From Infinite Decimals to Mixed Fractions
Date: 10/09/98 at 19:00:50 From: Jessi Michelle Subject: A challenge from a teacher I was given a challenging problem by my math teacher just to see if I could find a way to figure it out. The problem is to write this fraction as a mixed number: .131313... + .555... -------------------- .161616... - .222... I think that the top can first be simplified to .6868.... After that I'm stuck because of the 2 away from 1 thing in a repeating decimal. It seems like it would be easier to change the decimals to fractions, but I don't know how. Help!
Date: 10/13/98 at 11:10:28 From: Doctor Nick Subject: Re: A challenge from a teacher Hi Jessi - Yes, converting the decimals to fraction is the way to go. There is a neat trick you can use to convert repeating fractions like this. I'll give you a couple of examples. The main trick is to multiply the number by a power of 10 (10, 100, 1000, 10000, etc.) that you pick to get the right effect. For instance, if x = 0.55555..., then 10x = 5.5555555..., and so 10x - x = 5. That is, 9x = 5, so x = 5/9. The trick is to pick a power of 10 that makes the repeating pattern in the decimal expansion "line up" so it disappears when you subtract. Here is another example: if x = 0.1616161616..., then 100x = 16.16161616161616..., and so 100x - x = 16, Then 99x = 16, which implies x = 16/99. This works even with decimals that don't repeat right away: x = 0.71232323232323... 100x = 71.23232323232323... 100x - x = 99x = 71.23 - 0.71 = 70.52 x = 70.52/99 = 7052/9900 = 1763/2475 That's a little more complicated than the other cases, but the method still works. Now, convert all the decimals in your problem to fractions, and simplify to a single fraction, and you'll have it. Have fun, - Doctor Nick, The Math Forum http://mathforum.org/dr.math/
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