Browsing articles in "GRE Algebra"
Apr
22
2013

# Kaplan GRE Grudge Match, Vol 1: Pros and Cons of GRE Calculator

This week we have something special for our readers.

Whether you’ve seen us in class, in a free event, on this blog, or elsewhere, Kaplan always presents a unified front, as though we agree on everything. In fact, the smart, opinionated people who work for Kaplan disagree about lots of things, and today we’re offering a glimpse of our intellectual disputes.

Kaplan’s GRE bloggers Boris and Teresa take to their chairs to debate a common topic: does a GRE calculator make the test easier, or harder? Hear their arguments, pro and con, in the video below!

As always, let us know what you think in the comments below!

Apr
17
2013

# Prime Factors and Exponents – A Secret to Quantitative Comparisons

Knowing how to break a number apart into its factors is a useful skill that will serve you well on the GRE. However, sometimes you’re better served by not actually taking the time to determine what a number’s specific factors are. When is this a more useful approach? Take this Quantitative Comparison, for example:

We’re being asked to compare the number of non-prime, positive integers greater than 1 that are factors of Q, to 24. Do we care about what the actual factors are? Not at all. We could pick values for a, b, and c, and see how many factors of Q we can get. However, there’s a quick trick that will save us time and ensure that we don’t accidentally miss any factors.

If we only care about finding the number of factors that a given number has, then all we need to do is look at the exponents of its prime factors. In this case, we know that a, b, and c are prime, so we don’t need to regroup or simplify them at all.

All we do is add 1 to each of the exponents, and multiply the results. The exponents on a, b, and c are 2, 3, and 1, respectively. If we add 1 to each of these, we’ll get 3, 4, and 2. When we take the product, we get 24.

So can we compare this value of 24 to the value in Quantity B and move on? Not quite – remember, we specifically want to compare the non-prime factors that are greater than 1. Well, let’s figure out how many of the factors do not fit into that category. 1 certainly does, and the only primes that go into Q were given to us in the question stem:  a, b, and c. So out of the 24 total factors of Q, 4 of them don’t fall into the “prime and greater than 1” category. This leaves 20 total factors, which means that our two quantities are in fact equal.

To recap the steps for finding the total number of factors of a given integer:

1) Break the integer apart into its prime factors, written with exponents (eg a3b2c).

2) Add 1 to each of the exponents.

3) Multiply the resulting terms.

Simple, yet effective – that’s my favorite kind of math trick! What are your favorite math tricks? Let us know in the comments!

Apr
10
2013

# Study-Planning for the GRE Once Your Class Has Ended

As teachers, my colleagues and I spend a lot of time discussing the importance of study plans with our students – specifically, how do you continue working once you’ve completed the class sessions and the accompanying assignments? The best study plan for you will vary depending on the specific topics on which you need to improve, but we do have a basic template that you can use to plan out your studies into two-week cycles:

Week One:

- Start every two week cycle with a full-length MST, to gauge your progress

- Spend the first three days after the test reviewing the answers and explanations to every problem, and doing the recommended assignments in the Smart Report. So doing the test, review, and recommended assignments can account for four days’ worth of work in each two-week cycle.

- Then spend the next two days reviewing anything else that you know you need to work on – it could be particular topics or question types, or it could be using Quiz Bank quizzes to work on pacing.  This piece will change every cycle, which is why I’m being deliberately vague.

- Take day seven off – you’d take rest days if you were training for a marathon, and this is no different.

Week Two:

- For the five out of the next six days (the first six days of week 2) you can organize your studying by topic or question-type – Monday could be algebra day, Tuesday reading comp day, etc.  Spend your time just building a 6-8-question quizzes in Quiz Bank and reviewing them thoroughly, and reviewing any book material or Lessons on Demand to review the content as necessary.

- Allot one day out of this six-day span to going back through the previous MST, to see if you remember how to solve the questions that you looked through on that test – it can be very easy to look at a problem a week after you did it and forget how to solve it.  Reviewing the problems twice will help ensure that you’re remembering approaches that will help you on the next test.

- Take day seven of week 2 off, then begin the cycle again with the next MST!

