wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> easy >> fractions....
        
(Message started by: Noke Lieu on Jun 24^th, 2009, 1:08am)

Title: fractions....
Post by Noke Lieu on Jun 24^th, 2009, 1:08am

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)

Title: Re: fractions...
Post by MathsForFun on Jun 27^th, 2009, 2:54am

They all result in numbers in the range 0..1 which can be written with four or fewer decimal digits.

Title: Re: fractions...
Post by Noke Lieu on Jun 28^th, 2009, 2:06am

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 =

Title: Re: fractions...
Post by Grimbal on Jun 28^th, 2009, 4:03am

1/6 x 4/3?

Title: Re: fractions...
Post by Noke Lieu on Jun 28^th, 2009, 6:10am

why yes... :D
that one slipped by...

though there are nicer ones than that.

Title: Re: fractions...
Post by chronodekar on Jun 29^th, 2009, 3:33am

What kind of pattern are we looking for? This one has me stumped... ???

-chronodekar

Title: Re: fractions...
Post by towr on Jun 29^th, 2009, 4:14am

on 06/29/09 at 03:33:23, chronodekar wrote:

What kind of pattern are we looking for? This one has me stumped... ???

Have you tried Noke Lieu's clues?
[hide]21 * 60 = 1260
41 * 35 = 1435[/hide]

If you try that with the original problem, you get things like [hide]1/4 x 8/5 = 18/45[/hide]

Title: Re: fractions...
Post by chronodekar on Jun 30^th, 2009, 2:40am

:o

<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

Title: Re: fractions...
Post by Noke Lieu on Jun 30^th, 2009, 2:55am

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' (http://mathworld.wolfram.com/AnomalousCancellation.html)

What gets me scratchinng my head is how do you find them elegantly... I can crunch it, but that's far from elegant.

Title: Re: fractions...
Post by MathsForFun on Jun 30^th, 2009, 3:25am

on 06/30/09 at 02:55:42, 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.

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)]

Title: Re: fractions...
Post by towr on Jun 30^th, 2009, 4:09am

on 06/30/09 at 03:25:53, MathsForFun wrote:

(%i29) solve([a/b * c/d = (a*10 + c)/(b*10/d)], [a]);

Shouldn't (b*10/d) be (b*10+d)?

I get [hide]1/a = 1/b - 10/c + 10/d[/hide]

Title: Re: fractions...
Post by pex on Jun 30^th, 2009, 10:19am

33/58 x 88/87 8)

Title: Re: fractions...
Post by towr on Jun 30^th, 2009, 10:43am

44/9 x 3/68 x 85/4 ::)

169442/3 x 9/91 x 31/4 :P

Title: Re: fractions...
Post by MathsForFun on Jul 1^st, 2009, 1:23am

on 06/30/09 at 04:09:28, towr wrote:

Shouldn't (b*10/d) be (b*10+d)?

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.

Title: Re: fractions...
Post by chronodekar on Jul 1^st, 2009, 10:10pm

on 06/30/09 at 02:55:42, Noke Lieu wrote:

Really cool link. Thanks for sharing !! :)

-chronodekar