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Mensa® Daily Puzzlers

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Problem Description

Given a positive decimal  number, convert it to an irreducible mixed or  proper fraction.  If the denominator of the fraction is larger than a specified maximum denominator, present the solution in the above format as a fraction which is the best estimate of the input value and display the error value.

Version 3 - convert both ways (decimal to fraction, and fraction to decimal) and add optional constraint to display decimal input in mixed fraction form with  denominators restricted to 16, 32, or 64.   Jump to download link

Background & Techniques

First, some terminology: A fraction, N/D, is proper if the numerator, N,  is less than the denominator, D.   (I remember which is which by associating the D in denominator with the number that is down in the fraction.   A mixed fraction is one whose value is greater than 1 and is expressed as an integer plus a proper fraction (X + N/D).  If the GCD, Greatest Common Divisor of N and D equals 1 then the fraction  is in lowest terms, irreducible.

This program probably has little use unless you are a programmer curious about how to convert a decimal number to a fraction or someone who needs help with a homework assignment .

We start by separating the integer part from the decimal part of the input number. Multiply the decimal part by a power of 10 large enough to convert it to an integer, i.e. 10 raised to the power of the number of digits in the fractional part. (For example, given  0.124, multiply by 103 = 124).

So we now have our initial try at a fractional form answer (124/1000). We'll reduce it to lowest terms by dividing numerator and denominator by the greatest common denominator of the two.  In our example, GCD(124,1000)=4 so the fraction reduces to 31/250.   If the denominator is less than or equal to the specified max denominator size, we're done.

Otherwise, we'll have to provide the closest possible estimate whose denominator is smaller than the maximum specified. We'll just try all denominators from 2 to the max specified and calculate the numerator which produce a value closest to the original decimal part. Numerator = ((original decimal part) x trial denominator) rounded to the nearest integer.

The error of this estimate is original decimal part - numerator / trial denominator. We'll save the numerator and trial denominator which produces the smallest absolute error and report that as the solution.

Assuming the maximum denominator for our example is 100, the closest fractional representation of 0.124  is 12/97 which in decimal form is 0.12371...,  less than .0003 from the  input value.

Addendum May 28, 2005:  It didn't take long for regular contributor Don Rowlett to point out that  I once had ignored the European by defined some constants with '.' for a decimal separator rather than a ','.  I think that is now corrected.  Don also added a trick for converting repeated decimals if one of the repeated cycles is given so we implemented the option for the user to specify whether the input decimal terminates or repeats forever.  This also required adding big integer arithmetic because even a single cycle of some repeated decimals are too long to be represented in normal integer arithmetic (for example 1/29 has the 28 digit repeating decimal  representation .0344827586206896551724137931 0344827 etc......... )  Compiling Version 2 of this program will require the Big Integer unit contained in our DFF Library file.

Addendum  February 7, 2006:  I recently decided to upgrade another [program (Cutlist) to add a "carpenter arithmetic" option for input and output displays.    Woodworkers, at  least us Yanks or Brits,  who use the English measurement system are stuck with measuring lengths in feet and inches (12ths of a foot).   Parts of an inch for wood workers are not measured in 10ths and hundredths - not challenging enough for us - we measure parts of an inch in 64ths and multiples of 64ths  (32nds, 16ths, 8ths, 4ths and halves).    I developed the procedures to do the necessary conversions in  both directions and added them to version 3 of this program.  I also removed the Big Integer capability because it  was an unnecessary complication when measuring in 64ths.  I'll leave the previous version   on the download  zipped source  file for anyone interested in exploring rational numbers with  cycle lengths greater than 18 digits.    I removed the GCD function that had been embedded in favor of the common library version in our Mathslib unit.  As a result, DFFLIBV04 or later version of the library file is required to recompile version 3.