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### Math Puzzles & Brain Teasers

Speeding Away

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The drive from Oakland to Pinewood was a tricky one. I covered the uphill distance of 70 miles at 42 miles per hour. The return journey from Pinewood to Oakland was downhill, and I managed to drive at 56 miles per hour. What was my average speed for the entire journey?

Solution:

It it important to note that

 Average speed = Total distance / Total time.

Total distance = 2 × 70 miles.

Time for uphill journey (from Oakland to Pinewood) = 70 / 42 hours.

Time for downhill journey (from Pinewood to Oakland) = 70 / 56 hours.

Total time = (70 / 42) + (70 / 56) = 2 × 70 / 48 hours.

Average speed = Total distance / Total time = 48 miles per hour.

Food for thought:

The common mistake made in solving this problem is to assume the average speed to be the arithmetic mean, i.e., (42 + 56)/ 2 = 49 miles per hour.

In this problem, the average speed is clearly not the arithmetic mean. Is it the geometric mean, the harmonic mean or the logarithmic mean? Note the harmonic mean is given by

2 × 42 × 56 / (42 + 56).

In fact, the distance between Oakland and Pinewood need not be specified in the problem statement. The average speed can be calculated without this piece of information.