# How to write the fraction 4 over 5 as the sum of two or more fractions?

You can do this a number of ways. The simplest is to decompose 4 into a sum.
4 = 1 + 3
4/5 = 1/5 + 3/5 Or
4 = 2 + 2
4/5 = 2/5 + 2/5
The ancient Egyptians (and some other civilizations) made use of what we now call "Egyptian Fractions" to write fractions with a numerator greater than 1. They would, for example, write 2/5 as the sum
2/5 = 1/3 + 1/15
They had a simple algorithm for finding such sums good with fractions of the form 2/n. In your problem, however, we have double this amount.
4/5 = 2(2/5) = 2(1/3 + 1/15) = 2/3 + 2/15
Applying the 2/n algorithm again, we find that
4/5 = 2/3 + 2/15 = (1/2 + 1/6) + (1/10 + 1/30)
Thus, a decomposition of 4/5 into Egyptian Fraction form is
4/5 = 1/2 + 1/6 + 1/10 + 1/30 ______
It is unlikely you are expected to know anything about Egyptian Fractions, but they are an interesting way to work your problem. (At the link above is a section on how to compute them when the denominator is prime (3, 5) and when it is not (15).)
thanked the writer.
Mark Mottian commented
I learnt about Egyptian Fractions at school; however my teacher never taught us a method of working it out. This was an extremely helpful answer!
Oddman commented
It isn't clear how the Egyptians might have written this. One can optimize the number of fractions in the sum, or the range of the denominators. Or one can avoid using the denominator of the sum. 4/5 = 1/2 + 1/4 + 1/20 = 1/2 + 1/5 + 1/10. It seems probable that fractions would be chosen to aid further calculation.