{"id":17028,"date":"2013-03-01T09:00:44","date_gmt":"2013-03-01T16:00:44","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=17028"},"modified":"2013-02-17T07:10:39","modified_gmt":"2013-02-17T14:10:39","slug":"simplifying-math-with-substitution-on-the-gmat","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/simplifying-math-with-substitution-on-the-gmat\/","title":{"rendered":"Simplifying Math with Substitution on the GMAT"},"content":{"rendered":"

\"\"<\/a>This post discusses a strategy for simplifying complex mathematical expression on the\u00a0GMAT Quantitative<\/a>\u00a0section.\u00a0 Consider the following very challenging Data Sufficiency question.<\/p>\n

 <\/p>\n

1. What is the value of x?<\/p>\n

\"smws_img1\"<\/a><\/p>\n

Take a moment to wrestle with this before reading this article, in which we will explain how to approach this.<\/p>\n

 <\/p>\n

Complex Equations on the GMAT can be simplified!<\/h2>\n

How simple or complex an algebraic expression looks make no difference to the arithmetic operating on it.\u00a0\u00a0 When a mathematician looks at the following four expressions \u2026<\/p>\n

\"smws_img2\"<\/a><\/p>\n

\u2026 the mathematician sees, despite the apparent diversity, an underlying similarity: all four of these are in the form \"thing cubed minus five.\"\u00a0 Because all four have the same form, that means if we can figure out something important about the first, we could probably use that to figure out something important about the others.\u00a0 The first two expressions here are out the outer limit of what the GMAT will give you, and the latter two are well beyond GMAT math, but I list them here simply to demonstrate important skill of seeing essential mathematical patterns despite differences in complexity level.<\/p>\n

 <\/p>\n

The question: preliminary observations<\/h2>\n

With the DS question above, it's reasonably easy to see --- both statements are going to allow multiple possible values for x.\u00a0 The second statement, with a little factoring, becomes (x + 2)*(x \u2013 1) = 0, so the solutions are x = {+1, \u20132}.\u00a0 Two possible values, so Statement #2 by itself is not sufficient.\u00a0 Even without tackling the first one, it looks like this will also have more than one possible solution, which means, by itself, it wouldn't be sufficient either.\u00a0 Right away, without doing much at all in the way of calculations, we can eliminate choices\u00a0(A)<\/strong>\u00a0&\u00a0(B)<\/strong>\u00a0&\u00a0(D)<\/strong>, leaving only\u00a0(C)<\/strong>\u00a0&\u00a0(E)<\/strong>.\u00a0 Even if we could do nothing else, we would have excellent odds at this point if we simply\u00a0guessed from the remaining two answers<\/a>.\u00a0 Let's talk, though, about tackling Statement #1.<\/p>\n

 <\/p>\n

The benighted approach<\/h2>\n

Someone who is fairly comfortable with individual math rules, but who is not accustomed to thinking like a mathematician, is likely to multiply everything out in Statement #1.\u00a0 First of all, there's the danger of making in mistake in squaring the expression in the first term, but let's assume our foot soldier gets that part right.\u00a0 Then we would have \u2026.<\/p>\n

\"smws_img3\"<\/a><\/p>\n

Well, we could simplify that further, but notice: this orgy of algebra already has passed a few spots at which algebraic mistakes would be likely to happen.\u00a0 Even now, we have quartic (i.e. an equation with a fourth power), which is a bit more difficult than anything the GMAT will ask of you.<\/p>\n

 <\/p>\n

The u-substitution method<\/h2>\n

Whenever the same algebraic expression appears in two or more terms, we can always use the letter\u00a0u<\/strong>\u00a0as a kind of mathematical abbreviation.\u00a0 To use the mathematical term, we are substituting u for the repeated algebraic expression.\u00a0 Here, to simply the first statement, I am going to use the u-substitution \u2026<\/p>\n

\"smws_img4\"<\/a><\/p>\n

\u2026 because that is the expression repeated in two terms of statement #1.\u00a0 With this substitution, statement #1 becomes \u2026.<\/p>\n

\"smws_img5\"<\/a><\/p>\n

Now, we have found the possible values of u --- we have \"solved for u.\"\u00a0 The prompt, though, is asking for the value of x.\u00a0 Given these possible values of u, what are the possible values of x?<\/p>\n

\"smws_img6\"<\/a><\/p>\n

Thus, statement #1 allows for four different values of x: {+1, \u20131, +2, \u20132}.\u00a0 As seen above, statement #2 allows for two different values of x: {+1, \u20132}.\u00a0 Combined, there are still possible values of x: {+1, \u20132}.\u00a0 Thus, even with both statements combined, we are not able to determine a unique value of x and give a definitive answer to the prompt question.\u00a0 Even combined, the statements are insufficient.\u00a0\u00a0 Answer =\u00a0E<\/strong>.<\/p>\n

 <\/p>\n

Summary<\/h2>\n

Think like a mathematician.\u00a0 Don't simply plod through step-by-step ---- step back and look at the patterns, and use u-substitutions to make the patterns clear.\u00a0 This is an absolutely invaluable approach in +700-level GMAT Quantitative problems.\u00a0\u00a0 Here's a further question for practice<\/p>\n

2.\u00a0\u00a0https:\/\/gmat.magoosh.com\/questions\/126<\/a><\/p>\n

This post was written by Mike McGarry, GMAT expert at Magoosh<\/a>, and originally posted here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post discusses a strategy for simplifying complex mathematical expression on the\u00a0GMAT Quantitative\u00a0section.\u00a0 Consider the following very challenging Data Sufficiency question.   1. What is the value of x? Take…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,736],"tags":[],"class_list":["post-17028","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-quant-gmat","entry"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\n\t\t\n