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Topic: fractions.... (Read 693 times) |
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Noke Lieu
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fractions....
« on: Jun 24th, 2009, 1:08am » |
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What's special about these fractions?
1/4 x 8/5
1/2 x 5/4
1/9 x 9/5
4/9 x 9/8
(sorry that I've been away, if anyone noticed- there's been a bit of change at work. It's now less easy for me to drop in)
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| « Last Edit: Jul 9th, 2009, 5:46am by Noke Lieu » |
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MathsForFun
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Re: fractions...
« Reply #1 on: Jun 27th, 2009, 2:54am » |
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They all result in numbers in the range 0..1 which can be written with four or fewer decimal digits.
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Noke Lieu
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Re: fractions...
« Reply #2 on: Jun 28th, 2009, 2:06am » |
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There *ahem* might be a little more to it than that.
Otherwise we could include 1/2 x 1/2. Which we can't, though it is half way there.
Perhaps a clue?
21 * 60 =
41 * 35 =
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Grimbal
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Re: fractions...
« Reply #3 on: Jun 28th, 2009, 4:03am » |
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1/6 x 4/3?
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Noke Lieu
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Re: fractions...
« Reply #4 on: Jun 28th, 2009, 6:10am » |
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why yes... 
that one slipped by...
though there are nicer ones than that.
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chronodekar
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Re: fractions...
« Reply #5 on: Jun 29th, 2009, 3:33am » |
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What kind of pattern are we looking for? This one has me stumped... 
-chronodekar
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towr
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Re: fractions...
« Reply #6 on: Jun 29th, 2009, 4:14am » |
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on Jun 29th, 2009, 3:33am, chronodekar wrote:| What kind of pattern are we looking for? This one has me stumped... |
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Have you tried Noke Lieu's clues?
21 * 60 = 1260
41 * 35 = 1435
If you try that with the original problem, you get things like 1/4 x 8/5 = 18/45
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chronodekar
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Re: fractions...
« Reply #7 on: Jun 30th, 2009, 2:40am » |
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<hits palm on face>
towr,
I have no idea how I missed THAT !! It's just too obvious now!! 
Still, thanks a lot for the clarification.
-chronodekar
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Noke Lieu
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Re: fractions...
« Reply #8 on: Jun 30th, 2009, 2:55am » |
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don't worry- it's only good because it's so un-obvious.
For a related bit of trivia, have a look for 'anomolous cancellation'
What gets me scratchinng my head is how do you find them elegantly... I can crunch it, but that's far from elegant.
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MathsForFun
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Re: fractions...
« Reply #9 on: Jun 30th, 2009, 3:25am » |
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on Jun 30th, 2009, 2:55am, Noke Lieu wrote:| What gets me scratchinng my head is how do you find them elegantly... I can crunch it, but that's far from elegant. |
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There probably isn't any way - though "solving" the equation reduces the number of nested "for" loops needed from 4 to 3 - hence reducing the total number of loops from 9^4 to 9^3:
(%i26) solve([a/b * c/d = (a*10 + c)/(b*10/d)], [a,b,c,d]);
(%o26) [[a = -%r2*%r3^2/(10*%r3^2-10*%r2),b = %r1,c = %r2,d = %r3]]
Hence:
(%i29) solve([a/b * c/d = (a*10 + c)/(b*10/d)], [a]);
(%o29) [a = -c*d^2/(10*d^2-10*c)]
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| « Last Edit: Jun 30th, 2009, 3:28am by MathsForFun » |
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towr
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Re: fractions...
« Reply #10 on: Jun 30th, 2009, 4:09am » |
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on Jun 30th, 2009, 3:25am, MathsForFun wrote:| (%i29) solve([a/b * c/d = (a*10 + c)/(b*10/d)], [a]); |
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Shouldn't (b*10/d) be (b*10+d)?
I get 1/a = 1/b - 10/c + 10/d
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| « Last Edit: Jun 30th, 2009, 4:15am by towr » |
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pex
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Re: fractions...
« Reply #11 on: Jun 30th, 2009, 10:19am » |
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33/58 x 88/87 
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MathsForFun
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Re: fractions...
« Reply #13 on: Jul 1st, 2009, 1:23am » |
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on Jun 30th, 2009, 4:09am, towr wrote:| Shouldn't (b*10/d) be (b*10+d)? |
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Yes - that was a blunder by me.
(%i3) solve([a/b * c/d = (a*10 + c)/(b*10 + d)], [a]);
(%o3) [a = b*c*d/((c-10*b)*d+10*b*c)]
So the brute force solution would require 9^3=729 loops, and in each case, the test would be whether the expression b*c*d/((c-10*b)*d+10*b*c) yields an integer.
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chronodekar
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Re: fractions...
« Reply #14 on: Jul 1st, 2009, 10:10pm » |
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on Jun 30th, 2009, 2:55am, Noke Lieu wrote:don't worry- it's only good because it's so un-obvious.
For a related bit of trivia, have a look for 'anomolous cancellation'
What gets me scratchinng my head is how do you find them elegantly... I can crunch it, but that's far from elegant. |
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Really cool link. Thanks for sharing !! 
-chronodekar
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