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BMAD

SBT Fraction

Question

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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8 answers to this question

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hint

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modified Stern-Brocot Tree to third sequence.
3rd sequence  34/55   157/254  123/199 212/343  89/144  233/377  144/233  199/322 55/89

since 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 

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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.

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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,

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An observation without an explanation

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...:-(

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An observation without an explanation

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maybe this have something to do with math golden ratio, 
which it close to 1.6180339887...

2/3 = 1.5
3/5 = 1.6666....
5/8 = 1,6
8/13 = 1.623
.....
144/233 = 1.618055556....
233/377 = 1.618025751....

Edited by jasen
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Thanks, BMAD, I've never heard of Stern-Brocot before, a fascinating topic!

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