Henry Ernest Dudeney/Modern Puzzles/43 - The Two Ships/Solution

Modern Puzzles by Henry Ernest Dudeney: $43$

A correspondent asks the following question.

Two ships sail from one port to another -- $200$ nautical miles -- and return.

The Mary Jane travels outwards at $12$ miles an hour and returns at $8$ miles an hour,

thus taking $41 \tfrac 2 3$ hours on the double journey.

The Elizabeth Ann travels both ways at $10$ miles an hour, taking $40$ hours on the double journey.

Now, seeing that both ships travel at the average speed of $10$ miles per hour, why does the Mary Jane take longer than the Elizabeth Ann?

The error lies in assuming that the average speeds are equal.

They are not.

The first ship does a mile in $\tfrac 1 {12}$ of an hour outwards and $\tfrac 1 8$ of an hour homewards.

Half of the sum of these fractions is $\tfrac 5 {48}$.

Therefore the ship's average speed for the $400$ miles is a mile in $\tfrac 5 {48}$ of an hour.

The average speed of the second ship is a mile in $\tfrac 1 {10}$ of an hour.