Keep it Simple

Catherine and Poppy from Stoke by Nayland Middle School made a good start on this problem, and Kijung from Wind Point Elementary School found that:

Not all of Jamie's examples were right.

To be correct, one of the unit fractions must have a denominator which is 1 more than the denominator of the original unit fraction, and the other unit fraction must have a denominator which is the product of the other two denominators:

$$ \frac{1}{n} = \frac{1}{n+1}+\frac{1}{n(n+1)}$$

Here are some other examples that work:

$ \frac{1}{5} = \frac{1}{6}+\frac{1}{30}$
$ \frac{1}{6} = \frac{1}{7}+\frac{1}{42}$
$ \frac{1}{105} = \frac{1}{106}+\frac{1}{11130}$

$\frac{1}{8}$ can also be expressed as the sum of two unit fractions in several ways:

$\frac{1}{8} = \frac{1}{9} +\frac{1}{72}$
$\frac{1}{8} = \frac{1}{10} +\frac{1}{40}$
$\frac{1}{8} = \frac{1}{11} + \frac{1}{n}$ is not possible
$\frac{1}{8} = \frac{1}{12} +\frac{1}{24}$

Felix from Condover Primary acutely observed that unit fractions with denominators which are prime numbers can only be written in one way as the sum of two distinct unit fractions.

Can unit fractions with composite denominators always be written in more than one way as the sum of two unit fractions?