Egyptian fractions are unit fractions, that is fractions with a numerator of 1. We are going to investigate methods of reducing unit fractions to the sums of different smaller unit fractions as we see in this example

$\frac{1}{2}= \frac{1}{3}+\frac{1}{6}\\ \frac{1}{3}=\frac{1}{4}+\frac{1}{12}\\ \frac{1}{4}=\frac{1}{5}+\frac{1}{20}...$

Try some for yourself. Does this always work? Could you explain to someone else how to use this method?

Perhaps we can find other ways to reduce unit fractions to the sums of different smaller unit fractions.

For example $\frac{1}{6}= \frac{1}{7}+\frac{1}{42}\\ \frac{1}{6}=\frac{1}{8}+\frac{1}{24}\\ \frac{1}{6}=\frac{1}{9}+\frac{1}{18}\\ \frac{1}{6}=\frac{1}{10}+\frac{1}{15}\\ \frac{1}{6}=\frac{1}{11}+\frac{1}{14}+\frac{1}{231}....$

Can all unit fractions be made up in more than one way like this? Why? Try some examples of your own to test out your ideas.

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