An Easy, Yet Impressive Mental Math Trick You Don’t Know Yet (vol.2)

Elizabeth Westphal
3 min readAug 13, 2022

What if I asked you to convert the repeating decimal 0.636363… into a fraction? How fast can you do it? Can you do it at all? Give me a few minutes of your time and you will learn how to calculate this in a few seconds.

Let’s take a look at the method using the following example:

Step 1: Find the repeating unit. Our repeating unit is 63.

Step 2: How many digits are in the repeating unit? Our unit is 63 which has two digits. Easy, right?

Step 3: Take the repeating unit, and divide it by 9, 99, or 999… etc, depending on the number of digits in the repeating unit. Ours is two digits, so we divide 63 by 99

Step 4: This gives us a fraction which is equivalent to the decimal. The problem? It’s not in its lowest form. Reduce.

At a glance (if you remember your times tables) you will be able to determine that both the numerator and denominator are multiples of 9.

I invite you to try the same exercise with 0.33333…

Okay weird. Why 9's? Is 9 some sort of magic number?

While dividing the repeating part by 9’s is a neat little math trick, there is more going on behind the scenes.

Let’s assign our repeating decimal a variable, say x. That means we can represent 0.636363… as x, and we can represent 63.63636363… as 100x.

What happens when we subtract the first equation from the second equation? Can you see where I’m going with this?

We subtracted the equations to eliminate the repeating decimal, and look what we have: 99. Perform the same operation with 0.3333… and you’ll have an equation with 3 = 9x. Now we need to isolate the value of x:

Reduce as shown above, and you’re done! Want to learn more easy tricks to speed up your mental math? Check out vol.1 of this series, or follow to keep up with new releases. Thanks for reading!