BMAD Posted March 14, 2014 Report Share Posted March 14, 2014 There are two fractions, 34/55 and 55/89. We are looking for a third fraction of positive integers a/b, where 34/55>a/b>55/89 and 55<b<89. What is the smallest b where this is possible? 1 Quote Link to comment Share on other sites More sharing options...
0 jasen Posted October 21, 2015 Report Share Posted October 21, 2015 On 10/20/2015 at 6:26 PM, BMAD said: hintHidden Contentmodified Stern-Brocot Tree to third sequence.3rd sequence 34/55 157/254 123/199 212/343 89/144 233/377 144/233 199/322 55/89since it use fibbonaci number, which is relatively prime, so the fraction is already in simple form.so there is no fraction with 55<b<89 Quote Link to comment Share on other sites More sharing options...
0 k-man Posted March 14, 2014 Report Share Posted March 14, 2014 Reveal hidden contents The smallest b for a fraction in this range is 144 and the fraction is 89/144. I couldn't find any fractions with b<89 that would fall in the range between 55/89 and 34/55. Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted March 14, 2014 Author Report Share Posted March 14, 2014 On 3/14/2014 at 6:46 PM, k-man said: Reveal hidden contents The smallest b for a fraction in this range is 144 and the fraction is 89/144. I couldn't find any fractions with b<89 that would fall in the range between 55/89 and 34/55. Could you form a generalization as to why? the title is a hint at this, Quote Link to comment Share on other sites More sharing options...
0 CaptainEd Posted October 19, 2015 Report Share Posted October 19, 2015 An observation without an explanation Reveal hidden contents 34, 55, 89, and 144 are consecutive Fibonacci numbers. Presumably the next such fraction (in between 55/89 and 89/144) would be 144/233. Dunno why...:-( Quote Link to comment Share on other sites More sharing options...
0 jasen Posted October 20, 2015 Report Share Posted October 20, 2015 (edited) On 10/19/2015 at 4:10 PM, CaptainEd said: An observation without an explanationHidden Contentmaybe this have something to do with math golden ratio, which it close to 1.6180339887... Reveal hidden contents 2/3 = 1.53/5 = 1.6666....5/8 = 1,68/13 = 1.623.....144/233 = 1.618055556....233/377 = 1.618025751.... Edited October 20, 2015 by jasen Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted October 20, 2015 Author Report Share Posted October 20, 2015 hint Reveal hidden contents Stern-Brocot Tree Quote Link to comment Share on other sites More sharing options...
0 CaptainEd Posted October 21, 2015 Report Share Posted October 21, 2015 Thanks, BMAD, I've never heard of Stern-Brocot before, a fascinating topic! Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted October 22, 2015 Author Report Share Posted October 22, 2015 No problem ! Quote Link to comment Share on other sites More sharing options...
Question
BMAD
There are two fractions, 34/55 and 55/89. We are looking for a third fraction of positive integers a/b, where 34/55>a/b>55/89 and 55<b<89. What is the smallest b where this is possible?
Link to comment
Share on other sites
8 answers to this question
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.