The 5 lb. Book: Quantitative Comparisons
Our latest book, the 5 lb. Book of GRE Practice Problems, just hit the shelves! The book contains more than 1,100 pages of practice problems (and solutions), so you can drill on anything and everything that might be giving you trouble.
We’ve already tried out a regular math problem. This time, we’re going to try out one of the weird Quantitative Comparison problems. If the question type looks unfamiliar, or if you just haven’t had much practice with QC yet, you might want to check out this introductory article first.
Alright, are you ready to try one? Give yourself about 1 minute and 15 seconds to do this problem. (Remember that these times are averages, not limits “ you can choose to take a bit longer, but don’t go beyond about 30 seconds longer than the average. At that point, all the extra time is telling you that you don’t really know how to do this one.)
0 < a < < 9
Quantity A Quantity B
9 “ a b/2 – a
© ManhattanPrep, 2013
Yuck. That inequality thing at the top doesn’t look fun. It might have been fine if it said 0 < a < b < 9, but I'm not really sure how to think about that b/2 piece.
Let’s see. So a itself is between 0 and 9. What about b/2? Here’s a cool little trick: when we have a multi-part inequality (an inequality with 3 or more pieces), we can just chop out two parts as long as we keep the correct relationship. So let’s look just at the last two parts: b/2 < 9.
I know what to do with that: multiply both sides by 2 to get b < 18. Okay, so b is between 0 and 18 and a is between 0 and 9. In addition, b/2 has to be bigger than a.
Wow. I haven’t even gotten to Quantities A and B yet! That does seem like a lot of work before we even begin solving, but these steps really are necessary. Understanding the given info (above the quantities) can make our lives much easier as we get deeper into the problem.
Turns out, there are two really elegant ways to solve this problem. (I put that word in quotes because it sounds funny to most people to talk about an elegant solution to a standardized test problem “ but we dorks in the standardized test industry really do mean it when we talk about an elegant solution. : ) )
Elegant Solution #1
In QC, we can manipulate the two columns against each other in almost all of the ways that we can manipulate an equation “ with one restriction. For instance, we can add the same amount to both sides, or subtract the same amount from both sides.
The restriction is the same one that comes into play when manipulating inequalities: if we multiply or divide an inequality by a negative, we’re supposed to flip the sign. When dealing with QC, don’t multiply or divide both Quantities unless you know that the number or variable is positive.
Compare the two quantities. Are there any direct connections between the two? Yes, both contain the variable a. Add a to both sides. What happens? The a‘s cancel out in both quantities! Then, Quantity A becomes just 9 and Quantity B becomes b/2. Can we tell which is bigger?
Yes, we can! That initial inequality at the top told us that b/2 < 9. Quantity A (9), then, must be the bigger value. The correct answer is A. Elegant Solution #2
Alternatively, draw a number line. Label 0 and 9 on the number line. In between, place a somewhere and also b/2 (just remember that a needs to be placed to the left of b/2).
Now, how could Quantity A, 9 “ a, be represented on the number line? If you’re not sure, test out some real numbers. Let’s say it really said 9 “ 5. The answer is 4 and the answer represents the distance between 9 and 5 on the number line. (Draw it out if you’re not sure.) Similarly, 9 “ a also represents the distance between 9 and a on the number line.
What about Quantity B? Even though we’ve got two variables now, the same principle applies: b/2 – a represents the distance between b/2 and a on the number line.
Just based on the above drawing, it looks like 9 “ a is longer. But is that always true? Is this the only way to draw this stuff?
We do have some leeway in how we place a and b/2 relative to the 9 but not that much. No matter what we do, b/2 has to be closer to a and 9 has to be further away from a. If that’s the case, then the distance from 9 to a is always going to be larger and the correct answer is (again!) A.
There you have it: two elegant solutions for this QC problem. Which one is better? Neither one is inherently superior. Choose whichever method makes the most sense or seems the most intuitive to you!
Key Takeaways for Quantitative Comparisons:
(1) If information is given up front (that is, above the two Quantities), figure out whatever you can about that info before you begin tackling the Quantities themselves. Understanding the given information before you start solving can make all the difference in your ability to get the question right or to do so in an efficient way.
(2) Remember the second word of QC: comparisons. Note that we didn’t solve for a and b here (and, in fact, there’s no way to do so!). Rather, we found a way to compare the two Quantities directly; this is the kind of solution we actually want.
(3) In most cases, there are multiple ways to get to a quant solution. If you don’t think you’ve found an elegant solution yet (or if you’re not entirely happy with the official solution provided in the answer key), keep hammering away. Part of doing well on a test like the GRE is figuring out the most effective and most efficient way for you.
© ManhattanPrep, 2013