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Egyptian Fractions printable sheet
The ancient Egyptians didn't write fractions with a numerator greater than 1 - they wouldn't, for example, write $\frac{2}{7}$, $\frac{5}{9}$, $\frac{123}{467}$.....
Instead they wrote fractions like these as a sum of different unit fractions.
There are several NRICH problems based on Egyptian fractions. You can start by exploring unit fractions at Keep it Simple
In this problem we are going to start by considering how the Egyptians might have written fractions with a numerator of 2 (i.e. of the form $\frac{2}{n}$).
For example
$\frac{2}{3} = \frac{1}{3} + \frac{1}{3}$ (but since these are the same, this wasn't allowed.)
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{4} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{5} + \frac{1}{20} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{3} + \frac{1}{6} + \frac{1}{30} + \frac{1}{20} + \frac{1}{12}$
or
$\frac{2}{3} = \frac{1}{4} + \frac{1}{12} + \frac{1}{7} + \frac{1}{42} + \frac{1}{31} + \frac{1}{930} + \frac{1}{21} + \frac{1}{420} + \frac{1}{13} + \frac{1}{156}$
and so on, and so on!!
You might want to check that these are correct.
(If you can't see how these have been generated, take a look at Keep it Simple )
For $\frac{2}{3}$ that's quite easy.........$\frac{2}{3} = \frac{1}{2} + \frac{1}{6}$
But is it always so easy?
Try some other fractions with a numerator of 2.
Can they also be written as the sum of just two different unit fractions?
You might want to explore fractions of the form $\frac{3}{n}$, $\frac{4}{n}$, $\frac{5}{n}$...... and think about how the Egyptians would have represented these, using sums with the least number of unit fractions.
You might like to take a look at a follow up problem, The Greedy Algorithm