If a cyclist races at 30 miles per hour for 1/2 of the distance of a race, and 45 miles per hour for the final 1/2 of the distance, what is her average speed in iles per hour, for the entire race

  36 mph

Step-by-step explanation:

The appropriate relations are ...

  time = distance/speed

  speed = distance/time

Suppose the race has a total length of 360 miles. Then half the race is 180 miles.

The cyclist's time on the first half is (180 mi)/(30 mi/h) = 6 h.

The cyclist's time on the second half is (180 mi)/(45 mi/h) = 4 h.

The total time is 10 h for the total distance of 360 mi, so the average speed is ...

  (360 mi)/(10 h) = 36 mi/h

The length of the race doesn't matter. The above numbers make the math convenient. The average is computed from ...

  1/((1/2)/30 + (1/2)/45) = 2/(1/30 +1/45) = 2·30·45/(30+45) = 36

This is the harmonic mean of the numbers 30 and 45.

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step-by-step explanation: