Fraction Puzzle Using Digits from 0 to 9Date: 02/04/2008 at 19:50:14 From: Gabi Subject: fractions less then 1/2 Using all ten numbers 0 through 9, make 5 fractions less then 1/2. I only could come up with 1/4, 2/6, 3/9. I couldn't find fractions for the 0,5,7 and 8 even trying to change the 1/4, 2/6 and the 3/9. Date: 02/04/2008 at 21:09:11 From: Doctor Ian Subject: Re: fractions less then 1/2 Hi Gabi, This is the kind of problem where you can stumble around forever unless you set out with some kind of plan. I'd probably make a table, like this: 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Anything divided by itself is 1, so those can't be possibilities: 1 2 3 4 5 6 7 8 9 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 1 divided by anything greater than 2 is less than 1/2, so let's mark those as good, with an x. (And we'll mark 1/2 as '-', so we know we've already considered it.) 1 2 3 4 5 6 7 8 9 1 - - x x x x x x x 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 2 divided by anything greater than 4 is less than 1/2: 1 2 3 4 5 6 7 8 9 1 - - x x x x x x x 2 - - - x x x x x 3 - 4 - 5 - 6 - 7 - 8 - 9 - We can do the same kind of reasoning for 3, 4, and so on: 1 2 3 4 5 6 7 8 9 1 - - x x x x x x x 2 - - - x x x x x 3 - - - - x x x 4 - - - - - x 5 - - - - - 6 all these - - - - would be 7 greater than 1 - - - 8 - - 9 - How does this help? Well, it gives us a complete list of all the possible fractions that could be less than 1/2: 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 2/5, 2/6, 2/7, 2/8, 2/9 3/7, 3/8, 3/9 4/9 So now we have to find a set of 5 of these that use up all the digits. That's not necessarily easy, but we can be systematic about it, and that's half the battle. The first thing I notice is the diagonal: 1/3, 2/5, 3/7, 4/9 That's almost a solution right there: * 1 2 3 4 -, -, -, - We've used everything but 0, 6, and 8. 3 5 7 9 We've used 3 twice. * Can you see how to get from here to a complete solution? Keep in mind that 0 over anything is 0, which is less than 1/2. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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