# 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?

Aug 26, 2016

The Average Speed is $48 \text{mph}$.

#### Explanation:

Suppose that the distance between New York City and Boston is

$x$ miles.

Time reqd. for this trip, at the rate of $40 \text{mph, is,} \frac{x}{40}$ hours.

The return journey is at the rate of $60 \text{mph, so,} \frac{x}{60}$ hours for this

trip.

Thus, total distance of $2 x$ miles was covered in

$\frac{x}{40} + \frac{x}{60}$ hours.

Hence, the Average Speed for the complete journey is given by,

Distance/Time, i.e.,

$\frac{2 x}{\frac{x}{40} + \frac{x}{60}} = \frac{2}{\frac{1}{40} + \frac{1}{60}}$ mph.

Those who are familiar with Harmonic Mean will, at a glance, say

that the Average Speed for the trip is not the Arithmetic Mean

but the Harmonic Mean of the speeds.

Numerically, the Average Speed is $\frac{2 \cdot 60 \cdot 40}{60 + 40} = 48 \text{mph.}$

Enjoy Maths.!