Deriving the formula for harmonic mean

Pre-College Topics High School Life
1

<p>Specifically in reference to its application for finding
1) average rate of travel when the same distance is undergone at 2 different rates
2) total distance traveled at 2 different rates when the total travel time is 1 hr</p>

<p>Basically how do you get from
r1<em>t1 + r2</em>t2 with t1 + t2 = 1
to
2(r1*r2)/(r1 + r2)?</p>

<p>where r1 = rate 1, t1 = time spent at rate 1, etc.</p>

<p>Deriving this formula is helpful for truly understanding it and being able to apply it.</p>

2

<p>bump bump bump</p>

3

<p>1.)</p>

<p>Suppose you travel d distance at r1 rate followed by d distance at r2 rate. </p>

<p>Generally, rate*time = distance so we spend two times in our situation above: one at r1 and one at r2. So,</p>

<p>t1 = d/r1 and t2 = d/r2</p>

<p>Total time = t1 + t2=d/r1 + d/r2
Total distance = d+d = 2d</p>

<p>Average rate = total distance/total time = 2d/( d/r1 + d/r2 ) = 2r1r2/(r1 + r2)</p>

<p>2.) We don’t have enough information to come up with an expression. You have to give a proportion of the two times to each other, or the two rates to each other.</p>

4

<p>what’s a harmonic mean</p>

5

<p>^ [Harmonic</a> Mean – from Wolfram MathWorld](<a href=“http://mathworld.wolfram.com/HarmonicMean.html]Harmonic”>Harmonic Mean -- from Wolfram MathWorld)</p>

6

<p>oh, so like resistors in parallel?</p>

7

<p>@ firelight: Thanks! Actually the same formula is used for both 1) and 2).</p>

8

<p>

</p>

<p>This is what I was thinking.</p>

<p>Is this for SAT problems like train leaves from Philadelphia another leaves from New York blah blah blah?</p>