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<\/p>\nWhat are rates and ratios? How do rate and ratios problems differ? Hint: Not that much.<\/p>\n
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In fact, rates are just ratios in disguise.\u00a0Here are a four GMAT practice problems exploring rates and ratios.\u00a0 Remember:\u00a0no calculator<\/a>!<\/p>\n 1) Someone on a skateboard is traveling 12 miles per hour.\u00a0 How many feet does she travel in 10 seconds?\u00a0 (1 mile = 5280 feet)<\/p>\n (A) 60 <\/p>\n 2) At 12:00 noon, a machine, operating at a fixed rate, starts processing a large set of identical items.\u00a0 At 1:45 p.m., the twenty-first item has just been processed, and 15 have not yet been processed.\u00a0 At what time will all 36 items be processed?<\/p>\n (A) 2:25 pm <\/p>\n 3) An importer wants to purchase\u00a0N<\/strong>\u00a0high quality cameras from Germany and sell them in Japan.\u00a0 The cost in Germany of each camera is\u00a0E<\/strong>\u00a0euros.\u00a0 He will sell them in Japan at\u00a0Y<\/strong>\u00a0yen per camera, which will bring in a profit, given that the exchange rate is\u00a0C<\/strong>\u00a0yen per euro.\u00a0 Given the exchange rate of\u00a0D<\/strong>\u00a0US dollars per euro, and given that profit = (revenue) \u2013 (cost)<\/strong>, which of the following represents his profit in dollars?<\/p>\n (A) N(YC \u2013 DE)<\/p>\n (B) ND(YC \u2013 E)<\/p>\n (C) ND((Y\/C) \u2013 E)<\/p>\n (D) N((Y\/C) \u2013 DE)<\/p>\n (E) ND(Y \u2013 E)\/C<\/p>\n <\/p>\n 4) Machine A and machine B process the same work at different rates.\u00a0 Machine C processes work as fast as Machines A & B combined.\u00a0 Machine D processes work three times as fast as Machine C; Machine D’s work rate is also exactly four times Machine B’s rate.\u00a0 Assume all four machines work at fixed unchanging rates.\u00a0\u00a0 If Machine A works alone on a job, it takes 5 hours and 40 minutes.\u00a0 If all four machines work together on the same job simultaneously, how long will it take all of them to complete it?<\/p>\n (A) 8 minutes <\/p>\n Solutions will come to these at the end of the article. Can’t contain your excitement? Click here<\/a> to skip to the explanations.<\/p>\n Ratios<\/strong> are fractions<\/a>.\u00a0 When we have an equation of the form fraction = fraction<\/strong>, that’s called a proportion<\/strong>.\u00a0\u00a0 By far, the hardest part of dealing with a proportion is what you CAN and what you CAN’T cancel in a proportion.\u00a0 Many students are quite confused on this issue.<\/p>\n First of all, let’s be clear that cancelling is simply division.\u00a0 If I start with the fraction 24\/32, and I “cancel the 8’s” to get 3\/4, what I have really done is divide both the numerator and the denominator by 8.\u00a0 Similarly, if I have 5\/35, and I cancel the 5’s, in the numerator, I am left with 1: the simplified version is 1\/7.\u00a0 Too many student have the na\u00efve view that canceling means “going away” or some other fairy-godmother operation.\u00a0 Instead, cancelling is a card-carrying legitimate mathematical operation, the operation of division.<\/p>\n Clearly, it’s always legitimate to cancel in the numerator and denominator of the same fraction, the same ratio, so of course we can do that on each side in a proportion.<\/p>\n We might call that “vertical canceling” in a proportion: that’s 100% legal.\u00a0 The one that often surprises folks is what we might call that “horizontal canceling” in a proportion, which looks like this:<\/p>\n Canceling a common factor from a<\/strong> & c<\/strong> would simply involve dividing both sides of the equation by the same number, a 100% legal move.\u00a0\u00a0 Similarly, canceling a common factor from b<\/strong> & d<\/strong> would simply involve multiplying both sides of the equation by the same number, another totally legal move.\u00a0 Even though “horizontal canceling” across the equal sign may look suspect, it’s totally valid.<\/p>\n Now, the one that causes real problems is what we might call “diagonal canceling,” because so many students seem to be under the impression that is this OK, but in fact, it’s 100% illegal and incorrect.<\/p>\n I suspect people confuse this with “cross-canceling” in the process of multiplying fractions.