Hot Stuff!!!!

If you are in Georgia, it should be easy to find the fraction and decimal that says, "I am hot."

(Reprinted with permission from Mathematical Challenges for the Middle Grades, copyright 1991 by the National Council of Teachers of Mathematics. All rights reserved.)

Each of the letters represents a different digit from 0-9. Find the fraction.

I began this search the easiest way possible...I only had 9 numbers to work with, so why not begin just sticking in digits in some sort of systematic fashion. For example, I = 1, A = 2, M = 0. After keeping my numerator as 1 and constantly changing the numbers in my denominator consecutively, I found that 1/27 = .037037037. I continued...and found 1/37 = .027027027. I figured immediately that I must be on to something, but just to prove to myself, I went through all the twos, i.e. 2/10, 2/21, 2/22...2/27...2/37. Again, I saw the same type of repetition in decimals. The fractions 2/27 = .074074074 and 2/37 = .054054054...oh goodness...this one actually works. I wondered if another fraction would work and still fit in the same criteria. I continued...3/27 = well, to say the least it was a repeating decimal however, it did not work, but 3/37 = .081081081 did work although it did not fit my "different digits" criteria. Okay, let's go to the fours. The fraction 4/27 = .148148148 and 4/37 = .108108108...wow, this one works too!!! So now I am assuming it works with multiples of 2...NOT!!! The fractions 6/27 = .22222 and 6/37 = .162162162 did not work they did not fit the criteria. Now, I figured maybe, just maybe it worked by finding the powers of 2 in the numerator and having 37 as the denominator...let's see...8/37 = .216216216. Hot dog! It did work!!!

So my answers to are 2/37, 4/37 and 8/37.

I guess...I AM HOTHOTHOT!!!