BMAD 62 Report post 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 Share this post Link to post Share on other sites

0 jasen 4 Report post Posted October 21, 2015 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 Share this post Link to post Share on other sites

0 k-man 26 Report post Posted March 14, 2014 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. Share this post Link to post Share on other sites

0 BMAD 62 Report post Posted March 14, 2014 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, Share this post Link to post Share on other sites

0 CaptainEd 19 Report post Posted October 19, 2015 An observation without an explanation34, 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...:-( Share this post Link to post Share on other sites

0 jasen 4 Report post Posted October 20, 2015 (edited) An observation without an explanationHidden Contentmaybe this have something to do with math golden ratio, which it close to 1.6180339887...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 Share this post Link to post Share on other sites

0 BMAD 62 Report post Posted October 20, 2015 hintStern-Brocot Tree Share this post Link to post Share on other sites

0 CaptainEd 19 Report post Posted October 21, 2015 Thanks, BMAD, I've never heard of Stern-Brocot before, a fascinating topic! Share this post Link to post Share on other sites

0 BMAD 62 Report post Posted October 22, 2015 No problem ! Share this post Link to post Share on other sites

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?

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