Real Mathematics – Numbers #5

The word fraction has a Latin root “fractio” which means “to break”. Let’s take a slightly different approach for fractions. Instead of breaking, we will use “folding”.

Fractions with Papers

In order to understand the four arithmetical operations in fractions, we can use a standard A4 paper. First we will fold the paper into two halves. Every half represents the fraction ½. We can continue folding one of the pieces in two halves. In the end we will have the following papers which we can use for four mathematical operations:

a. 1
b. 1/2+1/2
c. 1/2+1/4+1/4
d. 1/2+1/4+1/8+1/8
e. 1/2+1/4+1/8+1/16+1/16


We are aware of the fact that all five of the versions are equal to one another and they each add up to 1. If we take a look at versions b and c we can conclude that ½ is equal to ¼ + ¼. If we substitute 1/2s in the version b we can find the following result:

1 = ½ + ½ = ¼ + ¼ + ¼ + ¼.


Let’s continue from the previous. If we subtract ¼ from both sides of ½ = ¼ + ¼, then we get:

½ – ¼ = ¼ + ¼ – ¼

½ – ¼ = ¼


Let’s start with another A4 paper and take its whole as 1. This time we will fold the paper and use the marks on it.


Assume that we are trying to find ½ * ¾.

Check the first fraction and fold the paper into two halves since it is ½.


Then check the second fraction and fold the paper into four equal parts.


Since second fraction is ¾, mark 3 parts of the paper.


Now unfold the A4 paper completely. It shows that there are 8 equal parts and 3 of them are marked. Hence solution is 3/8.


One wonders…

  1. Using the paper method show the connection(s) between 1/8 and 1/32.
  2. Find 1/8 – 1/32 with the paper method.


Imagine a gambling game in which you don’t have to gamble your money. In the worst case scenario, you will win nothing. A dream for gamblers, isn’t it?

Let’s say you get 128.000 dollars and 6 red and black cards (3 for each). Here is how the game goes: In every hand you must bet half of your total money. When you pick a red card, you will get twice what you played. When you pick a black card, you will lose all the money you bet in that hand.

When the game is finished (after all 6 cards are played) if you end up with more than 128.000 dollars you will get that surplus.

What is your strategy to win? Explain your answer.

M. Serkan Kalaycıoğlu

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