Quantitative Comparison: Secrets of Surface Area and Volume in a Comparison Problem

It’s time to once again practice a problem for the GRE exam. This time, the professional GRE preppers at Testmasters bring you a step-by-step solution to solving surface area, volume, and quantitative comparison problems.

First, a brief note about quantitative comparison problems. For these problems, the questions can come from any area of math, but you will have to determine which quantity is larger, if both are equal, or if a relationship cannot be determined with the given information. Sometimes, you will have to calculate values for each quantity to compare them, but other times, you will not be able to find exact values because there may be variables involved. In these instances, it helps to know what happens when certain math operations are performed.

Today’s problem is a quantitative comparison problem that uses surface area and volume of a cube. The GRE does not provide any formulas, so you will have to know them before taking the exam. If you write the formulas down every time you use them, memorizing them becomes much easier.

Below are two quantities. If quantity A is larger, the answer is A. If quantity B is larger, the answer is B. If they are both equal, the answer is C, and if a relationship cannot be determined, the answer is D.

The length of a side of cube A measures 4, and the length of each side of cube B measures 3.

Quantity A

The ratio of the surface area of cube A to the volume of cube A.

Quantity B

The ratio of the surface area of cube B to the volume of cube B.

First, find the surface area of each of the cubes. The surface area is equal to sa formula. Plug in the side length for each of the cubes into the formula.

cube a sa

cube b sa

Now, recall that the volume of a cube is volume formula. Plug in the side lengths for each of the cubes into this volume formula to find the volumes of the respective cubes.

cube a v

cube b v

Because the question asked for the ratio of each cube’s surface area to its volume, divide the surface area of cube A by its volume and repeat the process for cube B.

cube a sa over v

cube b sa over v

Because the ratio of the surface area of cube B to its volume is greater than the same ratio for the surface area and the volume of cube A, the answer is (B).

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GRE Vocab and the Seven Deadly Sins: Part VII – Pride

An early sixteenth century Dutch depiction of the seven deadly sins, by a follower of Heironymus Bosch.

Finally, we come to the end of our series with the last and deadliest of the seven deadly sins: pride. While in English the word “pride” can often have a positive connotation, as in “the parents are proud of their child,” this kind of pride is something else altogether. In Latin, this sin is referred to as superbia, which perhaps gives a clearer indication of its nature than does the English equivalent. Pride is defined by the Catholic church as the belief that one is innately superior to others, especially in the sense that the sinner feels that he or she does not have to act with regard to the well-being of others because he or she is “better” than they are. Pride is thus a sin of selfishness and arrogance that leads one to feel that one is above the rules and that other people don’t matter. You begin to see why this is the worst one. According to Catholic theology, pride is the worst sin of all because it is the source of the other sins. It was, after all, pride that caused the angel Lucifer to rebel against God and become Satan, the Devil himself.

There are many excellent potential GRE vocab words that have to do with the sin of pride, including superbity, vainglory, hubris, haughtiness, hauteur, superciliousness, amour-propre, conceitedness, narcissism, and condescension. Continue reading “GRE Vocab and the Seven Deadly Sins: Part VII – Pride” »

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Quantitative Reasoning Sample: Making Percents Less Perplexing

For the GRE’s quantitative reasoning section, you will have to show your ability to work with both fractions and percents. Occasionally, problems will combine both, but because percents and fractions are closely related, even problems that use both can be solved. The GRE guides at Testmasters bring you today a walkthrough of a common type of percent and fraction problem you could see on the exam.

In an undergraduate program at a university, there were 300 graduates one year. By the time the class reunion rolled around 10 years after graduation, of those 300 students, 1 3 went on to get their masters degrees, and of those who earned a masters, another 25% continued onto higher education and got their doctorate degrees. Of the 300 original graduates from the undergraduate university program, how many had a doctorate degree at the reunion?

First, break this problem down into its components. Any time you see the word “of” in a fraction or percent problem, it indicates multiplication. Start with the initial amount of graduates and find the number who got their masters. Because the problem indicates that 1 3  of the original 300 got a masters degree, turn this into a math equation by replacing the word “of” with a multiplication sign. This makes sense, because in a fraction problem, fraction times the whole is equal to the part.


1 3 times 300

This means that at the reunion, 100 had at least a masters degree.

