<p>You drive half the distance to work at 30 mph. At what speed must you drive the second half of the distance in order to average 60 mph for the entire trip?</p>
<p>Excellent question!</p>
<p>Humm - this comes to mind:</p>
<p>[The</a> Impossible Dream-Man of La Mancha - YouTube](<a href=“The Impossible Dream-Man of La Mancha - YouTube”>The Impossible Dream-Man of La Mancha - YouTube)</p>
<p>:)</p>
<p>^Wow, the hair on Peter O’Toole is weird.</p>
<p>This problem reminds me of the farmer who tries to get a ladder to fit into a barn by running into it at half the speed of light.</p>
<p>I got 0 for some reason… Does this mean that the answer cannot exist?</p>
<p>Teleport. That’s what I do.</p>
<p>I believe the answer is 90 mph? 30+90 = 120/2 = 60 average.</p>
<p>This problem is indeed impossible. You could get very close to 60 mph average but never be able to double the original speed of the first leg. </p>
<p>As far as the formula, it comes in handy to show the impossibility:</p>
<p>2.30.x / x+30 = 60
60x = 60 * (x+30)
x = x+30</p>
<p>Now, if the question had been about averaging 50 mph, the answer would have been</p>
<p>2.30.x / x+30 = 50
60x = 50 * (x+30)
60x = 50x + 1500
10x = 1500
x = 150</p>
<p>
</p>
<p>2.30.90 / 30+90 =
2.30.90 / 120 =
90/2 = 45 </p>
<p>:)</p>
<p>@AnAlien</p>
<p>You are computing the arithmetic mean, but the correct formula to use here is the harmonic mean. On this forum, the harmonic mean formula has been renamed Xiggi’s formula. I conjecture that within 50 years this new name will become standard terminology.</p>
<p>An example:</p>
<p>[Averages</a>, Arithmetic and Harmonic Means](<a href=“http://www.cut-the-knot.org/arithmetic/HarmonicMean.shtml]Averages”>Averages, Arithmetic and Harmonic Means)</p>
<p>And here’s another obnoxoius viariation on the theme:</p>
<p>You ride your bike at 30 km/hr from your house to the base of a big hill. Then you ride at 20 km/hr going up the hill. Then you zip back down the hill at 60 km/hr. And then you go back home at 30 km/hr again. </p>
<p>Your average speed for the trip is…?</p>
<p>a) 30 km/hr
b) 35 km/hr
c) 110/3 km/hr
d) 40 km/hr
e) Can’t be determined without more information</p>
<p>Wow, will the harmon… I mean the Xiggi’s forumula be tested on the SAT?</p>
<p>
</p>
<p>It is always nice to remember that this is the SAT, and that there are always different approaches to the same problem. When it comes to this type of problems, anyone who can move forward WITHOUT having to start with the basic d=rt formula will come ahead. </p>
<p>In this case, there are three approaches. </p>
<ol>
<li><p>Make the hill 60 miles. That is total 240 m. Add the hours for each leg (2,3,1,2) and divide 240 by 8. A = 30.</p></li>
<li><p>Apply the formula twice. The 30 and 20 yields 24. The 60 and 30 yields 40. The formula 2.24.40 / 24+40 = 30. </p></li>
<li><p>Realize that the 20 and 60 averages to 30 (2.20.60 / 20+60 = 30) and that the other two 30 will not change the average.</p></li>
</ol>
<p>@AnAlien. My estimate is that about 1 out of every 5 SATs have a level 5 problem where a quick application of Xiggi’s formula will get you the answer in under 10 seconds. </p>
<p>Other problems will appear where Xiggi’s formula CAN be used, but it may not be the quickest way. I had a nice thread going with an example of this, and a really nice discussion, but unfortunately the moderators deleted it.</p>
<p>As discussed a few days ago, a problem of average speeds is now included in the latest "blue book.’ The writers added an 11th test on the DVD and that test is a version of the January 2008. The problem was the last of the grid-in (18 I believe.)</p>
<p>Let me say first that I didn’t write this one. It’s not from an SAT either. It was from a workshop I went to a long time ago and I adapted it. And when I first worked on it, I did exactly what I discourage others from doing: I assigned variables to the distances and then plugged and chugged through the algebra. It works, but it takes too long. And when I saw the answer, I was annoyed.</p>
<p>I think that this problem illustrates the importance of “number awareness” although admittedly on a subtle level. The Xiggi’s third solution shows what I mean. But you have to have noticed that the 20 and 60 have the harmonic mean of 30 and very few will pick that up. Still, when you read an SAT problem, remember that there are no coincidences. Numbers are chosen by the writers for a reason.</p>
<p>
</p>
<p>Yes, that’s correct. Out of 19 QAS tests since March 2005, that problem is the only one I know of that can be solved with a direct application of the xiggi formula. So my estimated xiggi fraction is lower than DrSteve’s … Haven’t seen Oct 2011 though :)</p>
<p>Perhaps, but the CCers in January 2008 might have been smiling. :)</p>
<p>Yes. Perhaps I should think of resistors in parallel as xiggi resistors from now on. :)</p>