Sometimes an arithmetic procedural error produces a correct answer.
An accidental cancellation is reducing a fraction by canceling individual digits in the numerator and denominator.

In some cases the result is equal to that obtained by a correct process.
Examples:
16/64 = 16/64 = 1/4;
26/65 = 26/65 = 2/5;
266/665 = 266/665 = 2/5;
49/98 = 4/8 = 1/2 etc.

Try to find two-digit cases in bases other than base 10,(e.g. 13/32 = 1/2 ( the only solution in base 4).

To begin with there are certain trivial examples that I'm sure are meant to be ignored:

If the unit position of each of numerator and denominator is zero. That's why 20/30 is not a solution in base 4.

If the unit position matches the tens (or equivalent in other bases) position in both the numerator and the denominator. So 22/33 is not a solution in base 4.

A different triviality, that can't come up in base 4 is exemplified by the decimal 21/42, where each digit is individually the same ratio of numerator to denominator. The only attestation to this is that all the decimal examples shown involve different positions being struck in the numerator compared with the denominator: in the two-digit cases the 10's digit in one and the units digit in the other.

To avoid trivial variations (a/b vs b/a), only proper fractions are shown. Rather than arbitrarily exclude base 10, that base is also included.

DECLARE FUNCTION alp$ (x!)
CLS
OPEN "cancfall.txt" FOR OUTPUT AS #2
FOR b = 2 TO 16

FOR a1 = 1 TO b - 1
 FOR a2 = 1 TO b - 1
  FOR b1 = 1 TO b - 1
    FOR b2 = 1 TO b - 1
      IF a1 <> a2 OR b1 <> b2 THEN
       IF a1 <> b1 OR a2 <> b2 THEN
        num = b * a1 + a2: den = b * b1 + b2
         IF den >= num THEN
              IF a2 * den = b1 * num THEN
                PRINT alp$(a1); alp$(a2); "/"; alp$(b1); alp$(b2); "="; alp$(a2); "/"; alp$(b1); TAB(12); b
                PRINT #2, alp$(a1); alp$(a2); "/"; alp$(b1); alp$(b2); "="; alp$(a2); "/"; alp$(b1); TAB(12); b
              END IF
              IF a1 * den = b2 * num THEN
                PRINT alp$(a1); alp$(a2); "/"; alp$(b1); alp$(b2); "="; alp$(a1); "/"; alp$(b2); TAB(12); b
                PRINT #2, alp$(a1); alp$(a2); "/"; alp$(b1); alp$(b2); "="; alp$(a1); "/"; alp$(b2); TAB(12); b
              END IF
         END IF
       END IF
      END IF
    NEXT
  NEXT
 NEXT
NEXT

NEXT b

CLOSE

END

FUNCTION alp$ (x)
  alp$ = MID$("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ", x + 1, 1)
END FUNCTION

"cancellation" base
    13/32=1/2   4 
    13/43=1/3   6 
    14/32=1/2   6 
    15/53=1/3   6 
    25/54=2/4   6 
    13/54=1/4   8 
    15/32=1/2   8 
    15/64=1/4   8 
    17/74=1/4   8 
    26/54=2/4   8 
    37/76=3/6   8 
    14/43=1/3   9 
    14/86=1/6   9 
    17/53=1/3   9 
    25/76=2/6   9 
    28/43=2/3   9 
    28/86=2/6   9 
    13/65=1/5   10 
    15/75=1/5   10 
    16/32=1/2   10 
    16/64=1/4   10 
    16/96=1/6   10 
    17/85=1/5   10 
    19/95=1/5   10 
    26/65=2/5   10 
    27/54=2/4   10 
    38/76=3/6   10 
    39/65=3/5   10 
    49/98=4/8   10 
    13/76=1/6   12 
    14/54=1/4   12 
    14/A8=1/8   12 
    15/43=1/3   12 
    15/86=1/6   12 
    17/32=1/2   12 
    17/64=1/4   12 
    17/96=1/6   12 
    19/53=1/3   12 
    19/A6=1/6   12 
    1A/74=1/4   12 
    1B/B6=1/6   12 
    25/98=2/8   12 
    26/76=2/6   12 
    28/54=2/4   12 
    28/A8=2/8   12 
    2A/43=2/3   12 
    2A/86=2/6   12 
    2B/B8=2/8   12 
    37/A9=3/9   12 
    39/76=3/6   12 
    3B/B9=3/9   12 
    4A/98=4/8   12 
    5B/BA=5/A   12 
    13/87=1/7   14 
    15/97=1/7   14 
    17/A7=1/7   14 
    18/32=1/2   14 
    18/64=1/4   14 
    18/96=1/6   14 
    18/C8=1/8   14 
    19/B7=1/7   14 
    1B/C7=1/7   14 
    1D/D7=1/7   14 
    26/87=2/7   14 
    29/54=2/4   14 
    29/A8=2/8   14 
    2A/97=2/7   14 
    39/87=3/7   14 
    3A/76=3/6   14 
    4B/98=4/8   14 
    4C/87=4/7   14 
    5C/BA=5/A   14 
    6D/DC=6/C   14 
    14/65=1/5   15 
    14/CA=1/A   15 
    16/43=1/3   15 
    16/86=1/6   15 
    16/C9=1/9   15 
    17/75=1/5   15 
    17/EA=1/A   15 
    1A/85=1/5   15 
    1B/53=1/3   15 
    1B/A6=1/6   15 
    1D/95=1/5   15 
    25/BA=2/A   15 
    27/76=2/6   15 
    27/EC=2/C   15 
    28/65=2/5   15 
    28/CA=2/A   15 
    2B/DA=2/A   15 
    2C/43=2/3   15 
    2C/86=2/6   15 
    2C/C9=2/9   15 
    2E/75=2/5   15 
    2E/EA=2/A   15 
    38/A9=3/9   15 
    3C/65=3/5   15 
    3C/CA=3/A   15 
    3D/B9=3/9   15 
    49/DC=4/C   15 
    4A/BA=4/A   15 
    4E/76=4/6   15 
    4E/EC=4/C   15 
    13/98=1/8   16 
    15/54=1/4   16 
    15/A8=1/8   16 
    15/FC=1/C   16 
    17/B8=1/8   16 
    19/32=1/2   16 
    19/64=1/4   16 
    19/96=1/6   16 
    19/C8=1/8   16 
    19/FA=1/A   16 
    1B/D8=1/8   16 
    1D/74=1/4   16 
    1D/E8=1/8   16 
    1F/F8=1/8   16 
    26/98=2/8   16 
    2A/54=2/4   16 
    2A/A8=2/8   16 
    2A/FC=2/C   16 
    2E/B8=2/8   16 
    37/DC=3/C   16 
    39/98=3/8   16 
    3B/76=3/6   16 
    3B/EC=3/C   16 
    3F/54=3/4   16 
    3F/A8=3/8   16 
    3F/FC=3/C   16 
    4C/98=4/8   16 
    5D/BA=5/A   16 
    5F/98=5/8   16 
    6E/DC=6/C   16 
    7F/FE=7/E   16 

In all these proper fractions the "cancelled" digit is the units position of the numerator and the tens (or equivalent) position of the denominator.  Of course it would be the other way around when the numerator and denominator are interchanged to make the corresponding improper fraction.

Posted by Charlie on 2013-01-18 16:55:30