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Dana's Quant Strategy suggestions

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DanaJ Site Admin
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Dana's Quant Strategy suggestions

Post Wed May 20, 2009 11:56 am
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  • Lap #[LAPCOUNT] ([LAPTIME])
    I've decided to start this thread hoping that it might help some community members with their quant performance. It's not designed as something 100% solid, it's just going to be a collection of my thoughts on various topics or problems that we face when trying to Beat The GMAT. I'm not a GMAT expert, but I am considering teaching/tutoring some day, so any feedback is more than welcomed.

    Thought I'd start with something I hate, so here goes...


    NUMBER PICKING STRATEGY

    I personally don't like this one. There's a good reason for this: most of the time it's a risky strategy. If you're not paying attention, the set of numbers you've selected might turn the whole thing around! I honestly prefer ye old algebraic method of solving things, since it's 100% safe (if employed correctly, of course).
    However, there are a few instances when number picking is the recommended strategy.


    COUNTEREXAMPLES
    This is actually the only instance when I fully support number picking. Counterexamples are extremely useful in two instances:
    - DS questions, when you're not really interested in solving a particular problem, but in establishing whether the info provided is or is not sufficient
    - PS questions when an algebraic approach would take up too much time OR you're not really sure of how to solve a question by using the classical method

    I'll try to illustrate each situation with a few official questions.

    I. DS-ing

    If n is an integer, is n + 2 a prime number?
    1. n is a prime number
    2. n + 1 is not a prime number
    We analyze the stmt 1 first. Here comes your first number picking tip: try to pick the smallest numbers that fit the description. If you go for bigger numbers, your calculations might suffer. In this case, let's pick 2 and 3. 2 + 2 = 4 and 4 is not a prime number. 3 + 2 = 5 and 5 is indeed a prime number. So 1 is insufficient. If I were to pick 29 and 31, for instance, I'd get similar results, but there's no need to move too far away from zero.
    Stmt 2 is up next: in this case, let's pick 3 and 7. 3 + 1 = 4 is not a prime number, but 3 + 2 = 5 is a prime number. 7 + 1 = 8 is not a prime number and 7 + 2 = 9 is not a prime number. So 2 is insufficient as well.
    Put the two stmts together to get the same thing: use counterexamples 3 and 7.

    If n is a positive integer, is 150/n an integer?
    1. n < 7
    2. n is a prime number
    Search for a counterexample for stmt 1. Start from 0: 150 is divisible by 2, 150 is divisible by 3, but 150 is not divisible by 4. So 1 is out.
    For stmt 2, we stray a bit further from zero with say 7: 20*7 = 140, so 150 will not be divisible by this prime number. But pick either 2 or 3 and you get a divisibility.
    Since we've eliminated choices A, B and D, time to go for choice C: if we take the two stmts together, then we get three prime numbers that are smaller than 7: 2, 3 and 5. All divide 150 evenly, so here's your answer.

    II. PS-ing

    Which of the following describes all values of x for which 1 - x^2 ≥ 0?
    A. x ≥ 1
    B. x ≤ -1
    C. 0 ≤ x ≤1
    D. x ≤ -1 or x ≥ 1
    E. -1 ≤ x ≤ 1
    This is an easy one. Most test-takers will not hesitate in solving it algebraically, but here goes: for A, pick x = 2: x^2 = 4 and 1 - 4 = -3, which is definitely smaller than 0. For B, pick -2 with the same results. Since D can be eliminated on the same examples, you're basically left with two choices: C and E. Here's where your real skills kick in: you know that the square of an integer is also the square of the integer's opposite. This means that, if a certain a is in the interval that you're looking for, -a will also be in there. So that means that choice E is indeed your answer.

    If x is an integer and y = 3x + 2, which of the following CANNOT be a divisor of y?
    A. 4
    B. 5
    C. 6
    D. 7
    E. 8
    This question is most easily solved when noticing that y is not a multiple of 3. This means that it's also not a multiple of 6, so C is your answer. But if you don't notice this, then start picking numbers. For 4, so for 8 = 3*2 + 2. For 5, go for 18 = 3*6 + 2. For 7, pick 14 = 3*4 + 2. For 8, pick 8 itself = 3*2 +2. As you can see, all numbers are pretty close to zero, so as a general rule don't stray to far from it!

    The examples above bring us to the second number picking tip: counterexamples work best with divisibility and intervals. If you see this type of problem and you can't solve it algebraically, then try number picking.


    FRACTIONS/PERCENTAGES
    Another relatively safe bet for number picking is percentages and NOT TOO TANGLED fractions. Percentages are OK as far as this strategy is concerned because a percentage is basically a fraction with 100 as the denominator, so picking that one is a pretty solid approach.

