Cryptarithm with a Repeating Decimal

Date: 11/25/2004 at 12:25:36
From: Caroline
Subject: repeating decimals

I don't know how I would do this cryptarithm:

If FOCE / VOLVO = 0.COFFEECOFFEE....., find the value of COFFEE.

Note that the number zero is before the decimal, and not the letter
"O".  I need to figure out a six digit repeating decimal that equals a
fraction with four digits in the numerator and five in the 
denominator.  Is there a formula for this, or is it just guess and
check?  I don't understand how to do it.

Date: 11/28/2004 at 00:19:11
From: Doctor Greenie
Subject: Re: repeating decimals

Hi, Caroline --

Cool problem!!

But very advanced for a 13-year-old, unless there is some solution
method much easier than the one I found.

The first thing to notice about this problem is that the repeating
decimal repeats in groups of 6.  That means that we must have

  FOCE/VOLVO = COFFEE/999999

Since these are equivalent fractions, we can solve the problem if we
can find an integer x such that

  (a) FOCE * x = COFFEE  and
  (b) VOLVO * x = 999999

We don't have many clues to go on with (a).  However, with (b), we
have the potential to get somewhere.  It helps to know that

  999999 = 1001*999
    1001 = 7*11*13
     999 = 27*37

We want a factor of 999999 which fits the pattern VOLVO.  Fortunately,
we can quickly find two possibilities, using the factorizations of
1001 and 999 shown above.  We could have either

  (1)  VOLVO = 27*1001 = 27027

or

  (2)  VOLVO = 37*1001 = 37037

In case (1), according to (b) above, we will have

  VOLVO * 37 = 999999

and so we will have to have

  FOCE * 37 = COFFEE

Similarly, in case (2), we will have

  VOLVO * 27 = 999999

and so we will have to have

  FOCE * 27 = COFFEE

Note to start out that in both cases--"VOLVO" = 27027 or "VOLVO" =
37037--the "O" is 7 in both cases.  So to solve case (1), we will
need to solve

    F7CE
      37
  ------
  C7FFEE

or to solve case (2) we will need to solve

    F7CE
      27
  ------
  C7FFEE

There is no obvious reason I see for investigating one or the other of
these possibilities first; I happened to choose case (1).

The first thing to notice in this case is that in the units column we
have E times 7 giving final digit E; a quick check shows the only
possible digit for E is 5.  So now this case becomes

    F7C5
      37
  ------
   ???X5
  ????5
  ------
  C7FF55

This shows that the digit "X" preceding the final "5" in the first
line of the multiplication must be a "0"; that in turn leads to the
conclusion that the digit "C" in "F7C5" is 1.  So now this case becomes

    F715
      37
  ------
   ??005
  ??145
  ------
  17F455

Finally, we know the digit "F" in the product is the same as the next
digit, so F is 4.  So this case is now finished; we have

    4715
      37
  ------
   33005
  14145
  ------
  174455

A quick check shows that this solution fits the required pattern.  And
going back to the original problem, a quick check with the calculator
on my PC shows

  4715/27027 = 0.174455174455174455...

When we try solving case (2) the same way, we are very shortly forced
to have both C and E equal to 5, so there is no solution to the
problem using case(2).

I hope all this helps.  Please write back if you have any further
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 11/28/2004 at 20:24:07
From: Caroline
Subject: Thank you (repeating decimals)

Doctor Greeenie,

Thank you so much.  I think I understand how to reason such problems
out now.