tag:blogger.com,1999:blog-7399127082527147995.post6218465355396919473..comments2023-07-11T11:33:04.292-04:00Comments on Time, Effort, and Focus: this week's MGRE Math Beast ChallengeKevin Kimhttp://www.blogger.com/profile/01328790917314282058noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-7399127082527147995.post-41454499548707381842012-02-21T03:19:10.633-05:002012-02-21T03:19:10.633-05:00This one is a simple "plug-and-chug," as...This one is a simple "plug-and-chug," as they say: simply plug in 2 for "x," and see what happens. If you really want to, you can plug in 10 for x as well, just to make sure you're on the right track.<br /><br />For Quantity A, if you plug in a 2, then you get:<br /><br />1/(1-(1/(1+(1/2))))<br /><br />=<br /><br />1/(1-(1/(3/2)))<br /><br />= <br /><br />1/(1-(2/3))<br /><br />=<br /><br />1/(1/3)<br /><br />= 3<br /><br />For Quantity B:<br /><br />1/(1+(1/(1-(1/2))))<br /><br />= <br /><br />1/(1+(1/(1/2)))<br /><br />= 1/(1+2)<br /><br />= 1/3<br /><br />Obviously, A is greater than B.<br /><br />Just to be sure, plug in a 10 for x. In that case,<br /><br />Quantity A = 11<br /><br />Quantity B = 9/19<br /><br />At this point, it's obvious that, as x gets larger, Quantity B remains a fraction less than 1, whereas Quantity A keeps increasing as x increases. A will always be greater than B.<br /><br />For fun, I tried plugging in 10^6 for x. The result was still in favor of Quantity A. If you were to graph Quantity B as a function, you'd see that, as x increases in value, the function approaches 1/2 but can never reach it: 1/2 is, in fact, an asymptote! We can prove this by setting Quantity B equal to 1/2. If you do the math, you get the absurd result that -1 = 0. Such absurdities alert us to the presence of an asymptote. <br /><br />So: f(x) can never equal or surpass 1/2. <br /><br />f(x) < 1/2 as x approaches ∞.Kevin Kimhttps://www.blogger.com/profile/01328790917314282058noreply@blogger.com