Math in Critical Reasoning Questions
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. (Theres math on Critical Reasoning?!?) Without further ado, heres 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 cant actually conclude anything definitive about the real numbers when were 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, dont 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 havent 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, theres 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
[pmath](0.15F)/0.15[/pmath] > [pmath](0.07T)/0.15[/pmath]
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 Chriss 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 whats 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 theres 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 isnt necessarily the case that a particular proportion tells us something about real numbers (or vice versa).