{"id":11097,"date":"2012-08-21T14:06:25","date_gmt":"2012-08-21T21:06:25","guid":{"rendered":"https:\/\/magoosh.com\/gre\/?p=11097"},"modified":"2019-04-18T10:58:57","modified_gmt":"2019-04-18T17:58:57","slug":"gre-division-mixed-numerals-and-negatives","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/","title":{"rendered":"GRE Division, Mixed Numerals, and Negatives"},"content":{"rendered":"

\"GRE
\nThis is a post to clarify a potentially confusing passage in the OG.\u00a0 In the GRE OG 2e<\/a>, on p. 134, as well as in ETS’s GRE Mathematical Conventions PDF<\/a> on p. 5, point 9, we find a paragraph discussing quotients and remainders that utilizes the formula n = qd + r<\/em>. This formula pertains to the situation where an integer n<\/em> is divided by an integer d<\/em>. q<\/em> represents the quotient, and r<\/em> represents the non-negative remainder. The paragraph then goes on to discuss the results of the following expressions: 20 \u00f7 7, 21 \u00f7 7, and \u221217 \u00f7 7.<\/p>\n

Well, using their formula, we can verify those examples:<\/p>\n

a) 20 = (2)(7) + 6<\/p>\n

b) 21 = (3)(7) + 0<\/p>\n

c) \u201317 = (\u20133)(7) + 4<\/p>\n

All true.\u00a0 So far so good.\u00a0 The problem comes when we want to use all of this to convert an improper fraction to a mixed numeral.\u00a0 The guiding formula is what I will call the “Division Equation”:<\/p>\n

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

This equation is true 100% of the time.\u00a0 With example (a), if we just plug everything into this formula we magically change the improper fraction into a mixed numeral.<\/p>\n

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

Voila! The correct mixed numeral magically results.\u00a0 With example (b), we don’t need to worry about mixed numerals, because the divisor goes into the dividend evenly, and thus the result of the division is an integer: 21\/7 = 3.\u00a0 With example (c), things get tricky:<\/p>\n

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

The Division Equation above still works, but for negatives, we can’t just dump the numbers in and magically get the correct negative mixed numeral result.\u00a0 This has to do with the way ETS<\/a> has chosen to define remainders for negative numbers.\u00a0 Let’s explore this.<\/p>\n

 <\/p>\n

Remainders for negative numbers<\/h2>\n

In the passage from p. 134 quoted above, every math book on planet Earth, and probably most written by aliens on other planets, would agree 100% with what ETS said about quotients and remainders for positive numbers.\u00a0 All that is absolutely undisputable.\u00a0 The vast majority of math books deliberately ignore the entire topic of what happens to quotients and remainders with negative numbers, because of the mathematical trade-offs that arise. \u00a0Most math books have the good sense to avoid this troublesome topic, but ETS brashly charges in and firmly establishes a position.\u00a0 (To follow knowledge like a sinking star beyond the utmost bound? or, fools rush in where angels fear to tread?)<\/p>\n

The disadvantage<\/em> of defining remainders of negative numbers in this way is that makes finding negative mixed numerals more difficult: we can no longer use the Division Equation — we’ll get back to that point below.\u00a0 The advantage<\/em> is: when you divide by a certain divisor, say 7, and talk about all the numbers that have a remainder of 4, those numbers are equally spaced on the number line in both the positive and negative direction.\u00a0 The numbers which, when divided by 7, have a remainder of 4 are:<\/p>\n

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

Notice that \u201317 appears in that pattern, and follows the same pattern as the familiar positive numbers which, when divided by 7, have a remainder of 4.\u00a0 It’s very important: if ETS mentions a dividend which, when divided by 7, has a remainder of 4, don’t automatically assume the dividend is positive: it could be negative, and it would follow this sort of pattern.<\/p>\n

 <\/p>\n

Thoughts on mixed numerals<\/h2>\n

What exactly is a mixed numeral?\u00a0 That is to say, what do we mean when we write, say:<\/p>\n

\"2{1\/3}\"<\/p>\n

What this really means is:<\/p>\n

\"2{1\/3}<\/p>\n

In other words, any mixed numeral implicitly contains addition between the integer and the fraction.<\/em><\/strong>\u00a0 That is the BIG<\/strong> idea of mixed numerals.<\/p>\n

It’s relatively easy to see that, for positive numbers, the Fraction Equation will result in a whole number quotient plus a fraction less than one, which automatically fits the pattern for a mixed number.<\/p>\n

What’s going on with a negative mixed number? Well, that’s a little different:<\/p>\n

\"-2{1\/3}<\/p>\n

The negative sign in front of the whole mixed number applies to both terms and distributes, so you get subtraction instead of addition.\u00a0 If we are using ETS’ definition of remainder for negative numbers, then the Fraction Equation still works, but it doesn’t help us convert a negative improper fraction to a negative mixed number, because it results in adding <\/em>the fraction instead of subtracting<\/em> it.\u00a0 Again, this part is the disadvantage of the convention ETS is following on this particular topic.\u00a0 The Fraction Equation, so helpful for converting positive improper fractions to positive mixed numerals, is ostensibly useless in helping us analogously with negative numbers.<\/p>\n

