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GRE prep | Quantitative | GRE Challenge

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GRE Students, take your marks... Get set...

Last week we introduced a four-question Data Interpretation Challenge Set. As we noted then, per numerous student reports, data interpretation problems have been an area of unexpected difficulty on recent GRE examinations. To prepare effectively for the GRE, you must:

  • Master fundamental math and verbal concepts.
  • Practice and implement effective problem solving and reasoning skills.
  • Challenge yourself to tackle borderline problems, the difficult problems that will make or break your desired GRE score.

All three of these steps are essential, but they are not as obvious as one might think. Some students feel comfortable memorizing facts. Either on their own or in a class/tutoring context, they wish primarily to know which equations to commit to memory, which vocabulary list is the best, etc. These are critical skills, but without implementation and strategy, this strategy alone will lead students to run out of time or feel overwhelmed by a surfeit of disorganized knowledge.

Others memorize the facts and core knowledge and work on learning problem solving skills and shortcuts. These steps in tandem lead to much greater success; accuracy and time improve; patterns and common structures become more familiar; students achieve significant, reliable score improvements and gain confidence.

However, the third step, the challenge step, is where you can break through the final wall to master the GRE. Using your core math knowledge and appropriate problem-solving skills, push yourself to solve problems that you find daunting or impenetrable. Each time you discover on your own a solution to such a problem, you shatter a metaphorical ceiling in your score, and there is no limit to your performance.

To conclude this lengthy preface, please attempt the four problems from last week’s set before you consult the solutions below. Remember, the solutions are each just one possible approach to each of these problems. Any way that you find that gets you to the right answer is a valid solution, and if your approach gets you to the correct answer faster than the approaches outlined below, your approach is the best approach for you. 


Get the Pictures

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I like to compare Charts and Graphs problems to Reading Comp for Quant. The comparison is apt because just as with Reading Comp problems, it’s often a good idea to get a sense of what’s going on before you dive into the problems. Let’s take another look at the data:

The top left chart gives two data points, “Annual Total Days Spent on Recreation,” Hunting and Fishing. Numbers are given in millions.

Top right is “Total Annual Expenditures on Recreation” with three data points: Hunting, Fishing, and Unspecified. Dollar amounts are given in billions.

The bottom is a Venn Diagram with overlapping groups of anglers (people who fish) and hunters. The actual numbers are given as 33.1 million anglers and 13.7 million hunters. There is a total of 37.4 million people who either hunt or fish.

Each time you attempt a charts or graphs problem, you must determine which diagrams are applicable.


My Cousin Venny

Let’s look at the first problem.

HF problem 12.pngLook familiar? We discussed a similar situation not long ago on another blog post. In this case, let’s determine which charts or graphs to use and RECORD WHAT YOU KNOW™.

Since we’re talking just about number of people who fish or hunt, we’re looking only at the bottom chart. Now let’s record what we know. On our scratch paper, we might jot down:

  • Anglers: 33.1 million
  • Hunters: 13.7 million
  • Total: 37.4 million

These numbers don’t seem to add up! 33.1 + 13.7 ≠ 37.4. What are we missing? The people who both hunt and fish. Maybe we don’t remember exactly what to do in a situation like this. No problem! Just start doing what you can with the numbers. Keep your momentum moving forward. Maybe 33.1 + 13.7 ≠ 37.4 so what do they add up to? 46.8 million (don’t forget the units). Clearly we can’t have a bigger total than the total, but maybe we can just see if there’s some way to combine numbers to get rid of double counting. Which two numbers might be helpful to combine. We might not be sure, but maybe we can use this 46.8 million sum with the other total, 37.4 million. What should we do? Should we add, subtract, multiply, divide? I’d go with subtraction. Do 37.4 – 46.8 or 46.8 – 37.4. We get ±9.4. This seems to go along with answer choice (B). Should we go with that? Yes! Why not. We’ve done some work. Our choices were reasonable and logical. (B) is in fact the correct answer.

Occasionally when you’re taking the GRE, it will be necessary to take these “leaps of faith,” but they are actually just leaps of faith in yourself, belief in yourself that you’ve exercised good reasoning skills and come up with a choice that is likely the correct answer. Will you get burned sometimes? Sure. But overall you will earn more points and your score will be higher than if you ponder every question looking for the “proper” solution.