    The organizers of a fair projected a 25 percent increase in attendance this year over that of last year, but attendance this year actually decreased by 20 percent. What percent of the projected attendance was the actual attendance?
    A. 45%
    B. 56%
    C. 64%
    D. 75%
    E. 80%
    Say that last year, 100 people attended the fair. This means that this year, we were expecting 125 people, but only 80 showed up. How much is 80 out of 125? Well, it's (80/125)*100 = 1600/25 = 64, with C the correct answer.

    In a certain city, 60 percent of the registered voters are Democrats and the rest are Republicans. In a mayoral race, if 75 of the registered voters who are Democrats and 20 percent of the registered voters who are Republicans are expected to vote for Candidate A, what percent of the registered voters are expected to vote for Candidate A?
    A. 50%
    B. 53%
    C. 54%
    D. 55%
    E. 57%
    We'll assume that our city has 100 voters, so 60 are Democrats and 40 are Republicans. 75% of Democrats or 45 of them vote for A, while 8 Republicans also vote for A. This makes 45 + 8 = 53 votes for A, or 53% of registered voters.

    Picking numbers when you have slightly different fractions is tricky. Even though this strategy might produce the correct result, it should be used with caution. Even experts might not follow the exact proportions required by the problem!

    At a loading dock, each worker on the night crew loaded 3/4 as many boxes as each worker on the day crew. If the night crew has 4/5 as many workers as the day crew, what fraction of all the boxes loaded by the two crews did the day crew load?
    A. 1/5
    B. 2/5
    C. 3/5
    D. 4/5
    E. 5/8
    So the night workers load three quarters of the number of boxes that the day workers load. This means that you need to pick a number of boxes loaded by day workers in such a way that it's divisible by 4. Go for the smallest: say each day worker loads 4 boxes. This means that night workers load 3 boxes. Then comes your next tricky choice: the number of workers. Again, you're interested in picking a number of day workers that's divisible by 5, so pick 5. You have 5 day workers and 4 night workers. Now, this means that your day workers will load 5*4 = 20 boxes, while your night workers will load 3*4 = 12boxes. This brings us to 32 boxes in total, out of which 20 were loaded by the day crew. 20/32 = 5/8, so the answer's E.

    I admit that these are not the most challenging examples one can think of, but unfortunately I guess I need to work on finding the harder ones in the OG books... Will try to do better next time!



    Last edited by DanaJ on Tue Sep 07, 2010 8:58 am; edited 2 times in total

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    Post Wed May 20, 2009 12:04 pm
    Moving this post to the GMAT Math section. Amazing post Dana! I am going to include this in the GMAT Resources Directory right now: http://www.beatthegmat.com/resources.html

    Thanks!

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    ssmiles08 Master | Next Rank: 500 Posts
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    Post Wed May 20, 2009 1:21 pm
    Great thread Dana!

    I usually run into a bit of trouble when I am solving rate/distance type of problems (like 2 people meeting at a certain point or one train is moving than the 2nd train and they catch up)

    Do you have any techniques that would help with those types of problems?

    DanaJ Site Admin
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    Post Wed May 20, 2009 9:33 pm
    @ssmiles08: will try to approach that one soon.

    But for the moment, here's my take on the quantity vs. quality issue in GMAT prep and how to analyze your mistakes. It's actually something I wrote a little while back, responding to a PM received by one of the forum members.

    First off, the idea that "quality beats quantity" is, IMHO, something to be applied only to the GMAT. AGAIN, this is a personal opinion that stems from personal experience. Throughout my educational "career", I've noticed that practicing is more important that truly and deeply understanding something, from an "I need to score an A+ in this test" point of view, of course - and I suppose this is what you want as well, "score an A" in the GMAT. You're not really trying to get your Ph.D. in maths, so you don't need to understand all that underlying philosophy...

    Let me give you a practical example that's easier to understand: the maths teacher would come to class and teach us about Pythagoras's theorem (the one about the right triangle - pretty basic geometry). I didn't bother going too deep about the theorem in itself, like memorizing it right off the bat or trying to remember how to demonstrate it. Instead, I'd focus on the EXAMPLES that followed - and we usually had at least 5, with increasing difficulty. It's the examples that solidified my knowledge of the formula and really put an abstract thing to work.

    HOWEVER:

    The problem is that the OG contains about 100 or so small types of problems and you only get one or two examples per type. There is a difference between "small types" and "general types". A general type means number theory or geometry; a small type means divisibility by 3 or the length of an arc.

    The fact that you only get one or two examples per small type is really bad, IMHO, for someone who's not so good in maths. Don't bother looking for a book that gives you those extra 5 or 6 examples for each type: there isn't any and I doubt anyone has ever had the idea to write such a book - it would probably be 5 times as thick as the OG. Test prep companies do try to provide those extra examples, but as far as I could tell none has gone as far as to write such a mammoth. Besides, we all know that only the OG provides 100% GMAT-style questions. Since you've got very little to work with, you need to make the most of it.

