{"id":137,"date":"2015-12-04T16:04:31","date_gmt":"2015-12-05T00:04:31","guid":{"rendered":"https:\/\/magoosh.com\/praxis\/?p=137"},"modified":"2019-01-30T20:30:52","modified_gmt":"2019-01-31T04:30:52","slug":"fractions-on-the-praxis-core-mathematics-test-part-two","status":"publish","type":"post","link":"https:\/\/magoosh.com\/praxis\/fractions-on-the-praxis-core-mathematics-test-part-two\/","title":{"rendered":"Fractions on the Praxis Core Math Test: Part Two"},"content":{"rendered":"

In the previous post<\/a>, we discussed the basics of fractions as well as fraction addition and subtraction.\u00a0 As promised, in this post, we will discuss fraction multiplication and division.\u00a0 Have courage, friends!\u00a0 With study and practice, you can become a pro with fractions!<\/p>\n

We’ll start with a couple practice problems.<\/p>\n

\"fotpcmtp2_img1\"<\/a><\/p>\n

\"fotpcmtp2_img2\"<\/a><\/p>\n

Solutions will follow the article.<\/p>\n

 <\/p>\n

Multiplication<\/h2>\n

The rule for multiplying fractions is very intuitive.\u00a0 We just multiply across, numerator times numerator over denominator times denominator:<\/p>\n

\"fotpcmtp2_img3\"<\/a><\/p>\n

The “why” here gets very interesting.\u00a0 It turns out that this rule, multiplying across, is just one<\/em> of the many things we are allowed to do.\u00a0 A mathematician looking at (2\/7) times (3\/5) would see a product of four different factors: (2), (1\/7), (3), and (1\/5).\u00a0 These four factors can be rearranged in any orders and groups, according to the Commutative and Associative laws.\u00a0 Thus, in general,<\/p>\n

\"fotpcmtp2_img4\"<\/a><\/p>\n

Yes, it is perfectly fine to multiply across, and it’s also important to recognize these many other arrangements as entirely mathematically equivalent.<\/p>\n

 <\/p>\n

Division<\/h2>\n

Dividing fractions is also relatively easy.\u00a0 Dividing by a fraction is the the same a multiplying by the reciprocal of the fraction.\u00a0 The reciprocal<\/strong> of the fraction is the fraction “flipped over”: the old numerator becomes the denominator, and the old denominator becomes the numerator.\u00a0 For example,<\/p>\n

3\/5 and 5\/3 are reciprocal<\/p>\n

2\/7 and 7\/2 are reciprocals<\/p>\n

The product of any fraction times its reciprocal is 1.\u00a0 The technical way to say that is that the reciprocal of any fraction is the multiplicative inverse of the fraction.<\/p>\n

Dividing by a fraction is equivalent to multiplying by the fraction’s reciprocal.\u00a0 Thus, dividing by 3\/11 is equivalent to multiplying by 11\/3.\u00a0 Dividing by 8\/5 is the same as multiplying by 5\/8.\u00a0 You get the idea.\u00a0 If you are able to multiply fractions, then you are able to divide them.<\/p>\n

 <\/p>\n

Caveat #1: integers and fractions<\/h2>\n

Folks sometimes get stymied if they have to multiply an integer and a fraction, or divide either a fraction by an integer or an integer by a fraction.\u00a0 For all these cases, it helps to remember that any integer can be written as that integer over one.<\/p>\n

\"fotpcmtp2_img5\"<\/a><\/p>\n

This is a direct consequence of the fact that any integer divided by one is equal to that integer: it doesn’t change a number to multiply or divide it by one.\u00a0 Thus, any integer can be written as a fraction, and the reciprocal would simply be one over the integer: the reciprocal of 15 is 1\/15.\u00a0 Thus, we can do all combinations.<\/p>\n

\"fotpcmtp2_img6\"<\/a><\/p>\n

This last one was left as an improper fraction (i.e. the numerator is bigger than the denominator).\u00a0 This is fine for now.\u00a0 In the next post, I will discuss the issue of improper fractions vs. mixed numerals.<\/p>\n

 <\/p>\n

Caveat #2: canceling<\/h2>\n

In all the examples in this discussion so far, I have given only examples in which canceling would not be an issue.\u00a0 I avoided it so far because it is such an important topic that it deserves its own treatment.<\/p>\n

We have already discussed the fact that any number over itself divides to one.\u00a0 We already discussed how we can use that fact to construct fractions such as 3\/3 or 17\/17 that have a numerical value of one: such fractions play a role in creating common denominators.<\/p>\n

Many times, especially if the numerator and denominator are larger numbers, we can cancel a common factor in the numerator and denominator of the same fractions.\u00a0 This is called simplifying.\u00a0 When we simplify fully, so that no more simplifying is possible, this is called putting the fraction is simplest terms<\/strong>.\u00a0 It is an extremely valuable mathematical habit to put into simplest terms every fraction that crosses your path!!<\/p>\n

