Fractions are Hard!

4.3 A Curious Fraction Tree

Here is something fun to think about.  Consider the following “fraction tree:”



Do you see how it works?  Each fraction has two “children.” The left child is always a number smaller than 1 and the right child is always a number larger than 1 and the box in the upper right shows how to construct the two children from a given parent.


a) Continue drawing the fraction tree for another two rows.


b) Explain why the fraction \(\dfrac{13}{20}\) will eventually appear in the tree. (It might be easier to figure out what the parent of \(\dfrac{13}{20}\) must be by first noticing that is a left child. Next, what is its grandparent? Its great grandparent?)


c) Explain why the fraction \(\dfrac{13}{20}\) cannot appear twice in the tree.


d) Will the fraction \(\dfrac{456}{777}\) eventually appear in the tree? Could it appear twice?


In general: Explain why each and every fraction of the form \(\dfrac{a}{b}\) with \(a\) and \(b\) positive integers sharing only \(1\) as a common factor is sure to appear precisely one time in the tree.


COMMENT: To learn more about this tree and the astounding results young high-school students discovered about it, have a look at Chapter 21 of Mathematics Galore! The First Five Years of the St. Mark’s Institute of Mathematics published by the Mathematical Association of America, 2012.

Please join the conversation on Facebook and Twitter and kindly share this page using the buttons below.

Share on FacebookTweet about this on Twitter




Take your understanding to the next level with easy to understand books by James Tanton.



Guides & Solutions

Dive deeper into key topics through detailed, easy to follow guides and solution sets.


light bulb


Consider supporting G'Day Math! with a donation, of any amount.

Your support is so much appreciated and enables the continued creation of great course content. Thanks!


Ready to Help?

Donations can be made via PayPal and major credit cards. A PayPal account is not required. Many thanks!