• 16-19

Task: Claculate the thickness of an A4 size paper folded 42 times.

this was the written response of a Grade 9 student

Problem Solving Strategies

What would be the thickness of a piece of paper if it was folded in half 42 times?

Have you ever tried folding over a piece of A4 paper in half over and over again to see how thick it will become? When you do so, you might notice how quickly it grows from less than a millimeter’s thickness to twice the thickness of a notebook. But I could bet you’d never manage to fold it over more than six or seven times. But consider you would manage to fold it in half forty-two times – can you guess what total thickness you would get to then?

In actual fact, you can find out without having to fold over a piece of paper 42 times. The first step to this is finding the paper’s thickness. You might think the paper is too thin to measure, but you can in fact find the approximate thickness using only a ruler.

Think about what happens when you fold a piece of paper once. It is almost like putting another piece of paper with the same thickness onto the original paper – thus you have two layers of paper. When you fold all of this again, it is like putting another two layers of paper onto the other two layers. So basically, every time the paper is folded, the number of layers is doubled, and thus the sum thickness is doubled.

When you have folded the paper a 4 to 5 times, you will notice that the paper has significantly increased in thickness, and it is now so thick that you can easily measure it using a normal ruler.

When having folded the paper 5 times, it will be about 4 mm thick. But to find out how thick a single layer of paper would be, we have to find out how many layers of paper actually are piled so that we can find out how many millimeters are added per piece of paper. There is a number of ways to do this:

The simplest way is probably to count the layers one by one. It is sometimes the best way to be completely sure, though you should make sure not to miscount.

Consider that every time you fold the paper, you double the number of layers which are piled. In that case we can start with one layer, which is doubled with the first fold, meaning we have two layers. With the second fold, two is doubled to make four layers. With the third fold we get eight layers, and with the fourth fold we reach sixteen layers. Finally, when folding the sixteen layers to double the number of layers, we conclude that with five folds we will pile thirty-two layers of paper.

We can write the method above in mathematical language, making the process a lot faster. If we call the thickness of a single layer of paper n we can say that folding the sheet – that is to say when doubling the thickness of the sheet – the thickness is now two times n, which is written as 2n. Thus we can also say that when folding the paper two times – that is to say doubling the total thickness again – the total thickness is now two multiplied by two multiplied by n, which we write as 2 × 2 × n or 22 n. Thus we can also say that when folding a paper 5 times, the thickness is now 2 × 2 × 2 × 2 × 2 × n or 25n (two to the power of two times n, n being the original thickness). This is equal to 32n: thirty-two layers of paper.

So in both methods 2 and 3 we have proven that the number of layers you need to pile to make a 4 mm thickness worth of paper is 32. So how does this help us find the thickness of a single sheet? It’s very simple. As already explained, when we know how many layers you need to make a thickness of 4 mm, you can find out how much thickness is added for each sheet of paper by dividing the four millimeters by thirty two – which is the same as saying “what times thirty-two is equal to 4 mm”? This is written as

n = 4mm /32 = 0.125mm

Therefore we know that the thickness of a single sheet of paper is 0.125 mm.

Then how do we find out how what thickness worth of paper we will reach if we could fold a paper 42 times?

Let’s once again consider what happens when folding a piece of paper over and over again: The number of layers of paper is doubled every time. This means that when folding a piece of paper 42 times, we have to double the thickness 42 times as well. Therefore we can say that our answer must be the thickness of a single sheet (which is 0.125 mm) which is multiplied by two 42 times.

As it would be to long writing an equation that goes n × 2 × 2 × 2 × 2 × 2 × 2 × 2…. and so on until we have 42 twos in the equation, we can simply write 242 × n, or 242 n where the number to the top right of the 2 represents the number of times a two is multiplied with another two; in this case 42 times.

As we already know the value of n, we can substitute it into the equation so our final equation becomes

T = 242 × 0.125 mm where T represents the total thickness worth of paper when folding it 42 times.

Using a calculator, we can quickly find out that 242 (two to the power of forty-two or two multiplied by itself 42 times) equals 4 398 046 511 000, can develop the equation to

T = 4398046511000 × 0.125 mm = 549 755 813 900 mm

Therefore we know that the total thickness worth of paper when a 0.125 mm thick sheet is folded 42 times is 549 755 813 900 mm. As a million millimeters equal one kilometer, we can convert the total thickness of 549 755 813 900 mm to 549 755.813900 km – more than enough to cover the mean distance between the moon and the earth.