 <\/p>\n

Negative mixed numerals<\/h2>\n

The question arises, then: how do we convert a negative improper fraction to a negative mixed numeral?\u00a0 For example, suppose we have the fraction \u201317\/7.\u00a0 Again, as the OG p. 134 tells us, “when \u201317 is divided by 7, the quotient is \u20133, and the remainder is 4.”\u00a0 That will be consistent with the Fraction Equation, but, as we have seen above, that in turn will not lead to a correct mixed numeral.\u00a0 Therefore, everything OG says about quotients and remainders for negative number is useless if what we want to find is a negative mixed numeral.\u00a0 What do we do?<\/p>\n

Actually, what you do is remarkably simple: pretend the negative sign isn’t there for the moment, and just convert the positive improper fraction to positive mixed numeral!<\/p>\n

Now, we are dividing +17 by 7: of course, the quotient is 2 and the remainder is 3, so the Fraction Equation conveniently gives us<\/p>\n

\"{17\/7}<\/p>\n

So far, so good.\u00a0 As a result of working with the positive numbers, we know:<\/p>\n

\"{17\/7}<\/p>\n

But what we wanted was \u201317 divided by 7, not +17 divided by 7.\u00a0 How do we get that?\u00a0 We get that simply by multiplying the previous equation by a negative sign on both sides.<\/p>\n

\"-{17\/7}<\/p>\n

Again, to convert negative improper fraction to a negative mixed numeral: just pretend the improper fraction is positive, do the conversion with positive numbers, and then simply make the output negative when you are done.<\/p>\n

 <\/p>\n

Practice questions<\/h2>\n

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

 <\/p>\n

Practice question solutions<\/h2>\n

1) Here, we must follow the ETS convention.\u00a0 According to this convention, when a negative number is divided the remainder must be positive.\u00a0 What is the multiple of 6 that is less than \u201323 and closest to \u201323?\u00a0 That would be\u00a0 \u201324, so 6 goes into\u00a0 \u201323\u00a0 negative four times, with a remainder of (\u201323) \u2013 (\u201324) = +1.\u00a0 This checks out with the ETS’s equation: n = \u201323, d = 6, q = \u20134, and r = 1, and it’s true that<\/p>\n

\u201323 = (6)( \u20134) + 1<\/p>\n

So the quotient is \u20134 and the remainder is +1.\u00a0 Answer = E<\/strong>.<\/p>\n

2) Here, we will first change it to a positive improper fraction.\u00a0 When +23 is divided 6, the quotient is 3 and the remainder is 5, so:<\/p>\n

\"{23\/6}<\/p>\n

Now, just multiply both sides by a negative sign.<\/p>\n

\"-{23\/6}<\/p>\n

Answer = B<\/strong>.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

This is a post to clarify a potentially confusing passage in the OG.\u00a0 In the GRE OG 2e, on p. 134, as well as in ETS’s GRE Mathematical Conventions PDF on p. 5, point 9, we find a paragraph discussing quotients and remainders that utilizes the formula n = qd + r. This formula pertains […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[41,25],"tags":[],"ppma_author":[12267],"class_list":["post-11097","post","type-post","status-publish","format-standard","hentry","category-arithmetic","category-math"],"acf":[],"yoast_head":"\nGRE Division, Mixed Numerals, and Negatives - Magoosh Blog \u2014 GRE\u00ae Test<\/title>\n<meta name=\"description\" content=\"In this post, we\u2019ll take a look at a paragraph discussing quotients and remainders that utilizes the formula n = qd + r from the GRE OG 2e.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"GRE Division, Mixed Numerals, and Negatives\" \/>\n<meta property=\"og:description\" content=\"In this post, we\u2019ll take a look at a paragraph discussing quotients and remainders that utilizes the formula n = qd + r from the GRE OG 2e.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GRE\u00ae Test\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Magoosh\/\" \/>\n<meta property=\"article:published_time\" content=\"2012-08-21T21:06:25+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-04-18T17:58:57+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gre\/files\/2012\/08\/image-gre-header-GREdivision.jpg\" \/>\n<meta name=\"author\" content=\"Mike M\u1d9cGarry\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGRE\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGRE\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mike M\u1d9cGarry\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\"},\"author\":{\"name\":\"Mike M\u1d9cGarry\",\"@id\":\"https:\/\/magoosh.com\/gre\/#\/schema\/person\/320346c205075513344435baf9b0521b\"},\"headline\":\"GRE Division, Mixed Numerals, and Negatives\",\"datePublished\":\"2012-08-21T21:06:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\"},\"wordCount\":1099,\"commentCount\":31,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gre\/#organization\"},\"articleSection\":[\"GRE Arithmetic\",\"GRE Math\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\",\"url\":\"https:\/\/magoosh.com\/gre\/gre-division-mixed-numerals-and-negatives\/\",\"name\":\"GRE Division, Mixed Numerals, and Negatives - 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