tag:blogger.com,1999:blog-7399127082527147995.post905814869188822971..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.comBlogger3125tag:blogger.com,1999:blog-7399127082527147995.post-80843940805006185792012-04-10T21:11:15.204-04:002012-04-10T21:11:15.204-04:00I just want to know where Jack lives that he has t...I just want to know where Jack lives that he has to deal with such an insane tax system.Charleshttp://www.liminality.orgnoreply@blogger.comtag:blogger.com,1999:blog-7399127082527147995.post-54112337571658811422012-04-10T08:55:46.920-04:002012-04-10T08:55:46.920-04:00Kevin, you are correct. Another methodology derive...Kevin, you are correct. Another methodology derives the same result:<br /><br />Lower tax percentage of 15%:<br />.15 time X = 8700<br />X = 8700 / .15<br />X = 58,000<br /><br />Upper tax percentage of 35%:<br />.35 times X = 8700<br />X = 8700 / .35<br />X = 24,857.14<br /><br /><br />Both amounts are higher than the exclusion range, so you add the total amount of the range to both the upper and lower tax rates:<br />9800 - 5200 = 4600<br /><br />58,000 + 4,600 = 62,600<br />24,857.14 + 4,600 = 29,457,14<br /><br />All choices between these amounts, inclusive, are possible income amounts that yield an $8,700 tax liability.<br /><br />That would be the ones you identified in your solution.<br /><br />DaveDavehttp://mutzu501atgmail.comnoreply@blogger.comtag:blogger.com,1999:blog-7399127082527147995.post-89536696312615443382012-04-10T00:33:26.845-04:002012-04-10T00:33:26.845-04:00This is one of those GRE problems for which multip...This is one of those GRE problems for which multiple responses are possible. I have a bad feeling that my own answer is wrong, because I ended up choosing four of the seven possibilities. Let me take you through my reasoning, though, and you can tell me whether I'm off my rocker.<br /><br />Let's start with what we know and what we can derive.<br /><br />I = total income (the thing we want to find out)<br /><br />E = exclusion amount, which is a range:<br /><br />$5200 < E < $9800<br /><br />T = income tax = $8700<br /><br />I' = net income (i.e., I - E)<br /><br />We can render the tax as an inequality:<br /><br />0.15•I' < T < 0.35•I'<br /><br />or<br /><br />0.15•I' < $8700 < 0.35•I'<br /><br />which is<br /><br />0.15(I - E) < $8700 < 0.35(I - E)<br /><br />It seems obvious to me that the upper-range figure for the exclusion amount should go on the left: by subtracting it from I, we end up with a smaller number, and thus we can define the lower end of the tax range. So let's plug in $9800 for E on the left end of the inequality, and $5200 for E on the right end:<br /><br />0.15(I - 9800) < 8700 < 0.35(I - 5200)<br /><br />This distributes out:<br /><br />0.15I - 0.15(9800) < 8700 < 0.35I - 0.35(5200)<br /><br />We can solve for I for the left-hand inequality:<br /><br />0.15I - 1470 < 8700<br /><br />0.15I < 10,170<br /><br />I < 67,800<br /><br />Be careful now: this gives us the <i>upper</i> range of Jack's salary, because it's the <i>lower</i> end of the tax spectrum. Let's turn, now, to the right-hand side of the inequality.<br /><br />8700 < 0.35I - 1820<br /><br />10520 < 0.35I<br /><br />approx. 30,057 < I<br /><br />So the range for I, if I've done this correctly, would appear to be:<br /><br />$30,057 < I < $67,800<br /><br />By that reasoning, I conclude that the answer to the question is:<br /><br />$33,200<br />$43,300<br />$53,400<br />$63,500<br /><br />If I've gone wrong somewhere, here's your chance to tell me.Kevin Kimhttps://www.blogger.com/profile/01328790917314282058noreply@blogger.com