From the archives:
http://talk.collegeconfidential.com/sat-preparation/68210-xiggis-sat-prep-advice-p57.html - Post 854
Humm, I think I have done this a few times. There is really not much to learn from seeing how this very effective SAT tool is derived directly from the well-known formula … d = rt.
As we know the total average speed = Total distance/Total time
This is s = d/t. For the people who prefer the d=rt, simply substitute r for speed.
A. Let’s establish that
d = distance traveled in each direction
The total distance traveled is d+d or 2d
B. Let’s establish that
t1 = time spent on first leg
t2 = time spent on return leg
Total Time = t1+t2
C. By formula, speed = distance/time but also time = distance/speed
S1= d/t1 and t1 = d/S1
S2 = d/t2 and t2 = d/S2
Now, let’s start (following A and B above)
Total average speed = Total Distance/Total time or
Total average speed = d + d / t1+t2
Total average speed = 2d / t1+t2
Substituting according to C yields
Total average speed = 2d / (d/S1) + (d/S2)
Getting rid of the common “d” gives
2 / (1/S1) + (1/S2) or
2 / (1.S2/S1S2) + (1.S1/S1S2) or
2 / (S1+S2 / S1S2) or
2 * S1S2 / S1 +S2
And here we have it
Total average speed = 2 * S1*S2 / S1 +S2
++++++++++++
Piece of cake for my friend MITer94, but helpful to understand that there is no voodoo involved in the harmonic rate formula. It remains a very simple and elegant tool for the SAT.