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<\/a><\/p>\n3) The numbers m, n, and K are all positive integers.\u00a0 Given that m is a factor of K, that n is also a factor of K, and m < n, which of the following must also be a positive integer factor of K?<\/p>\n
4) Marcia took a trip consisting of three segments at three different speeds: she drove a distance of (5D) at a speed of (2v), then a distance of (4D) at a speed of (3v), then a distance of D at a speed of (6v).\u00a0 In terms of D & v, what was the total time of Marcia’s trip?<\/p>\n Answers and explanations will follow this article.<\/p>\n <\/p>\n It’s absolutely true that, when the GMAT gives a problem with variables in the answer choices, one possible approach is to pick numbers for the variables.\u00a0 Students sometimes ask, “How do I pick numbers? Can you give me a general strategy for picking good numbers in all cases?<\/i>”\u00a0 Well, let’s start with the second question, which has a crystal clear answer: “No<\/b>!”\u00a0 You see, there are several different kinds of questions in which picking numbers could be an appropriate strategy, and the best numbers to pick in any case depend much more on the specifics of the question.\u00a0 Some numbers that would be excellent choice in one question would be the worst possible choice in another question.\u00a0 For many questions, the best numbers to pick are inextricably tied up with the mathematics in the question itself.\u00a0 Some of this I will explain in the course of this article, and other things I will explain in the solution.<\/p>\n <\/p>\n First, let’s start with how you think about the word “number”.\u00a0 What pops into your head when I say that word?\u00a0 If what pops into your head is the set {1, 2, 3, 4, 5, \u2026}, then with all due respect, you are operating with a third-grade level definition of the word “number”, and the GMAT Quant section will mercilessly punish you for such na\u00efvet\u00e9.\u00a0 The word “number” is a broad category, encompassing everything on the number line: zero is a number; 4\/7 is a number; negative four is a number; negative one fifth is a number; pi is number; the square-root of 17 is a number; etc. etc. All of these possibilities must leap to mind when a problem mentions a “number”.\u00a0 It’s good to appreciate the categories of numbers, as well as their different properties<\/a>.<\/p>\n Number sense is the ability to see patterns among numbers and sense which ones are best for a problem.\u00a0 The person who asks the question at the top, how to pick numbers — what that person really needs is number sense!!\u00a0 In other words, the person who asks for a general rule for picking numbers is really, at a very profound level, asking the wrong question.\u00a0 Although that person might not even be familiar with the terms, what that person really needs to develop is number sense<\/a> and mathematical thinking<\/a> in general. These two posts say more about these larger issues, but I will say a few general things about picking numbers here.<\/p>\n <\/p>\n A strategy you must avoid is picking 100 for the starting value in every percent problem.\u00a0 Everyone has heard this strategy.\u00a0 Everyone who picks numbers does precisely this, as if it were a reflex.\u00a0 That, in a nutshell, is the problem.\u00a0 You see, if the test writers know<\/i> that everyone makes the same choice, then they design answer choices designed to trip up people who make that all-too-obvious choice.\u00a0\u00a0 Think a little out of the box here.\u00a0 Instead of 100, pick 500 or 1000 — you should be able to figure out round number percents with those relatively quickly.<\/p>\n Extending that strategy even a little further: if anything is a super-obvious choice to pick, then rather than pick that super-obvious thing, pick something a shade away from super-obvious.<\/p>\n If the question is one of a very typical type, a percent increase or decrease followed by another percent increase or decrease, then I would strongly advise against<\/i> picking numbers.\u00a0 Instead, I would recommend using multipliers<\/a> to handle percent changes.<\/p>\n <\/p>\n If different variable represent quantities of different units, then make sure you understand what combination of them will have the correct units for the answer choice.\u00a0\u00a0 In Physics, this thinking about the units of different quantities is called dimensional analysis<\/b>.\u00a0\u00a0 For example if J is in units of miles per gallon, and M is in units of miles, then to get units of gallons, I would need M\/J — if the question is asking for “how many gallons”, then the correct answer would have to be a number times M\/J.\u00a0\u00a0 Any other combinations (J\/M, M*J, etc.) are automatically incorrect.\u00a0 This helps you eliminate answers choices, which makes the process of plugging in numbers easier.<\/p>\n <\/p>\n Many different math questions have very predictable traps: the mistake that a large number of test takers routinely make, time and time again.\u00a0 GMAT problems always<\/span><\/i><\/b> have answer choices that reflect these traps: in some cases, it’s as if the GMAT simply sets up a trap answer like a big butterfly net, and they watch hordes of test takers run headlong into it.