# Maya drives from New York City to Boston at a rate of 40 MPH and drives at a rate of 60 MPH on the return trip. What was his average speed for the entire trip? Use the Harmonic mean to compute? Construct the HM geometrically?

The Average Speed is

Assume that Boston and New York City are separated by a distance of

trip.

Therefore, the Average Speed for the entire trip is provided by,

Time/Distance, that is,

People who are acquainted with Harmonic Mean will quickly remark

that the trip's average speed is not equal to its arithmetic mean

but the speeds' harmonic mean.

Have fun with math!

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The harmonic mean (HM) for a set of numbers is calculated by dividing the number of values by the sum of the reciprocals of those values.

In the case of Maya's trip from New York City to Boston and back, with speeds of 40 MPH and 60 MPH respectively, the harmonic mean can be calculated as follows:

Harmonic Mean = (Number of values) / (Sum of reciprocals of speeds)

For Maya's trip: Number of values = 2 (one for each leg of the journey) Speeds: 40 MPH and 60 MPH

Reciprocal of 40 MPH = 1 / 40 Reciprocal of 60 MPH = 1 / 60

Sum of reciprocals = (1 / 40) + (1 / 60)

Harmonic Mean = 2 / [(1 / 40) + (1 / 60)]

To construct the harmonic mean geometrically, you can imagine a rectangle where the length represents the distance traveled at 40 MPH and the width represents the distance traveled at 60 MPH. Then, you can calculate the harmonic mean as the reciprocal of the diagonal of this rectangle. This geometric interpretation corresponds to the reciprocal relationship between speed and time.

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