Not Continued Fractions
Age
14 to 18
Challenge level
Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?
Problem
- Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$
- Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.
Getting Started
What is the integer part of $N$?
Student Solutions
- The key here is that $x$ has to be the integer part of $N$ because the 'continued fraction' part of the expression gives a value less than one.
As $y$ and $z$ are positive integers (whole numbers), $y + 1/z > 1$ and $1/(y+1/z) < 1$ so we know that this must equal $3/7$ and $x = 1$.
Hence $y + 1/z = 7/3$. Again $y$ has to be the integer part of $7/3$ so $y = 2$ and $z = 3$.
- As in the first part, if $N = 8/5$, then we must have $x = 1$ and $y + 1/z = 5/3$.
To make $y$ and $z$ positive integers we must have $1/z < 1$ and $y = 1$.
It then follows that $1/z = 2/3$ so it is impossible to find positive integer values for $x$, $y$ and $z$.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
For experience of reasoning about the integer part of a number and working with fractions.
Possible approach
Challenge the students to invent their own problems of this type.
Key question
What is the integer part of $N$?