Question 1008835
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Let L be the distance Tim covered.

Since Tim drove half of the distance, {{{L/2}}}, at the speed of 60 mph, he spent for it the amount of time {{{L/2}}}/{{{60}}} = {{{L/(2*60)}}} hours.

Since Tim drove the other half of the distance. {{{L/2}}}, at the speed of 80 mph, he spent for it the amount of time {{{L/2}}}/{{{80}}} = {{{L/(2*80)}}} hours.

The average speed is the whole distance L divided by the time spent, which is {{{L/(2*60)}}} + {{{L/(2*80)}}}, or 

{{{L/(L/(2*60) + L/(2*80))}}} = {{{(4*60*80)/(2*60 + 2*80)}}} = {{{2*60*80)/(60+80)}}} = 68.571 {{{mi/h}}}.

The value {{{2/((1/u) + (1/v))}}} is called the <U>harmonic mean of values u and v</U>.

Thus the average speed in this problem is the harmonic mean of the given speed 60 mph and 80 mph.

See also the lesson <A HREF=http://www.algebra.com/algebra/homework/word/travel/Calculating-an-average-speed.lesson>Calculating an average speed: a train going from A to B and back</A> in this site.
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