Simple but effective – lather, rinse, and repeat your way to GRE success. Do you have any questions about setting up study plans? Do you have trouble sticking to the plans that you’ve set? Let us know in the comments!

Apr
1
2013

# The Trap of Knowing Too Much

Recently, a fellow instructor sent me a GRE problem with which his student was struggling, and it led to an interesting conversation. Here’s the problem:

Which of the following is equivalent to (√3 + 1) / (√3 – 1)?

(A) 3 – √3

(B) 2√3 – 2

(C) 3

(D) 2 + √3

(E) 4 + 2√3

Several of us instructors looked at the problem, and without even glancing at the answer choices, half of us said “Whichever choice is equal to 2.” But when we actually read the answer choices, we noticed that there wasn’t one that matched our prediction.

Where did we go wrong? It was hubris. We’ve all been teaching for long enough that we know the common math tricks inside and out – in this case, as soon as we saw (√3 + 1) and (√3 – 1), we noticed that they looked like the terms in the common quadratic equation (a + b)*(a – b), and assumed that we were supposed to multiply the terms together. In that case, the result would be 3 – 1, which equals 2. Of course, the problem asks us to divide the terms, not multiply them, but we’d read too quickly and made a fatal mistake. We were able to fix it, but only after we had 1) realized that no choice matched our result, and 2) gone back and reread the question. That’s a lot of time wasted, on a problem that – if read correctly on the first pass – can be solved with a couple of very straightforward calculations.

If it was so easy for a group of people who have been teaching the GRE for decades (albeit a group who was half-distracted with work) to read too quickly, just think how easy it is for the average test-taker to fall into a similar trap. So learn from our mistake: Know every shortcut and approach that you can possibly learn, but do not ever, under any circumstances, let that be a substitute for reading carefully and identifying exactly what a question is asking you to do.

What traps do you commonly fall into in your GRE practice? And what answer did you get to this quant problem? Let us know in the comments!

Feb
20
2013

# A Rates and Speed Problems Shortcut – Now That’s Sweet!

For anyone who ate too much chocolate last week and is looking for a sweet treat that won’t add to his or her waistline, look no further: How would you like a quick way to solve rate & speed problems that have lots of variables? It probably doesn’t sound quite as good as a box of truffles does, but truffles unfortunately aren’t going to get you into grad school.

The beautiful thing about rate problem is that they virtually always are written as a fraction. Think about it: Any rate – miles/gallon, cookies/jar, heartbeats/minute – is just one term divided by another term. So when you’re setting up a rate problem in which you have variables without real numbers, figuring out where each variable goes in the fraction – the numerator or the denominator – will often get you directly to the correct answer. Let’s look at an example to demonstrate this:

Sweetheart Candies sells boxes holding p pieces of chocolate each. The boxes are shipped in crates, each holding b boxes. What is the price charged per piece of chocolate, in cents, if Sweetheart Candies charges m dollars for each crate?

(A) 100bp/m

(B) 100m/bp

(C) bp/100m

(D) m/bp

(E) bp/m

What’s this question asking us for? We’re looking for the price per piece of chocolate, in cents. The keyword “per”, along with the fact that the answer choices are all fractions, clue us into the fact that this is a rate problem. We have three variables, so let’s see where we can put them in the fraction.

Anything that comes before the “per” will go in the fraction’s numerator. In this case, that’s the price. m is the only variable that has anything to do with money, so it must be on top. If we glance back up at the answer choices right now, which answer choices can we eliminate because they don’t correctly place m? That’s right: (A), (C), and (E) are now gone.

The only difference between (B) and (D), the two remaining choices, is that (B) multiplies m by 100. Is that extra 100 a necessary component of the rate that we’ve been asked to set up? Let’s think: The problem asks for the price, in cents, but m represents a number of dollars. So yes, we absolutely need to multiply m by 100 to get from dollars to cents. And voila – choice (D) is out, and we’ve determined that (B ) is the correct answer.

The next time that you’re confronted with a tough rate problem that involves multiple variables, don’t despair – just figure out whether each variable belongs on the top of the bottom of the fraction. On Test Day, you’ll have nailed points and moved on before your competition is even done reading the problem!

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