For example, the fraction 15\/25 has a common factor of 5 in both the numerator and denominator.\u00a0 When we cancel the factor of 5, we are left with 2\/3, a simplified fraction.<\/p>\n

Some people think that canceling means that factors simply “go away”—in fact, as reprehensible as this is, some math teachers even use this poor language.\u00a0 Nothing in mathematics simply “goes away.”\u00a0 Cancelling is fundamentally\u00a0an act of division<\/em><\/strong>.\u00a0 This becomes very significant when we have to simplify a fraction such as 8\/56.\u00a0 Clearly, both numerator and denominator have a factor of 8, because 56 = 8*7.\u00a0 Clearly, when the 8 in the denominator cancels, we are left with a 7 in the denominator.\u00a0 What happens in the numerator?\u00a0 Folks holding the erroneous “goes away” position might get confused here by the nothingness that they think results in the numerator once the 8 “goes away.’\u00a0 In fact, what has happened is division, and 8 divided by 8 is 1.\u00a0 We are left with a one in the numerator and a seven in the denominator: the fraction simplifies to 1\/7.<\/p>\n

We can simplify in a single fraction, but most of the individual fractions you will be given on the Praxis Core Mathematics Test already will be in simplest form.\u00a0 Canceling takes on a much larger role in the multiplication and division of fractions.<\/p>\n

Here we run into another poisonous term of which some math teachers are fond: cross-canceling.\u00a0 I will discuss below why I think this term is detrimental to understanding fractions.\u00a0 Here, we simply don’t need it.\u00a0 Suppose a problem resulted in the following multiplication:<\/p>\n

\"fotpcmtp2_img7\"<\/a><\/p>\n

What can we cancel?\u00a0 The rule is simple.\u00a0 When we are multiplying fractions, we can cancel any numerator with any denominator<\/em><\/strong>!\u00a0 For example we can cancel the factor of 7 in common between the 63 and the 49, simplifying the fraction on the right.<\/p>\n

\"fotpcmtp2_img8\"<\/a><\/p>\n

Now, we can cancel that other 7 in the denominator with the factor of 7 in 35 (remember that we are left with a one after we cancel the 7).<\/p>\n

\"fotpcmtp2_img9\"<\/a><\/p>\n

Now, we can cancel the factor of 9 in the 27 with the 9, and then we can complete the multiplication.<\/p>\n

\"fotpcmtp2_img10\"<\/a><\/p>\n

Another beautiful improper fraction: for now, we will leave it in that form.\u00a0 Notice that, once we canceled, the multiplication was ridiculously easy to perform.\u00a0 Notice that if we had begun by multiplying the uncanceled original numbers, we would have gotten huge numbers, 2205 in the numerator and 1323 in the denominator\u2014good luck canceling common factors from those gargantuans!\u00a0 Choosing to multiply uncanceled numbers is like banging in nails with your head: it is a spectacularly bad idea!\u00a0 One of the most productive mathematical habits you can cultivate is: always cancel completely before you multiply<\/em><\/strong>.\u00a0\u00a0 The value of this should be evident from the multiplication above.<\/p>\n

 <\/p>\n

Caveat #3: proportions<\/h2>\n

Another word for a fraction is a ratio<\/strong>.\u00a0 The two words mean the same thing mathematically, although ratio is more often used to denote a fraction of some real world or geometric quantities.\u00a0 A proportion<\/strong> is simply an equation of the form [fraction] = [fraction].\u00a0 Again, proportions are often set up to solve all kind of real world or geometric problems.\u00a0 Here, I will just discuss the pure mathematics of proportions themselves, not their myriad real world applications.<\/p>\n

For any proportion, a 100% bonafide all-the-time move is something called cross-multiplication<\/strong>.\u00a0 It works like this:<\/p>\n

\"fotpcmtp2_img11\"<\/a><\/p>\n

Each numerator is multiplied by the denominator on the opposite side.\u00a0 This is a totally legitimate, 100% correct move, and the advantage of it is that it eliminates fractions.\u00a0 If there is an algebraic variable somewhere in the proportion, then we go from an algebra problem with fractions to one with no fractions: always a step toward simplification!<\/p>\n

As with ordinary multiplication of fractions, it’s good to cancel before you multiply.\u00a0 The question is: exactly what are we allowed to cancel in a proportion?\u00a0 Of course, we can always do up-down cancellation within either single fraction.\u00a0 Let’s call that vertical cancellation: up-and-down in a single fraction.<\/p>\n

We also are allowed to cancel a common factor in both numerators: this is the equivalent of dividing both sides by the same number.\u00a0 We can also cancel a common factor in both denominator: this is the equivalent of multiplying both sides by the same number.\u00a0 Let’s call this horizontal canceling.\u00a0 These are the legitimate forms of canceling in a proportion:<\/p>\n

