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Who Am I?

I am a proper fraction.

The sum of my numerator and my denominator is 1 less than a perfect square.

The difference of my denominator and my numerator is 1 more than a perfect square.

The product of my numerator and my denominator is 1 less than a perfect square.

What fraction am I?

 


Hint: First, try some examples of proper fractions, so that you get a handle on the problem.

Now, focus on the nature of the possible squares involved...as they contribute some great information.

 


Solution Commentary: One solution is 3/5....but there are others!

So,to guide your investigation, suppose A/B is your desired fraction, where A+B = C2-1 and B-A = D2+1 and AB = E2-1. Now, here are some more important questions to try to resolve in order to generalize the problem:

  • Can A and B both be even? Both be odd? One be odd and one be even?
  • Which is greater, C or D? Can you tell anything about the value of E relative to the values of C or D?
  • Can C and D both be even? Both be odd? One be odd and one be even?
  • Can C = D?
  • Is it necessary that C - D = 2?
  • Since AB = E2-1, does it help to rewrite this expression as AB = (E+1)(E-1)?
  • Once you find some solutions, can you find a pattern in these solutions?
  • Have fun.......and what happens if the fourth clue was changed to "The product of my numerator and my denominator is 1 more than a perfect square"?