TLQP
Egyptian Fractions
Egyptian Fractions
The Egyptians of 3000 BC had an interesting way to represent fractions.
Although they had a notation for 1/2 and 1/3 and 1/4 and so on (these are called unit fractions since
they are 1/n for some number n), their notation did not allow them to write 2/5 or 3/4 or 4/7 as we would
today.
Instead, they were able to write any fraction as a sum of unit fractions where all the unit fractions
were different.
For example,
3
/4 = 1/2 + 1/4
6
/7 = 1/2 + 1/3 + 1/42
Your turn:
1. Write
5
as Egyptians might have done so.
8
Why use Egyptian fractions today?


For two very good reasons:
The first reason is a practical one.
Suppose you have 5 sacks of grain to share between 8 people, so each would receive 5/8 of a sack of
grain in terms of present-day fractions. How are you going to do it simply, without using a calculator?
You could try pouring the 5 sacks of grain into 8 heaps and, by carefully comparing them, perhaps by
weighing them against each other, balance them so they are all the same! But is there a better way?
We will see that using unit fractions makes this easier.
The second reason is that it is much easier to compare fractions using Egyptian fractions than it is by
using our present-day notation for fractions! For instance:
Which is bigger: 5/8 or 4/7?
Remember - you are not allowed to use your calculator to answer this! Again unit fractions can make
this much simpler.
So suppose Fatima has 5 loaves of bread to share among the 8 workers who have helped dig her fields this
week and clear the irrigation channels. Pause for a minute and decide how YOU would solve this problem before
reading on.....
First Fatima sees that they all get at least half a loaf, so she gives all 8 of them half a loaf each, with one whole
loaf left.
Now it is easy to divide one loaf into 8, so they get an extra eighth of a loaf each and all the loaves are divided
equally between the 5 workers. On the picture here they each receive one red part (1/2 a loaf) and one green
part (1/8 of
a loaf):
and 5/8 = 1/2 + 1/8
Follow up questions: You try.
2.
3.
4.
5.
Suppose Fatima had 3 loaves to share between 4 people. How would she do it?
...and what if it was 2 loaves amongst 5 people?
...or 4 loaves between 5 people?
What about 13 loaves to share among 12 people? We could give them one loaf each and divide the
13th into 13 parts for the final portion to give to everyone.
Try representing 13/12 as 1/2 + 1/3 + 1/?.
What does this mean - that is, how would you divide the loaves using this representation?
Was this easier?
Comparing Egyptian fractions
Which is larger: 3/4 or 4/5?
We could use decimals so that 3/4 =0.75 =75/100 whereas 4/5 =0.8 = 0.80 = 80/100 so we can see that 80
(hundredths) is bigger than 75 and we can now see that 4/5 is bigger than3/4.
Could you do this without converting to decimals?
We could try using ordinary fractions as follows:
What common fraction could we convert both 3/4 and 4/5 into? 20ths would do:
3
/4 = 15/20 whereas
4
/5 = 16/20
so again we can easily see that 4/5 is larger than 3/4.
Using Egyptian fractions we write each as a sum of unit fractions:
3
/4 = 1/2 + 1/4
4
/5 = 1/2 + 3/10 and, expanding 3/10 as 1/4 + 1/20 we have
4
/5 = 1/2 + 1/4 + 1/20
We can now see that 4/5 is the larger - by exactly 1/20.
Follow up. You try.
Use Egyptian Fractions to determine which is larger.
6. Which is larger: 4/7 or 5/8?
7. Which is larger: 3/11 or 2/7?