    HERE COMES THE QUALITY PART:

    Your material is very limited, so you need to squeeze the most out of it, meaning that you need to understand that type perfectly by using just one example or at best two. This is why you need quality over quantity in the GMAT: you need to see the "hidden" 5 or 6 examples in just one example.

    The person who sent me the PM asked for a checklist when making mistakes in quant... I'm the worst person to ask for such a thing; you should see me when I take a GMAT test: I write two legal size papers for 37 questions and most of the time it's graphs. This is because I'm more of a "keep it inside my brain" type of person. However, I took a solid half an hour before deciding what steps are appropriate and here is what I came up with (again, take this with a mountain of salt, since it's not "my style", so to speak):

    1. Re-read the question carefully.
    This sometimes happens to me: I get a wrong answer because I didn't bother to read the whole question or I misread it. Re-reading after you've realized that you've made a mistake is essential, since you might notice that your mistake stems from misunderstanding rather than poor skill.

    2. Try to solve it again BEFORE checking the explanation.
    Sometimes we need a different approach. This time you may find that it's better to do things differently or you might have a new idea that will yield the correct answer.

    3. Even if you've found the correct answer by re-doing the question, check the explanations thoroughly.
    You'd be surprised to notice that some of the explanations are indeed well written. This is not the case for all books; some prep companies have less than perfect content developers or even less talented "explanation providers". If you think your way is best, roll with it. You've got your own style! However, you might stumble upon an easier/faster way to do things, so be on the lookout for treasure!

    4. Dismantle the question.
    My uncle is a super handyman: he fixes anything from cars made in the 1980s to dual SIM cell phones. How is that possible? Well, when he was about 12 years old, he dismantled his bike and put it back together in the same day. It was his first experience of the sort: this made him the best bicycle guy on this side of the planet. He continued doing so with various other stuff until no new invention can catch him off guard when it comes to fixing it...
    This is what you need to do: just tear down that question! Analyze it by letters if that's what gets you to understand it thoroughly. Think of theorems applied and tested, of possible meanings and consequences of the fact that "a = 3k + 1", for instance. This will help you see the other 5-6 examples you need.

    5. If you end up with too many mistakes, redo the question set after a reasonable pause.
    If you end up with a below 75% hit rate (although this percentage is debatable - pick your own standard according to your own goals!), I'd suggest redoing that question set a week later or two weeks later. This is how you can test your progress: did all that analysis help or are you stuck in the same spots again? If it's the second case, try re-analyzing, and if you're still not satisfied, ask around the forum for help.

    kyabe Senior | Next Rank: 100 Posts Default Avatar
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    Post Sun May 24, 2009 3:13 pm
    Hi Dana,

    Many many thanks for these tips.. Looking forward for more of these Smile

    Cheers!!!

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    Post Mon May 25, 2009 5:00 am
    Hey,

    Thanks for your kind words. I'm going to update the thread this week-end, since |I currentyl have really limited free time (my finals are all over me). I'm thinking of some exponents/powers section and we'll se what else!

    DanaJ Site Admin
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    Post Fri Jul 10, 2009 7:05 am
    I know it's been a while since I've posted here, but I have some solid excuses... In May the dreadful finals started and lasted up until the 5th of June... Then I had to tutor a friend of mine who had trouble with most of the subjects we've studied this year, adding up to about 30 hours of tutoring over three weeks plus 10 hours of prepping... I must say I've learned my lesson: from now on, no more tutoring without negotiating a price! I was pretty upset when he didn't even offer to buy me a box of chocolates... Oh well... Now I'm currently helping my brother with his college admissions exams, so I'm still kindda busy. Will try to post more in August.

    This post is about memorizing stuff before the big day. One of the things that helped a lot during the GMAT was the fact that I could instantly access my database of memorized items to boost my speed. Since someone asked, here is the essential list of things, IMHO:

    I.POWERS

    1. Squares - up to 15 at least
    2^2 = 4
    3^2 = 9
    4^2 = 16
    5^2 = 25
    6^2 = 36
    7^2 = 49
    8^2 = 64
    9^2 = 81
    10^2 = 100
    11^2 = 121
    12^2 = 144
    13^2 = 169
    14^2 = 196
    15^2 = 225

    2. Cubes - up to 5 at least
    2^3 = 8
    3^3 = 27
    4^3 = 64
    5^3 = 125

    3. Other important powers
    2^5 = 32
    2^10 = 1024
    3^4 = 81
    3^6 = 729
    5^4 = 625

    4. A perfect square can have only the following units digit: 0, 1, 4, 5, 6, 9 - a number with 2, 3, 7 or 8 as units digit is NOT a perfect square. Also, note that whatever power of 1, 5 and 6 will always keep the units digit of the original.


