those things in your first post.
rahul on May 23rd, 2010 at 8:38 am
i m really new to the blogging and forum so i don't exactly know what to do ... the problem i 've posted here is from 1000 CR (test c).....about elimination i did eliminate A (for extreme ans) and C fro it isn't exactly supported by the evidence..... i m not exactly sure wat to do ...even D doesn't appeal to me.....pls help me if u can n i apologise for all the inconvenience i've caused
Stacey Koprince on May 23rd, 2010 at 1:58 pm
It's okay that you're new to the forums and not entirely sure what to do. Just go into the "forums" part of the site and look for the GMAT Verbal --> Critical Reasoning folder. When you get into that folder, you can create a new post.
Once you've created the post, it will have a URL (a web page address). Then you can copy the URL and go into the Private Messaging (PM) system - this is like an email system, except it's through the BTG web site. (You can find the link in the upper-right corner of any BTG web page - it'll either say that you have no new messages or that you have some number of new messages.)
Open up a new PM, address it to me, write a short message, and paste the URL of your post into the message. When I get it, I'll follow the link to find your post and then I'll respond.
rahul on May 24th, 2010 at 6:41 am
thanks a lot ....u r an angel