\"fotpcmtp2_img12\"<\/a> \"fotpcmtp2_img13\"<\/a><\/p>\n

All good so far, but a great deal of confusing arises from the deleterious idea of cross-canceling.\u00a0 You see, the proper domain of this suspect idea is the multiplication of fractions, and once there’s an equal sign between two fractions instead of a multiplication sign, all the rules are different.\u00a0 Folks who learn cross-canceling mechanically mistakenly apply it to the proportion situation, and the very name seems to reinforce this mechanical misapprehension.\u00a0 The following kind of canceling is 100% illegal and never valid in proportions.<\/p>\n

\"fotpcmtp2_img14\"<\/a> \"fotpcmtp2_img15\"<\/a><\/p>\n

Student who diligent learned cross-canceling are sometimes astounded to learn that this kind of canceling is illegal in a proportion, while the horizontal canceling above is legal.\u00a0 If this is your reaction, then your understanding has been damaged by the poisonous influence of the idea of cross-canceling.<\/p>\n

 <\/p>\n

Summary<\/h2>\n

If you had any moments of “aha” while you were reading this article, you may want to give the practice problems at the top another look before reading the explanations below.\u00a0 The final article in this series will explore the issue of improper fractions vs. mixed numerals.<\/p>\n

\"fotpcmtp2_img16\"<\/a><\/p>\n

 <\/p>\n

Practice problem explanations<\/h2>\n

1) Think of the 36 in the numerator as 36\/1, and we are dividing two fractions.\u00a0 This means we multiply the numerator fraction, 36\/1, by the reciprocal of the denominator.<\/p>\n

\"fotpcmtp2_img17\"<\/a><\/p>\n

Answer = (B)<\/strong><\/p>\n

2) This one contains a trap for people who mistakenly think that the idea of “cross-canceling” applies to proportions.\u00a0 We cannot<\/em><\/strong> cancel diagonally in a proportion.\u00a0 We can cancel a factor of 3 in the two numerators before we cross-multiply:<\/p>\n

\"fotpcmtp2_img18\"<\/a><\/p>\n

Answer = (E)<\/strong><\/p>\n

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

In the previous post, we discussed the basics of fractions as well as fraction addition and subtraction.\u00a0 As promised, in this post, we will discuss fraction multiplication and division.\u00a0 Have courage, friends!\u00a0 With study and practice, you can become a pro with fractions! We’ll start with a couple practice problems. Solutions will follow the article. […]<\/p>\n","protected":false},"author":26,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"footnotes":""},"categories":[2435],"tags":[],"ppma_author":[4911],"class_list":["post-137","post","type-post","status-publish","format-standard","hentry","category-praxis-math-practice"],"acf":[],"yoast_head":"\nFractions on the Praxis Core Math Test: Part Two - Magoosh Blog \u2013 Praxis\u00ae\ufe0f Test<\/title>\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\/praxis\/fractions-on-the-praxis-core-mathematics-test-part-two\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Fractions on the Praxis Core Math Test: Part Two\" \/>\n<meta property=\"og:description\" content=\"In the previous post, we discussed the basics of fractions as well as fraction addition and subtraction.\u00a0 As promised, in this post, we will discuss fraction multiplication and division.\u00a0 Have courage, friends!\u00a0 With study and practice, you can become a pro with fractions! 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Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations.","sameAs":["https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured"],"award":["Magna cum laude from Harvard"],"knowsAbout":["GMAT"],"knowsLanguage":["English"],"jobTitle":"Content Creator","worksFor":"Magoosh","url":"https:\/\/magoosh.com\/praxis\/author\/mikemcgarry\/"}]}},"authors":[{"term_id":4911,"user_id":26,"is_guest":0,"slug":"mikemcgarry","display_name":"Mike M\u1d9cGarry","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/6b06de81592cd77bb46aa560cc59aee179cba4d042835c3529221ea1b344cce0?s=96&d=mm&r=g","user_url":"","last_name":"M\u1d9cGarry","first_name":"Mike","description":"Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as \"member of the month\" for over two years at <a href=\"https:\/\/gmatclub.com\/blog\/2012\/09\/mike-mcgarrys-gmat-experience\/\" rel=\"noopener noreferrer\">GMAT Club<\/a>. Mike holds an A.B. in Physics (graduating <em>magna cum laude<\/em>) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's <a href=\"https:\/\/www.youtube.com\/c\/MagooshGMATChannel\/featured\" rel=\"noopener noreferrer\">Youtube <\/a>video explanations and resources like <a href=\"https:\/\/magoosh.com\/gmat\/whats-a-good-gmat-score\/\" rel=\"noopener noreferrer\">What is a Good GMAT Score?<\/a> and the <a href=\"https:\/\/magoosh.com\/gmat\/gmat-diagnostic-test\/\" rel=\"noopener noreferrer\">GMAT Diagnostic Test<\/a>."}],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/posts\/137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/users\/26"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/comments?post=137"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/posts\/137\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/media?parent=137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/categories?post=137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/tags?post=137"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/praxis\/wp-json\/wp\/v2\/ppma_author?post=137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}