    II. Primes - up to 30 at least
    2 - the ONLY even prime number
    3
    5
    7
    11
    13
    17
    19
    23
    29


    III. Quadratics
    (a + b)^2 = a^2 + b^2 + 2ab
    (a - b)^2 = a^2 + b^2 - 2ab
    (a + b)(a - b) = a^2 - b^2
    x^2 + (a + b)*x + ab = (x + a)(x + b) - UTTERLY INVALUABLE for solving quadratic equations


    IV. Progressions

    1. Arithmetic
    n-th element of a series: a1 + (n - 1)*r
    sum of n elements: n(a1 + an)/2

    2. Geometric
    n-th element of a series: b1*[q^(n-1)]
    sum of n elements: b1*[q^(n + 1) - 1]/(q - 1)


    V. Combinatorics
    Permutations of n objects n!
    Arrangements: of n object in k spots: n!/(n-k)!
    k-Combinations of n objects: n!/[(n-k)! * k!]


    VI. Geometry

    1.General
    Area of equilateral triangle: sqrt(3)*l^2/4
    Area of circle: Pi*r^2
    Circumference of a circle: 2Pi*r
    Area of trapezoid: (base + Base)*height/2
    Volume of an object:
    - right (ex. cube): base*height
    - triangular (ex. pyramid): base*height/3

    2. Right triangle:
    Any right triangle: the median drawn from the right vertex will be half the hypotenuse
    Isosceles: hypotenuse = side*sqrt(2)
    30-60 degrees: the side facing the 30 degree angle is half the size of the hypotenuse

    That's it for now, I guess. I do see myself editing this list a few times. Please feel free to post your own suggestions.

    fadibassil Junior | Next Rank: 30 Posts Default Avatar
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    Post Tue Sep 07, 2010 8:38 am
    Thank you for these posts.

    The example about the workers in the first post shows a question mark symbol instead of a value for the fraction and choice A. what should be the value in place of the question mark?

    Thanks again,

    Fadi

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    Post Tue Sep 07, 2010 8:59 am
    I think it's 3/4, but I don't have the OG with me right now so I can't double check!

    jsiverio Newbie | Next Rank: 10 Posts
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    Post Tue Oct 26, 2010 3:50 pm
    Thanks so much Dana this is great.

    One question though:

    Isn't this supposed to be the bonus "treasure hunt" for the day??

    vineetasaxena Newbie | Next Rank: 10 Posts
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    Post Tue Oct 26, 2010 8:38 pm
    I also think this is bonus post

    thp510 Senior | Next Rank: 100 Posts Default Avatar
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    Post Sat Oct 30, 2010 12:32 pm
    DanaJ wrote:
    V. Combinatorics
    Permutations of n objects n!
    Arrangements: of n object in k spots: n!/(n-k)!
    k-Combinations of n objects: n!/[(n-k)! * k!]

    DanaJ

    For the Arrangements.... if I have 4 chairs to put in 6 spots, does the equation look like this?: 4!/(4-6)!
    ...a negative 2! would be in the denominator. Is this correct?

    sweatitsmart Newbie | Next Rank: 10 Posts Default Avatar
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    Post Tue Mar 08, 2011 10:21 pm
    Hi Dhana,

    I would specifically want to comment on your statement "If you go for bigger numbers, your calculations might suffer." . I believe that the DS problem of prime numbers can be resolved in one flick using the number 31 which is not nearer to 0. Using numbers nearer to 0 only increase the time taken to resolve the problem and also might be misleading. This is just my opinion. However, it is quite nice to find such blogs because we get different perceptions. Thumbs up !

    DanaJ Site Admin
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    Post Wed Mar 09, 2011 4:44 am
    That's an interesting point - although IMHO using numbers that are too big will hurt your calculations. It's easier to figure out what 3*5 is than figuring out what 5*37 is, for instance.

    sweatitsmart Newbie | Next Rank: 10 Posts Default Avatar
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    Post Thu Mar 10, 2011 1:25 am
    thp510 wrote:
    DanaJ wrote:
    V. Combinatorics
    Permutations of n objects n!
    Arrangements: of n object in k spots: n!/(n-k)!
    k-Combinations of n objects: n!/[(n-k)! * k!]

    DanaJ

    For the Arrangements.... if I have 4 chairs to put in 6 spots, does the equation look like this?: 4!/(4-6)!
    ...a negative 2! would be in the denominator. Is this correct?
    No... Not at all!
    First you need to choose 4 spots out of 6 = 6C4 = 6!/(4!*2!)
    And then you can rearrange all of these 4 spots = 6!4!/(4!*2!) = 6!/2!

    So, I believe the answer is 6!/2!

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