A simple application problem involving a rational expression

Dan Meyer wants to know what to do with rational expressions. Here is an example:

harmonic_slide

Of course, the students have a ready answer to this.

But … the answer is 48 miles per hour. Experimentation with a meter stick and toy cars can eventually lead to the formula:

\frac{2}{\frac{1}{a} + \frac{1}{b}}

Where a is the first speed taken, and b is the second. This is the harmonic mean of the two numbers.

Simplifying:

\frac{2}{\frac{b}{ab} + \frac{a}{ab}}
\frac{2}{\frac{a + b}{ab}}
\frac{2ab}{a+b}

12 Responses

  1. Problem: what do you mean by “half”? A perfectly valid interpretation is spending one hour driving at 40 mph and another hour at 60. Then it takes a total of 2 hours to get there, for an average speed of 50 mph.

  2. Good point! I might consider leaving the text as it is and having the students argue that point before laying down the exact meaning.

  3. what is the solution of 90% and 30%

  4. How did you experiment with toy cars and a meter stick to derive that formula?

    • They can collect data and test out their guesses on a small scale (do the same problem on a 1 meter journey where the speed changes at the 50 cm part). The same thing can be done on paper but with props it is more fun!

      As far as deriving the formula goes, at some point they do need to get their hands dirty with the math, but they can do it by looking at a version of the formula with the numbers substituted in and change them to variables instead.

  5. could you work this out and show me exactly how you got 48 mph? because i do not completely understand how youve gotten this solution

  6. Divide 100 by 2 and that’s half of your trip. Use the formula Average Rate=Total distance/Total time, For the first half of the trip which is 50miles you would put, Average Rate= 50miles/40mph=1.25hours
    For the second half of the trip you would put Average Rate=50miles/60mph=.83hours. Add the hours together which equal 2.08 hours, Go back to the orignal formula, Average Rate=100miles/2.08=48.07=48mph. So the correct answer is 48mph

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