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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


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


  (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


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


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


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


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


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

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.
Associated Topics:
Middle School Puzzles

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