Rephrasing Data Sufficiency Questions
Data sufficiency problems can be a lot of fun because we dont actually have to solve all the way to the end of the problem. At the same time, data sufficiency problems can be maddening because of the way in which the information is worded. Often, especially on harder questions, the question stem or statements in a data sufficiency problem are worded in such a tricky way that were not sure of the significance of the information after weve read it.
Todays lesson is all about how to Rephrase the information in a more useful way. (For those who have taken or are planning to take our class, the Rephrasing lesson occurs during class 1, though Ive changed the order in which the types are presented in this article.)
What is Rephrasing?
Rephrasing is simply finding an easier, clearer way to represent the information given in a data sufficiency question or statement. Ideally, we want to understand the significance of the question before we begin to evaluate the statements; if we can get to the heart of the information presented in the question stem, then weve made our task a lot easier when we start evaluating the statements because we have an idea of what would be useful (and what wouldnt) when looking for sufficiency.
There are four main types of rephrasing. Some of these may seem more obvious to you; some may seem less so. The important thing to remember is that you do want to invest a little time upfront in order to rephrase the information so that you save time when youre evaluating the problem.
Rephrasing Type 1: Translate English Into Math
This is the most basic type of rephrasing. Data Sufficiency is a quant question type, yet they often present information in full sentences. Forget English! I want that info in terms of math. One GMATPrep question has the following question stem:
A certain jar contains only b black marbles, w white marbles, and r red marbles. If one marble is to be chosen at random from the jar, is the probability that the marble chosen will be red greater than the probability that the marble chosen will be white?
What does that mean in math? Probability is represented by a fraction. The numerator of the fraction shows the number of ways in which the desired outcome can happen, and the denominator shows the number of ways in which any outcome can happen. In this case, the probability that a red marble will be chosen can be represented as r / (b+w+r), where r represents the number of ways in which a red marble can be chosen, and b+w+r represents the total number or ways that anything can be chosen. Similarly, the probability that a white marble will be chosen can be represented as w / (b+w+r). The question asks whether the probability of the former is greater than the probability of the latter:
Is [pmath]r/(b+w+r)[/pmath] > [pmath]w/(b+w+r)[/pmath]?
Rephrasing Type 2: Focus on the Minimum Needed
Every time you do any kind of manipulation or rephrasing on data sufficiency, the next question to ask yourself is: okay, do I need all of that? Whats the minimum that I actually need to know here?
Consider the GMATPrep problem we examined above. We translated the question into this form:
Is [pmath]r/(b+w+r)[/pmath] > [pmath]w/(b+w+r)[/pmath]?
Is that the minimum we need to know in order to answer that question? Or can we strip this down further? Try it out and see what you think.
We can strip this down further. The denominators of the two fractions are identical. Further, we know that b, w, and r are all positive. We can, therefore, simply cancel out the denominators. The new question becomes:
Is r > w?
Which question would you rather answer: the original form above, or this minimum needed question? :)
Rephrasing Type 3: Unravel the Ball of Yarn
Lets say were given this question:
Is ax - x(b + a) = 3 + bx?
When I look at that equation, I can see that there are three variables. Beyond that, though, Im not really sure what Im looking at or what I should be looking for when I evaluate the statements. Because I notice that the variables are not already combined (that is, there are multiple instances of each variable in the equation), Im going to try to manipulate, or rearrange, the equation to see whether that might be useful. Specifically, Im going to try to get all of the like variables together bs with bs, xs with xs, and as with as. First, I dont like the parentheses, so lets get rid of them:
Is ax bx ax = 3 + bx?
Oh, great, now I can combine the ax terms:
Is bx = 3 + bx?
Ah, and now the bx terms:
Is -3 = 2bx?
And I know that algebra is all about trying to isolate the variables, so Im going to do that next:
Is [pmath](-3/2) = bx[/pmath]?
Thats all theyre asking me? Whether I can find one specific value for the expression bx? Great! Id much rather deal with that than the original equation. (Note: a very important change occurred from the original equation to the final equation something we definitely would want to notice on data sufficiency. Whats that change? See the end of the article for the answer.)
Rephrasing Type 4: Unscramble the Code
This is the toughest type of rephrasing to do; to use it effectively requires advanced exposure to and study of the material so that you learn how to unscramble the code before you ever sit down to take the test. This allows you to recognize the code when you see it on the test, similar to recognizing a word or a particular symbol. If you dont recognize the code, your chances of getting the problem right in 2 minutes drop significantly.
Coded questions and statements are telling us something about some fundamental property of math, but in you guessed it code. For instance, a question might ask whether x equals y. A statement might tell me that xy < 0. That isnt enough for me to tell what x and y are, so I guess that information isnt sufficient to answer the question, right?
Look again. The statement is telling me that x and y multiply to a negative number. What must be true of two numbers that multiply to make a negative? One must be positive and one must be negative. What was the question again? Does x equal y? No! I dont know what x and y are, but if one is positive and one is negative, then they cant equal each other.
Heres another one: is [pmath]y^2[/pmath] less than y? Whats this one really telling me?
Most of the time, when you square something, it gets larger. Square 3 and you get 9. Square 4 and you get 16. Square -1 (negative one) and you get 1 (positive one). But this question asks whether the value gets smaller when you square it, not larger. When does that happen?
When you square 0 or 1, the number stays the same; thats different than the above pattern but not quite what we want yet. When you square a fraction between 0 and 1, then the number gets smaller. Bingo! The question is really asking: is y between 0 and 1, or is 0 < y < 1? I dont want to have to figure that out during the test, so from now on, Im going to remember this:
If I get a question or statement about a number getting smaller when its squared, then I know were talking about fractions between 0 and 1.
I recommend keeping a list of unscramble the code rephrasings that you discover. If you struggle to remember any, make flash cards.
What to remember for rephrasing on data sufficiency:
- Do look for opportunities to rephrase information in the question stem or the statements. The time you invest to do so will save you time when evaluating the question, and you will also be more likely to answer correctly (because you actually know what to look for when evaluating the question).
- For unscramble the code rephrasings, study these ahead of time. Your goal with these is to be able to recognize the code so that you dont actually have to spend time unscrambling anything during the test.
*Answer to the question asked at the end of the Rephrasing Type 3 section: the critical thing to notice is that the variable a drops out of the equation. We dont need to know anything about a in order to answer this question. Thats really important to recognize on data sufficiency; if we think we need to know something about a, well probably get the question wrong.
Copyright note: GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.