<p>The formula in question is 2(rateA x rateB)/(rateA + rateB). Where does this formula come from? I think that knowing how its derived will help me better apply it in those kinds of problems.</p>
<p>1/r + 1/s = (s + r)/(rs)</p>
<p>It comes (among other places) in rate problems.</p>
<p>Consider the general round-trip problem. I drive from SF to LA at 60 miles per hour, and I drive back at 55 miles per hour. The cities are 400 miles apart. How long does the (driving part) of the trip take? From SF to LA: [400/60] hours; from LA to SF: [400/55] hours. So the total time is 400/60 + 400/55 = 400 (60 + 55)/[(60)(55)].</p>
<p>How about the average speed for the trip? That’s distance/time. [800/400] [(60)(55)]/[60 +55] or 2 [(60)(55)]/[60 + 55]</p>
<p>I wouldn’t recommend memorizing this formula or it’s usage.</p>
<p>This is called the harmonic mean formula, although on this forum we have renamed it Xiggi’s formula. The harmonic mean gives a different kind of “average” and has many uses. But the only use that I’ve seen appear on the SAT is to “average” two rates when the distance is the same. </p>
<p>Here is a derivation of this formula from the simpler formula d=r*t:</p>
<p>Now d=r1<em>t1 and d=r2</em>t2 (note we use the same d because the two distances are the same). Distances and times can be added (rates cannot), so that the total distance is 2d and the total time is t1+t2. Therefore, the average rate is (total distance)/(total time)=2d/(t1+t2)=2d/(d/r1+d/r2). I will now leave it to you to try to simplify this complex fraction to get the formula in it’s usual form. If you have trouble tell me and I will finish the simplification. </p>
<p>Note: I agree that MOST students shouldn’t bother with this formula. However, if you are going for an 800, then I believe very strongly that you SHOULD memorize this formula.</p>