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.
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