In this case, the work we did is very similar to the textbook approach. With two overlapping groups, the basic equation is:

  • Group 1 Members + Group 2 Members – Members of Both Groups = Total Members
  • Substituting our numbers gives us:
  • 33.1 + 13.7 – Both = 37.4
  • Both = 9.4

Our reasoning skills led us to the same solution. Well done! Let’s look at the next problem. 


Multiple Multiple Choice

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What do the square boxes mean? They always indicate the possibility of multiple correct answers; sometimes the question will indicate that you must choose exactly two/three/four correct answers; other times, as in this case, you are instructed to select ALL that apply. There will always be at least one correct answers, but it is possible for all the answer choices to be correct. You get credit for a correct answer if and only if you select all and only the correct answers.

Which charts are we using here? We see that we’re talking about “expenditures,” so that means we’re using the top right chart. We’re then talking about “average expenditure per hunter/angler,” which means we’d need more information from another chart. Remember, to get the average you take the total sum and divide by the number of items. For “expenditure per hunter/angler,” the total sum (dividend) is the expenditure. What’s the number of items? “Per hunter/angler” tells us that the divisor is the number of hunters or anglers.

Let’s take a moment to RECORD WHAT YOU KNOW™. This way we have a second to pause and assess what information we have and what to do with it. Sometimes problems can seem overwhelming, but when you put the information in front of you, you can at least determine one thing you can do to get you closer to the solution. What do we know?

  • Hunting Expenditure: ~$33.5 billion
  • Fishing Expenditure: ~$42 billion
  • Unspecified Expenditure: ~$14.5 billion
  • Hunters: 13.7 million
  • Anglers: 33.1 million

There are a couple things to note here. First, we must approximate. Charts and graphs don’t always give us exact values, so we must ballpark out estimates. We should aim to be ±3% from the actual amount. For instance, we’d probably be okay with $33-34 billion for the Hunting Expenditure, but $32 billion or $35 billion would probably be pushing it.

Second, we must keep track of units. Here, we’re looking at billions and millions. Whether we do anything with that right now is optional, but we must remember this distinction.

Now that we have our information on our scratch paper, we should SPLIT THE QUESTION INTO PARTS™ to attempt the question incrementally. Keep your momentum going by doing whatever you can with the information in front of you.

Remember, we were talking about averages: Expenditure per hunter and expenditure per angler. We can do a couple averages right now:

  • Hunting Expenditure: ~$33.5 billion ÷ Hunters: 13.7 million = Average expenditure per hunter

I’m going to deal with my units right now.

  • ~$33.5 billion = ~$33,500 million
  • Average expenditure per hunter = ~$33,500 million ÷ 13.7 million = ~$2445 per hunter

Notice that I’m using the calculator to remain roughly precise in my calculations, but I’m not losing any sleep over dropping 0.255474452554745 from my number, because, really, that’s not going to make any difference.

Let’s do the same calculation per angler.

  • ~$42 billion = ~$42,000 million
  • Average expenditure per angler = ~$42,000 million ÷ 33.1 million = ~$1269 per angler

The question asks about “range of differences between the average expenditure per hunter and the average expenditure per angler. Right now the difference between the averages is:

  • ~$2445 per hunter – ~$1269 per angler = $1176

However, what’s this talk of “range of differences?” This must have to do with the Unspecified Expenditures. The question asks us to assign the Unspecified Expenditures to either Hunting or Fishing. Let’s try both.

  • ~$14.5 billion = ~$14,500 million unspecified expenditure
  • Add this all to the hunting expenditure: ~$14,500 million + ~$33,500 million = ~$48,000 million
  • This gives us the MAXIMUM possible average expenditure per hunter = ~$48,000 million ÷ 13.7 million = ~$3503 per hunter

Now the same calculations for anglers.

