<p>Can anybody give me a brief explanation of it?</p>
<p>I cant get to understand it because it seems that it has more than one form.</p>
<p>Sometimes i see the formula multiplied by 2 and sometimes not.</p>
<p>Any help?</p>
<p>Xiggi’s formula is what is more commonly know as the formula for the harmonic mean. </p>
<p>A common mistake is to take the “average” of two rates by using the arithmetic mean. But for rates the harmonic mean should be used.</p>
<p>Average rate = 2(rate1)(rate2)/(rate1 + rate2)</p>
<p>Note that this formula works only when the two distances are the same. Here is a sample problem where Xiggi’s formula can be used to get the answer very quickly:</p>
<p>An elephant traveled 7 miles at an average rate of 4 miles per hour and then traveled the next 7 miles at an average rate of 1 mile per hour. What was the average speed, in miles per hour, of the elephant for the 14 miles? </p>
<p>It’s basically the harmonic mean of two numbers, and is often used to find the average speed if half the distance is traveled at rate r1 and half the distance is traveled at rate r2.</p>
<p>In the above case, if you travel distance d at rate r1 and distance d at rate r2, then overall, you travel 2d distance in time d/r1 + d/r2, so your average rate is 2d/(d/r1 + d/r2) = 2/(1/r1 + 1/r2), which is the harmonic mean of r1 and r2 (this generalizes to more rates). The above expression is equal to 2r1*r2/(r1 + r2).</p>
<p>In general, the harmonic mean of n numbers x1, …, xn is n/(1/x1 + 1/x2 + … + 1/xn).</p>
<p>Why do you need a formula for that? Don’t clog your brain with formulas, use logic. </p>
<p>v1 = d1 / t1
(4) = (7) / t1
t1 = 7 / 4</p>
<p>v2 = d2 / t2
(1) = (7) / t2
t2 = 7</p>
<p>velocity = (delta)position / (delta)t
velocity = (14) / [ (7/4) + 7]
velocity = 14 / (53/4)
velocity = 56 / 53 mi/h</p>
<p>^should be 35 instead of 53.</p>
<p>I agree, but it may be helpful to remember the harmonic mean aspect. After seeing such a rate problem many times, it is not really worth re-deriving the solution (however one must be careful that it only works when the distances are the same!). For example, a while back I had difficulty remembering the sum of a geometric series but I was able to derive it instantly, so eventually I committed to memory the formula for the sum so I don’t have to re-derive it again.</p>
<p>
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<p>Because you can use your logic to recognize a problem and apply a VERY simple and elegant solution that works every time on the SAT. </p>
<p>By the way, when the problem appears on the SAT on a multiple choice, do you realize you CAN answer it with logic in 2 seconds? I could also tell you that the answer is always B, but that would be too darn much logic for one day.</p>
<p>Anyway, learn the darn formula and be happy you did! </p>
<p>I got it.
Thank you all!</p>
<p>Just wanna make sure about something.If the numerator has the coefficient 2 then you will get the average of the whole trip,but if the coefficient is 1,then you will get the average of half of the trip.</p>
<p>Right? </p>
<p>@meumeu Definitely not! In a “Xiggi” problem you are given the averages for half the trip - and each half is generally different. Harmonic means do not behave like arithmetic means! If you half a distance you get half the distance travelled. If you half a time you get half the time it took. If you half the rate you get a meaningless number. </p>
<p>For example, if you were traveling 30 mph and then 60 mph, then Xiggi’s formula gives an average rate of 40 mph for the whole trip. Half of 40 is 20. This number means nothing! You were always travelling faster then 20mph.</p>
<p>Steve, the average speed is indeed 40 mph. But the problem could have asked what the DISTANCE was for either the round trip or one way. In the second case, one could use a numerator of … 1. It works this way with round trips of 1 hour. </p>
<p>As usual, 90 percent of the solution comes from reading the problem correctly and knowing how to solve it logically. </p>
<p>Humm, the original elephant problem can be solved with a different method as the distances are given. That one is basic arithmetic that is 14 / (35/4). Too simple. </p>
<p>Here is a different problem. </p>
<p>An elephant traveled from its cage to the circus at an average rate of 4 miles per hour and then traveled back at an average rate of 1 mile per hour. If the total trip took 1 hour,
Case 1 : What was the distance travelled?<br>
Case 2: What is the distance between the cage and the circus? </p>
<p>@DrSteve Thank you again for replying. When I said half of the trip, I meant the instance where the we calculate the average going from the circus to the cage only,not the whole trip(circus to cage and cage to circus).Of course, both routes have the same distance.Here is the question that really made me ask this specific question about the coefficient of the numerator. <a href=“Hard Math Problem =( - Test Preparation - College Confidential Forums”>http://talk.collegeconfidential.com/sat-act-tests-test-preparation/74410-hard-math-problem.html</a></p>
<p>@Xiggi Thank you again for replying.Case 2: We should use the coefficient 1(which equals 5/4).Case 2 : Coefficient 1(which equals 10/4).The thing that is confusing me even more now is the fact we used the same formula to find the distance and to find the average rate. :/</p>
<p>@Xiggi I’ve also found this forumla t[(r1r2)/(r2 + r1)] = d. The d in the equation is supposed to be the half of the distance and t is the time,which is in this case 1.</p>
<p>It looks like this last formula is easily derived from substituting the average rate (from Xiggi’s formula) into the usual formula d=rt as follows: </p>
<p>d=rt = [2(r1r2)/(r1+r2)]t</p>
<p>Since this is the distance for both parts of the trip, if we get rid of the 2, we get the distance for just the first (or second) part of the trip.</p>
<p>@DrSteve Oh, that was extremely helpful!</p>
<p>Again, the distance travelled in one hour is the same absolute number as the HoUrLy average speed for the trip! </p>
<p>The harmonic average formula works but one needs to pay attention to the problem statement. </p>