# Egyptian Fractions - How Many Ways?

You might want to take a look at Egyptian Fractions first.

Some unit fractions can be written as a sum of two different unit fractions in more than one way.
Here are the ways to write $\frac{1}{6}$:

$\frac{1}{6} = \frac{1}{7} + \frac{1}{42}$

$\frac{1}{6} = \frac{1}{8} + \frac{1}{24}$

$\frac{1}{6} = \frac{1}{9} + \frac{1}{18}$

$\frac{1}{6} = \frac{1}{10} + \frac{1}{15}$

($\frac{1}{6} = \frac{1}{12} + \frac{1}{12}$, but this has a repeat so it doesn't count.)

How many ways can you find to write $\frac{1}{8}$ as a sum of two different unit fractions?

Or $\frac{1}{9}$? Or $\frac{1}{10}$? Or $\frac{1}{11}$...

Charlie had a go at writing $\frac{1}{12}$ as the sum of two different unit fractions.

Can you find all the ways in which this can be done?

Click below to check you haven't missed any!

$\frac{1}{12}=\frac{1}{13}+\frac{1}{156}$

$\frac{1}{12}=\frac{1}{14}+\frac{1}{84}$

$\frac{1}{12}=\frac{1}{15}+\frac{1}{60}$

$\frac{1}{12}=\frac{1}{16}+\frac{1}{48}$

$\frac{1}{12}=\frac{1}{18}+\frac{1}{36}$

$\frac{1}{12}=\frac{1}{20}+\frac{1}{30}$

$\frac{1}{12}=\frac{1}{21}+\frac{1}{28}$

($\frac{1}{12}=\frac{1}{24}+\frac{1}{24}$)

Alison wondered whether there's a quick way to work out how many ways there are to write any unit fraction as the sum of two different unit fractions.

She noticed that when she wrote $\frac{1}{6}$ as the sum of two unit fractions, the denominators of the fractions were 1, 2, 3, 4, 6, 9, 12, 18 and 36 more than 6. She thought that was interesting...

Take a look at other examples of unit fractions written as the sum of two unit fractions.

What do you notice about the difference between the denominator of a unit fraction and the denominators of any pair of unit fractions that add up to make it?

Can you use this to determine how many ways there are to write any unit fraction as the sum of two different unit fractions?