Egyptian Fractions - Increasing the Numerator

This task is all about writing fractions of the form $\frac{2}{n}$ as Egyptian Fractions.

Can you find some different ways of writing $\frac{2}{3}$ as a sum of different unit fractions?

The easiest way would be to write $\frac{2}{3}=\frac{1}{3}+\frac{1}{3}$, but Egyptian Fractions can't include a repeated unit fraction.

We could always split the second $\frac{1}{3}$ into a sum of unit fractions...

$\frac{2}{3}=\frac{1}{3} + \frac{1}{4} + \frac{1}{12}$

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

$\frac{2}{3}=\frac{1}{3} + \frac{1}{6} + \frac{1}{30}+\frac{1}{20}+\frac{1}{12}$

But is there a way to write $\frac{2}{3}$ as the sum of just two unit fractions, without repeats?

Can all fractions of the form $\frac{2}{n}$ be written as a sum of two different unit fractions?

What about fractions of the form $\frac{3}{n}$? Or $\frac{4}{n}$? Or ... ?

How could you write these as Egyptian Fractions using the least number of unit fractions?

Share your thoughts and discoveries