\u00a0 I actually abhor that uses term, “cross-canceling”: I think this term causes dozens of times more harm than good.\u00a0 If we were to perform the canceling of a<\/strong> with d<\/strong>, that would essential be equivalent to dividing one side of an equation by a number and multiply the other side of the equation by the same number!\u00a0 That’s not allowed!\u00a0 We always have to do the same thing to both sides!\u00a0 This is why this kind of “diagonal canceling” in a proportion is always disastrously incorrect.<\/p>\n OK, that’s the relevant mathematics without the real world stuff involved!<\/p>\n Rates are ratios, that is, fractions.\u00a0 Any fraction with different units in the numerator and in the denominator is a rate: miles per hour, $ per pound, grams per cubic centimeter, etc.\u00a0 Most rate questions can be solved by setting up a\u00a0proportion.\u00a0 One common proportion type involves a (part)\/(whole) on each side:\u00a0for example, part of the job over all of the job, and part of the price or time over all of the price or time. In setting up any rate proportion, we have to make sure that units match: the same units in the two numerators, and the same units in the two denominators.\u00a0 The GMAT will expect you to know a few common unit changes (e.g. 1 hour = 60 min; 1 dozen items = 12 items, etc.); because some test-takers are familiar with metric and other are familiar with English, the GMAT most often would specify the conversion, as in #1 above.\u00a0 In any case, a GMAT rate problem often involves reconciling units differences in some way before we can do the math.<\/p>\n Another pertinent topic is that of work rates<\/a>.\u00a0 Suppose Machine P does a job in 3 hours and Machine Q can do the same job in 6 hours.\u00a0 How fast would it take both machines working together?\u00a0 You see, we can’t add or subtract the times it takes to perform jobs.\u00a0 What we can add are the work rates!\u00a0 It doesn’t matter that these work rates would have the ambiguous units of “job\/hour”\u2014it doesn’t matter as long as every rate in the problem has the same units.\u00a0 The rate of P, job per time, would be 1\/3, which means either one job every three hours or one-third of a job every hour: either is correct.\u00a0\u00a0 Similarly, the rate of Q would be 1\/6.\u00a0 We can’t add or subtract times, but we can add individual work rate to find a combined work rate.\u00a0 Adding fractions, we get (1\/6) + (1\/3) = (1\/6) + (2\/6) = 3\/6 = 1\/2.\u00a0\u00a0 The combined rate of P & Q working together is 1\/2, or one job per 2 hours.\u00a0 Thus, if P & Q were working together, it would take them just two hours to get the job done.\u00a0 That is the basic logic of work rates.<\/p>\n If you understand the rules of fractions and the concept of work rate, there’s nothing about rates and ratios you can’t understand.\u00a0 If you had any “aha” moments while reading this article, give the practice problems above another look before jumping in the solutions below.<\/p>\n 1) The speed is 12 mph.\u00a0 To change this to feet\/second, we need to multiply by (5280 ft\/mile), to cancel miles, and to multiply by (1 hour\/3600 second) to cancel seconds.<\/p>\n So, in 10 seconds, the skateboarder moves 176 feet.\u00a0 Answer =\u00a0(D)<\/strong><\/p>\n <\/p>\n 2) At 1:45, that is, 105 minutes after starting, the machine has completed 21\/36 = 7\/12 of the job.\u00a0 Let\u00a0T<\/strong>\u00a0be the whole time in minutes.\u00a0 For the total time, set up a proportion<\/p>\n Remember, with\u00a0proportions<\/a>, we can cancel a common factor in the two numerators; cancel the factor of 7.<\/p>\nRates and Ratios Practice Problems<\/h2>\n
\n(B) 88
\n(C) 120
\n(D) 176
\n(E) 264<\/p>\n
\n(B) 3:00 pm
\n(C) 3:27 pm
\n(D) 4:13 pm
\n(E) 5:15 pm<\/p>\n
\n(B) 17 minutes
\n(C) 35 minutes
\n(D) 1 hour and 15 minutes
\n(E) 1 hours and 35 minutes<\/p>\nRatios and Proportions<\/h2>\n
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<\/p>\nRates<\/h2>\n
Rate and Ratios Summary<\/h2>\n
![\"intersecting-nested-circles\"](\"https:\/\/magoosh-company-site.s3.amazonaws.com\/wp-content\/uploads\/sites\/3\/2016\/09\/12164618\/intersecting-nested-circles.png\")
\n<\/a><\/p>\nPractice Problem Explanations<\/h2>\n
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