Next, the problem says that of those who got a masters degree, 25% went on to get a doctorate. Rather than using 300 for the whole, the whole group in this part of the problem is the 100 who got a masters degree. With percent problems, the same formula used for fraction problems applies.

fraction part


At the 10-year reunion, 25 of the original graduates will have a doctorate.

Now, you can conquer percent and fraction problems by remembering the equations:

fraction part


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How is Graduate School Funded?

Today we break down the ins and outs of graduate school funding!

Today we break down the ins and outs of graduate school funding!

As you’re looking into graduate school, you may be wondering “How do finances work as a graduate student? How is graduate school funded?” Well don’t worry! Today we’ll be discussing exactly that!

How are doctoral programs funded?

In general, if you are accepted as a doctoral candidate, you will receive a stipend with which to do your work. Part of this money comes from the school’s coffers, but the majority stems from your research advisor’s grant funding. This stipend won’t be a king’s ransom, and indeed you’ll likely be living at or below the federal poverty line, but you shouldn’t have to take out loans to attend a doctoral program. The exact stipend you receive varies greatly by school and especially by geographic location, but in general, you will have enough money for rent, food, and basic expenses. Many graduate students receive a roughly $22,000 stipend, which is again just enough to keep you afloat, with some beer money on the side. The important thing is you will likely not have to take out loans to attend graduate school, so though you won’t be making bank or saving a ton during these years, at least when you graduate, you will have a >$0 net worth!

Should I pay to attend graduate school for a PhD?

In general, NO. Most, if not all, reputable PhD programs are fully funded, so if you are expected to take out loans, this is a huge red flag. If a graduate program does not have enough grant funding or resources to support a graduate student, it’s likely not a very strong program. Academia is all about the research, and if the professors aren’t pulling enough grant funding to support even the most meager of graduate student stipends, they probably aren’t doing high profile work or research that would assist you in the future. Be very, very careful with pay-for-PhD programs as attending these likely won’t further your future interests!

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5 Things to Judge a Graduate Program By, Besides Academics

Here are some criteria you can use to rank the graduate programs you've applied to

Here are some criteria you can use to rank the graduate programs you’ve applied to

We’re following up last week’s post on What Graduate Programs Should I Apply to? with some more qualitative things to look at. We’ve already covered the “hard” stuff like academics and publications, so here’s some other things you should consider when looking at potential graduate sites!

  • Location: This is of course a self-evident criteria for choosing schools, but think long and hard about the area each school is located in. Do you want to live in a small college town in the southwest, or do you want to live in a more urban area on the East Coast? Are you okay with commuting with a million other people every morning, or do you want to get stuck on in an endless concrete spaghetti bowl? You’re going to be living in this location for 5+ years, so make sure you love the place!
  • Availability of collaborations: Is the school  you’re applying to a large one? Does it have many researchers in a diverse ranges of specializations? These may be questions you want to consider when narrowing down schools to apply to. Though a doctoral program will essentially have you doing your own work, it’s important to have the ability to bring in other researchers from other departments to help if necessary. Diversity is the spice of life and research as well!
  • Proximity to desired institutions: In general, if you want to work at an East Coast university post-graduation, you’ll want to attend school in that region. Just as last week’s post touched on the importance of word-of-mouth recognition, so too do we suggest that word-of-mouth is highly location-centric. A lower-ranked local graduate program is likely more respected than a marginally higher-ranked graduate program located halfway across the country. The closer you are to the institution you want to work at, the better your chances at having a leg up!
  • Local life: What kind of activities do you like doing in your free time? It’s important to promote some semblance of work-life balance, so make sure the local area fits your personality! If you like running or mountain biking, look for a school located near natural formations, and if you like being disappointed by local sports teams, make sure you choose Houston or Cleveland. Just make sure you have an ability to let off steam and find a life outside of your research. You’ll need it.
  • Cost of living: While most schools provide a stipend to live on, the distance your stipend will go depends highly on location. Rent is more expensive in San Francisco than Madison, Wisconsin, and a week’s worth of groceries in NYC might cost the same as a month’s in Charlotte, NC. You’re going to be bemoaning your lack of money in grad school anyway, but location may be the difference between crying into 1-ply tissues or 2-ply.
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GRE Vocab and the Seven Deadly Sins: Part VI – Envy

An early sixteenth century Dutch depiction of the seven deadly sins, by a follower of Heironymus Bosch.