\u00a0 For every such trap you learn to recognize and avoid, you will be that much ahead of so many other test takers.<\/p>\n One big trap involves a sequence of percent increases & decreases<\/a>.\u00a0 There are a few common ones involving fractions<\/a>.\u00a0 Still more involve exponent properties<\/a> and powers<\/a>.\u00a0 Some very common ones involve incorrect variations on three basic algebra formulas<\/a>.\u00a0 Here’s a post on the most common math mistakes<\/a>.<\/p>\n <\/p>\n Avoid picking 1 as a value for a variable.\u00a0 The reason is: one has so many special properties that no other number has.\u00a0\u00a0 For example, all powers of 1 equal one: one is the only number that has this particular property.\u00a0 Think if you had to distinguish choices involving different powers of that variable: picking one for the value would make all the power equal the same thing, making it impossible to distinguish which answers should be eliminated.\u00a0 One is often the worst possible choice.<\/p>\n The only exception to this is — sometimes, it’s worthwhile to pick very easy numbers, like 1 and 2, just to do a very quick elimination of a couple of answers.\u00a0 If that’s your goal, 1 can be a perfectly acceptable choice on a first pass.\u00a0 It will never be a choice that eliminate everything, but if your goal is to begin by eliminating low-hanging fruit, then by all means pick it.<\/p>\n <\/p>\n Finally, let’s step back.\u00a0 Why do folks pick numbers for problems with variable?\u00a0 One reason is that sometimes, quite rarely, one can either eliminate many answers or even get to a single right answer very quickly by picking numbers.\u00a0 Most often, though, this is not the case.\u00a0 In fact, often it’s much quicker and much easier to do the algebra directly.\u00a0 After all, this is precisely why mathematicians invented algebra in the first place: it’s a powerful efficient problem solving system.<\/p>\n Most of the time, most habitual number-pickers are choosing this strategy, not because it’s necessarily faster or easier or more efficient or better in any particular way, but simply because these folks are absolutely petrified of algebra.\u00a0 For these folks, algebra is a fire-breathing dragon, and folks will do anything, no matter how awkward or cumbersome, simply to avoid facing that dragon.\u00a0\u00a0 If this describes you, then I have some hard medicine for you: do algebra<\/b>.\u00a0 From this point forward, in any problem with variable, you are forbidden to pick numbers, and you must practice the algebra.\u00a0 You see, for the GMAT, you will have to know some algebra, and the only way you get better at algebra is by practicing it at every opportunity you can get.<\/p>\n <\/p>\n If you had any flashes of insight while reading this article and exploring the links, you may want to give the questions at the top another look.\u00a0\u00a0 Here’s another problem of another sort, from inside Magoosh:<\/p>\n 5) http:\/\/gmat.magoosh.com\/questions\/123<\/a><\/p>\n Finally, if you have any questions, please let us know in the comments section at the bottom.<\/p>\n <\/p>\n 1) First of all, let’s talk about the trap.\u00a0 Even if you don’t learn how to solve this problem, learn to recognize this trap, possibly the single most enticing trap on the entire GMAT.\u00a0 When a percent increase or decrease is followed by another percent increase or decrease, you cannot simply add or subtract the two percents<\/b>.\u00a0 If you increase something by 40% and then decrease it by 30%, that is absolutely NOT<\/i><\/b> a 10% increase.\u00a0\u00a0 That is the trap answer to which hordes of GMAT test takers are drawn like bugs to a bug zapper<\/a>.<\/p>\n Problems are this sort are not ones on which I would recommend picking numbers at all.\u00a0 Instead, I would recommend using multipliers<\/a> for the percent changes.\u00a0 The multiplier for a 40% increase would be (1 + 0.40) = 1.40.\u00a0 The multiplier for a 30% decrease would be (1 \u2013 0.30) = 0.70.\u00a0 The net effect of both would be<\/p>\n Final price = (1.40)*(0.70)*P = 0.98*P<\/p>\n Answer = (E)<\/b><\/p>\n Even though it’s not as efficient, here’s a picking-numbers approach, just so you can see how the numbers play out. Let p = $1000.\u00a0\u00a0 This increases by 40%, to $1400.\u00a0 Now, this is going to decrease by 30% — but notice<\/b>!\u00a0 It decreases not by 30% of the original, $1000; rather, it decrease by 30% of the new price, $1400.\u00a0 After the 40% increase, the price of $1000 doesn’t even exist anymore: the only price that exists at that point is the $1400 price; that’s what decreases by 30%.\u00a0 Well, 10% of $1400 is $140, and so, 30% = 3*10% = 3*$140 = $420.\u00a0 The price decrease to $1400 – $420 = $980, which is 98% of P or 0.98*P.<\/p>\n<\/a><\/p>\n<\/a><\/p>\nPicking numbers<\/h2>\n
Numbers<\/h2>\n
Percent problems<\/h2>\n
Dimensional analysis<\/h2>\n
Know the traps<\/h2>\n
Avoid one<\/h2>\n
To pick or not to pick<\/h2>\n
Summary<\/h2>\n
Practice problem explanations<\/h2>\n