A couple of weeks ago I wrote a Revised GRE Math practice questions post. The math practice questions ranged from relatively easy to very difficult. At the time I provided answers but not explanations. One of my hopes was that students would be able to reason on their own why a certain answer is correct. I was also curious to see if anyone would post explanations.
Understandably, a few of you wanted explanations, whether you were stumped or because you were curious if your reasoning was valid. As for the second point, some one did post an explanation (thanks, Ahmad!)
Below are my explanations.
Question Type: Multiple Answer Questions (Choose all that apply)
What are all the possible solutions of | |x – 2| – 2| = 5?
If we focus just on the , we can see that the result must be positive. Stepping back and looking at the entire equation we substitute u for , to get . Solving for absolute value, we get the following:
Thus, and . Because u must be positive, we discount the second result. Next, we have to find in the original , which we had substituted with u. Replacing u with 7 we get:
A faster way is to plug in the answer choices to see which ones work.
Question Type: Multiple Choice
Concept: Symbolic Reasoning/Exponents
Level: 165 – 170
If is an integer which of the following must be an integer?
- None of the above
Let’s choose numbers to disprove each case. By the way, the word disprove is very important here – the question says ‘must’ so by picking numbers that prove the case, we are not necessarily proving that an answer choice must always be an integer.
For A. I can choose , and b is any integer. Because a is not an integer, A. is not correct.
For B. it’s a bit tricky. However, if you keep in mind that there are no constraints in the problem stating that a cannot equal b, we can make and .
For C. we can choose the same numbers to show that ab is not an integer.
For D. if and equals an integer, but does not.
Question Type: Numeric Entry
Concept: Prime Numbers/Factors
Level: 150 – 155
How many positive integers less than 100 are the product of three distinct primes? 
Let’s write out some primes: 2, 3, 5, 7, 11, 13, and 17.
I’m stopping at 17 because the smallest distinct primes, 2 and 3, when multiplied. by 17 give us 102. Therefore 13 is the greatest prime conforming to the question. Here is one instance. is greater than 100 so we can discount it.
Working in this fashion we can add the following instances:
Therefore, there are five instances.
Question Type: Quantitative Comparison
Concept: Exponents and Fractions
Level: 155 – 160
|Column A||Column B|
- The quantity in Column A is greater
- The quantity in Column B is greater
- The two quantities are equal
- The relationship cannot be determined from the information given
If x is less than 0 the answer is B. If x is , the answer is A. Therefore, the answer is D.
Question Type: Multiple Choice
Concept: Geometry/Variables in Answer Choices
Level: 160 – 165
A square garden is surrounded by a path of uniform width. If the path and the garden both have an area of x, then what is the width of the path in terms of x? (160 – 165)