In this, our latest post in the series GRE Vocab and the Seven Deadly Sins, we turn to the sin of envy and its corresponding heavenly virtue, kindness. According to Catholic theology, the sin of envy is defined as ill will for those whom one believes are better off than oneself. The Latin word the church uses for envy is invidia, which evolved into the Old French envie before entering English as “envy” in the late 1200s. There are a few good potential GRE vocab words related to envy that are worth mentioning, including invidious, covetous, begrudge, and jealousy.

Invidious, as you may have guessed, is derived from the Latin invidia. The reason it bears a greater similarity to its root than the word “envy” does is because it entered English straight from Latin in the first decade of the 1600s. While it originally simply meant envious, over time its meaning changed to causing or tending to cause not only envy, but also general animosity or resentment. For example: “The twins grew upset when their teacher invidiously compared their academic abilities.” In other words, they got mad when the teacher said one was smarter than the other, a comparison that would cause envy or ill will. Continue reading “GRE Vocab and the Seven Deadly Sins: Part VI – Envy” »

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Sample GRE Multiple Choice Math Problem – Too Big for the Calculator

Let's face it, you're nothing without your calculator.

Let’s face it, you’re nothing without your calculator.

On every GRE Math section, the test makers try to come up with a few extremely difficult problems that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following problem:

What is the remainder when 3^200 is divided by 5?

A) 0

B) 1

C) 2

D) 3

E) 4

At first you might think, “Hey, this is easy! I’ll just plug it into my calculator–”

3^200 = 2.656139889… x 10^95


My hammer totally isn't a crutch that has allowed me to forget my multiplication tables.

My hammer totally isn’t a crutch that has allowed me to forget my multiplication tables.

Question mark indeed. You see, these GRE test writers are truly diabolical. They know how much you love your calculator, so they’re always scheming up ways to separate you from it, in much the same way that Loki is always trying to separate Thor from his hammer. Of course, even without his hammer, Thor is pretty strong, and don’t worry – so are you! You’re just going to have to use your other secret weapon…your brain!

Let me show you how. In this case, they’ve made it so that you need to know the ones digit of an insanely large number that is too big for your calculator to display on your screen. However, we can still figure out what the ones digit of 3^200 is. Are you just going multiply it out by hand? No! Of course not! That would take too long. All you need to do is recognize a pattern. Consider the first few powers of 3:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729

3^7 = 2187

3^8 = 6561

3^9 = 19683

You may have noticed that the ones digits follow a pattern: 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1… over and over again. Every 4 powers the pattern repeats. Now we just need to figure out if 3^200 ends in 3, 9, 7, or 1. To do this, we just need to divide 200 by 4 and see what the remainder is. To see how this works, consider the following chart:

Consider: The Power is: Power/4 = Remainder = So ones digit =
3^1 = 3 1 1/4 = .25 1 3
3^2 = 9 2 2/4 = .5 2 9
3^3 = 27 3 3/4 = .75 3 7
3^4 = 81 4 4/4 = 1 0 1
3^5 = 243 5 5/4 = 1.25 1 3
3^6 = 729 6 6/4 = 1.5 2 9
3^7 = 2187 7 7/4 = 1.75 3 7
3^8 = 6561 8 8/4 = 2 0 1

200/4 = 50

200 divides evenly by 4, so there is a remainder of 0, which means the ones digit of 3^200 must be a 1. Any number that ends in a 1 has a remainder of 1 when divided by 5:

1/5 = 0 R1

11/5 = 2 R1

21/5 = 4 R1

Thus, the answer is choice B. If you know what to do, it takes only about 30 seconds to solve this problem. So you see, with practice, even the hardest problems on the GRE become easy. Check back here each week for more extra hard problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until  then, keep up the good work and happy studying!

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Testmasters GRE Online Course Sample Video & Testimonials

Testmasters has one of the most popular GRE online courses in the country. Testmasters GRE online course was developed by Testmasters’ most experienced instructors. As with the classroom course, the online course comes with a 10 point score increase guarantee!

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You can learn more about the Testmasters GRE online course here!

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