  • ~$14.5 billion = ~$14,500 million unspecified expenditure
  • Add this all to the fishing expenditure: ~$14,500 million + ~$42,000 million = ~$56,500 million
  • This gives us the MAXIMUM possible average expenditure per angler = ~$56,500 million ÷ 33.1 million = ~$1706 per angler

What can we do now that we’ve plowed through some calculations? Look at what we know:

  • Average expenditure per hunter = ~$2445 per hunter
  • Average expenditure per angler = ~$1269 per angler
  • MAXIMUM possible average expenditure per hunter = ~$3503 per hunter
  • MAXIMUM possible average expenditure per angler = ~$1706 per angler

The MAXIMUM possible averages are clear. That’s what we get when we assign all the unspecified funds to either hunting or fishing. The first averages we calculated are what happens if we assign none of the unspecified funds to hunting or fishing. In other words, these are the MINIMUM possible averages:

  • MINIMUM possible average expenditure per hunter = ~$2445 per hunter
  • MINIMUM possible average expenditure per angler = ~$1269 per angler
  • MAXIMUM possible average expenditure per hunter = ~$3503 per hunter
  • MAXIMUM possible average expenditure per angler = ~$1706 per angler

It’s time to put it all together. We’re looking for a range of possible differences. If we’re looking at MAXIMUM possible average expenditure per angler, then we must also be looking at MINIMUM possible average expenditure per hunter. The difference would be:

  • ~$2445 per hunter – ~$1706 per angler = ~$739

At the other extreme we get:

  • ~$3503 per hunter – ~$1269 per angler = ~$2234

Thus, the range of possible values includes all numbers between $739 and $2234. Answers (A), (B), (C), (D), and (E) are correct.


Mean, Median, Mode, (and Range)

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Another question about averages, in this case the arithmetic mean. Naturally, data interpretation problems involving charts and graphs frequently test students’ grasp of statistical “averages,” namely the mean, median, and mode. As we observed on the previous problem, the related concept of range also occurs frequently. In this case, the default average asked for is the arithmetic mean, again the sum of the items in a set divided by the total number of items in the set. Don’t forget to RECORD WHAT YOU KNOW™.

We’re asked about days spent hunting and fishing, so we’re using the top two graphs here. Let’s get the data down onto our scratch paper.

  • ~550 million days spent fishing
  • ~280 million days spent hunting
  • Fishing Expenditure: ~$42 billion
  • Hunting Expenditure: ~$33.5 billion

Compared to the previous question, this one might appear straightforward. It is! As such, it is a reminder that you are not required to do the questions in order. In fact, you should not, especially on a multi-question problem set such as this. Knock the easier ones out first. Then move on to the more challenging questions. Since we’re looking for average spent on days fishing and hunting, let’s do those calculations without forgetting to convert our units.

  • ~$42 billion ÷ ~550 million days = Average spent on fishing day
  • ~$42,000 million ÷ ~550 million days = ~$76 = Average spent on fishing day
  • ~$33.5 billion ÷ ~280 million days = Average spent on hunting day
  • ~$33,500 million ÷ ~280 million days = ~$120 = Average spent on hunting day

And now the difference:

  • $120 – $76 = $44

Here we’ve hit $44 on the nose, so answer choice (D) is correct.


I’d Rather Be Taking the GRE

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The final problem, like the first problem, is a one-chart affair, but here we rely on some additional “common” knowledge to solve the problem. Begin again by getting the relevant data and any other important facts onto our scratch paper.

  • ~550 million days spent fishing
  • ~280 million days spent hunting
  • 365 days in a year
  • Average life span is 72 years.

Time to SPLIT THE QUESTION INTO PARTS™. our scratch paper.

  • 72 × 365 = 26280 days in average life
  • ~280 million days + ~550 million days = ~830 million days annually spent hunting or fishing
  • ~830,000,000 ÷ 26280 = ~31582 

Notice again that calculator use is essential. Use your numeric keypad to speed things up! Our result is closest to 31,400, which means (E) is the credited response.


Uncharted Territory

Here are a couple key lessons you should take away from the above discussion:

  • Do not underestimate or “blow off” data interpretation. Practice, practice, practice.
  • Use your scratch paper effectively. Get your data in front of you.
  • Don’t get overwhelmed.
  • Proceed incrementally and keep your momentum going doing whatever calculation you can, even if you don’t see the “big picture.”
  • Stay positive and confident. You are equipped with the skills and knowledge to solve even the most difficult GRE problem.

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