SOLUTION TO AVERAGE OF AVERAGE SPEEDS

192 mph! The correct answer is 192 mph! Or, in "multiple choice terms", (D) 192mph. After flying four 100-mi courses at average speeds of 100 mph, 200 mph, 300 mph, 400 mph, THE AVERAGE SPEED OF THESE AVERAGE SPEEDS IS 192 mph!

Did you pick (B) 250 mph? I think I know why. But, first, some background on this problem.

I first encountered this problem in a delightful book, Facts Into Figures, by F. J. Moroney, a statistician in the British equivalent of AT&T. I don't know if the paperback is still in print. If so, I recommend buying, for its many goodies.

First, why the correct answer is 192 mph. Then, why so many pick 250 mph. (I did, too, the first time I saw the problem.)

(physics definition) AVERAGE SPEED = D/T (total distance traversed divided by total time traveling).

D (total distance traversed) = (100 + 100 + 100 + 100) miles = 400 miles.

Total time (T) involves ADDING FRACTIONS. (Did you just yecch! on my bib?)

  1. 100 miles W-E at 100 mph takes 1 hour.

  2. 100 miles N-S at 200 mph (twice the speed) takes 1/2 hour (half the 1st-course time).

  3. 100 miles E-W at 300 mph (thrice the first speed) takes 1/3 hour (one-third 1st-course time).

  4. 100 miles S-N at 400 mph (four times the first speed) takes 1/4 hour (one-fourth 1st-course time).

T (total time elapsed) = (1/1 + 1/2 + 1/3 + 1/4) hrs.

The common denominator (remember this?) of these fractions is the product of the denominators, 12 = 1 · 2 · 3 · 4.

Then, 1/1 = 12/12; 1/2 = 6/12; 1/3 = 4/12; 1/4 = 3/14. So, T = (12/12 + 6/12 + 4/12 + 3/12) hrs = (12 + 6 + 4 + 3)/12 hrs = 25/12 hrs -- total time, T, elapsed for flying the square course distance of 100 miles on a side (or 400 miles).

Remember? Average speed = D/T (total distance traversed divided by total time elapsed).

So, AVERAGE SPEED 400/(25/12) mph.

Remember that we INVERT A FRACTIONAL DENOMINATOR AND MULTIPLY: AVERAGE SPEED = 400/(25/12) mph = (400 · 12)/25 mph = 4800/25 mph = 192 mph. Charlotte?

Actually, there's a simple formula, which you should know, that easily yields this answer. But first a comment on the usual guess about this answer, and HOW ERRONEOUS it is.

The formula for calculating "percentage error" of a incorrect (I) answer by comparison with a correct (C) answer is: (|I - C|)/C %, where |...| means absolute value -- ignoring SIGN.

Then we have: (|250 - 192|)/192 % = (58/192) %, or, approximately 30% ERROR! WOW! The answer most people would give is over 30% incorrect! Why?

Most people -- including jh, that first time -- believe that "average" means "arithmetic mean". Elsewhere we learn that, in an arithmetic progression of numbers, the arithmetic mean is the NUMBER IN THE MIDDLE.) And most text books and reference books treat arithmetic mean as "the average". The word "average" is one of the buzziest of buzz-words in the English language.

But we've just seen why that usage is MISLEADING. It forces us to say that "Average speed [physics] is not the average [statistics] of speeds"!

Restating, the arithmetic mean of numbers is (100 + 200 + 300 + 400)/4 = 1000/4 = 250. But this means that the [physics] average speed (192) is not the "average" (arithmetic mean) of the speeds (250).

Either we must change the language in physics or the general language. And I opt for changing the general language: not only for this kind of problem but to promote a more general and useful reference for "average".

Moroney (in Facts from Figures) explains the origin of the word "average". It originated among Mediterrean merchants in Roman times -- from the Latin word "havaria", meaning "share". The merchants who shipped goods on the Mediterranean were plagued by thefts in port, or rats destroying goods, or pirates stealing goods at sea. Several merchants would make a pact that any merchant who suffered loss would be given a share from the goods of the others as compensation. It was a primitive form of insurance. And Moroney notes that AVERAGE means REPRESENTATIVE. Just as you choose a chairman or delegate or congressperson to REPRESENT YOUR GROUP, so you may choose A NUMBER TO REPRESENT A SET OF NUMBERS.

Now, SPEED IS A RATE -- THE RATIO OF DISTANCE TO TIME. Amd THE PROPER REPRESENTATIVE OF RATES IS THE HARMONIC MEAN. (Also, elsewhere, we seen that the harmonic mean is the middle number of a harmonic progression. Everyone should study MEANS in ARITHMETIC CLASS!) Here's the easy way to find the HARMONIC MEAN OF NUMBERS.

  1. WRITE THE RECIPROCAL OF EACH OF THOSE NUMBERS. (The previous "MEANS" hyperlink explains this operation. There are TABLES OF RECIPROCALS to faciliate this.)

  2. NOW CALCULATE THE ARITHMETIC MEAN OF THESE RECIPROCAL NUMBERS. This yields the value 1/h, where h is THE HARMONIC MEAN.

  3. Obviously, THE RECIPROCAL OF THAT ANSWER IS THE HARMONIC MEAN.

We apply this procedure:

  1. Given, 100 mph, 200 mph, 300 mph, 400 mph, we form the reciprocals of those "pure numbers": 1/100, 1/200, 1/300, 1/400

  2. This yields the ARITHMETIC MEAN: 1/4(1/100 + 1/200 + 1/300 + 1/400).

    • The COMMON DENOMINATOR is 1200, so 1/100 = 12/1200, 1/200 = 6/1200, 1/300 = 4/1200, 1/400 = 3/1200.

    • And we have: 1/4(12/1200 + 6/1200 + 4/1200 + 3/1200) = 1/4[(12 + 6 + 4 + 3)/1200 = 25/(4 · 1200) = 25/4800 = 1/192.

  3. Then, RECIPROCATING, h = 192 mph -- "right on the nose!" Or 0% ERROR.

As I implied above, I first calculated 250 mph, having been brain-washed about "average". In 35 years, I've yet to find a person who gets the correct answer -- usually opting for 250 mph.

However, I must give credit where credit it due. I often complain about "standardized tests" -- IQ, SAT, GRE, etc. But, recently, I read in a newspaper about a multiple choice problem concerning the average speed of two trains where the correct choice is their harmonic mean. So, maybe things are getting better!

For the IMPLICATION of

I suggest return to HARMONIC PROGRESSIONS.