# Ancient Egyptian Fractions

## Use unit fractions the Ancient Egyptian way to pay workers using unit fractions.

# INSTRUCTIONS

A UNIT fraction is a fraction with a numerator one(1). Instead of 3/4, the Ancient Egyptians would use the unit fractions 1/2 + 1/4. Fractions notation as we know it was not in use generally until after 1500 AD.

Think of slicing 3 cakes among 4 workers. We would slice each cake into 4 pieces and give each worker 3 of them. Each worker would get three 1/4 pieces. That means you will slice and move 12 pieces of cake.

With unit fractions you will slice two of the cakes into 2 pieces and the third into 4 pieces.Each worker would get one 1/2 piece and one 1/4 piece. Here you will slice and move 8 pieces. The unit fraction method could be a faster way to solve the problem.

After pressing the <START> button a timer will start. You will see a number of cakes and a number of buckets. First, press the OK button on the slicer to choose the number of slices for each cake. Then press the <SLICE> button to move the slicer onto the cake. You can change the number of cake cutters before you slice. You can then can drag each slice into a worker basket.

Do not put the same size slice into one basket. Each basket will contain UNIT fraction in different sizes. For example, you should not put two 1/5 pieces into the same basket. If you do, the game will stop and ask you to press the <RESET EXAMPLE>.

If you make a mistake, just press the <RESET EXAMPLE> button to do the same example over.

The slicer will move over to the last piece when you have one slice left. This will allow you to slice the final piece into more parts, if necessary. When all barley cakes are given, a scroll will open showing Ancient Egyptian script for the fractional pieces of bread.

The 1/15 in this image shows that the denominator 15 is below a mouth image. The inverted U is used for the number 10, and each of the 5 bottom marks represent one(1). Because the numerator is always one(1) only the denominator needs to be shown.How the ancient Egyptians solved the problem of 2 cakes to 5 workers:

First, cut the cake into enough pieces so that each worker will get a piece.

If you cut the cakes into 2 pieces for 1/2 will each worker get a piece? No.

If you cut the cakes into 3 pieces for 1/3 will each worker get a piece? Yes.

So you will slice the cakes into 3 parts so that each workers will get a 1/3 piece. This will leave one 1/3 piece. This 1/3 piece can be sliced into 5 parts. Each part is actually 1/15 slice (1/5X1/3). You will drag each smaller pieces into each worker basket. That gives each worker 1/3 slice and 1/15 slice. If you add 1/3 and 1/15 you will get 2/5. This way you would have to move 10 pieces.

Entering the the largest possible unit fraction that is smaller than the fraction you're working on is called the GREEDY ALGORITHM.

Compare this to a more modern way of slicing each cake into 5 pieces and giving each worker 2 of them. To do this you would also move 10 pieces.

The clock will start when you press the <START> button. The clock will stop when you give out all the cakes. If correct, the scroll will open so that you can see the results. You will be warned if you put more than one of the same slice into the same basket. You will then redo the example by pressing the <RESET EXAMPLE> button.

You will have 5 tries to pay the workers. The game will then stop after the fifth example. You will be given a score of the number of workers paid per minute. The <Report> button will open a dialog box that will ask for your name. You may, of course, enter any code for your name instead. The <Send> key on the dialog box will open a browser window with your report card.

Press the <START> button to start a new game.

Qualify as an ASSISTANT ANCIENT EGYPTIAN CAKE SLICER by achieving a score of more than 5 workers paid per minute.

Qualify as an ANCIENT EGYPTIAN CAKE SLICER by achieving a score of more than 8 workers paid per minute.

Press the <Report> button to make a report card. This button will open a dialog that asks you to submit your name. You may prefer to submit a code for your name. The <Send> key on the dialog box will open a new browser window with your printable report card.

For the experts:

Any unit fraction can be cut into smaller unit fractions. For example:

To give 4 cakes to 6 workers you would give each worker 1/2 cake and 1/6 cake. The 1/6 cake can can be sliced into 1/7 + 1/42. So instead of giving each worker 1/2 + 1/6 you could give 1/2 + 1/7+ 1/42.

You can qualify for the ANCIENT EGYPTIAN CAKE SLICER title if you can arrive at a denominator of 50 or more.

If you get a piece equal to or smaller than 1/100 the scroll will show a coiled rope for each hundred. Be careful, though, the computer cannot show pieces smaller than 1/300. You will have to <RESET> the example and cut into larger pieces. What worker is going to want such a small piece anyway? These crumbs probably were eaten by the Ancient Egyptian Paymaster.

## WHAT WILL I LEARN?

- What is meant by a unit fraction..
- How Ancient Egyptians wrote fractions.
- More practice with adding fractions.