Converting a Recurring Decimal into a Fraction

In a previous section of this unit we looked at converting a decimal to a fraction. This method would only work for terminating decimals. It is impossible to write an irrational number as a fraction, however with a few little tricks, we can write a recurring decimal as a fraction.

To do this, we want to get rid of the repeating part of the decimal. For example, if we had the decimal 0.1543434343434343.... (going on forever), we want to get rid of the "43434343....." part. To do this, we call our original number "x" (i.e. x = 0.1543434343434343.... (going on forever)). Next, we want to multiply by a power of 10 (i.e. 10, 100, 1000, 10000, etc) that has the same number of zeros as the number of repeating digits. In this case we have two repeating digits (4 and 3) so we will multiply by 100 (because it has two zeros).
This gives us 15.434343434343..... (going on forever). So from this, we can say that 100x = 15.4343434343.... (going on forever) because we multiplied our original number (which we called "x") by 100.
Now we get a bit fancy. We then take this 100x and subtract our original x, so we get:

100x - x = 15.434343434343.... - 0.15434343434343....

Now if we do the left hand side of this sum we get 99x
If we do the right hand side of this sum, we see that the repeating pattern (the 43434343.... part) actually disappears (gets cancelled out), leaving us with 15.28

So what we now have is:

99x = 15.28

Solving this equation then gives us:

x = 15.28/99

x = 382/2475


Now this may seem quite complicated, but with a little practice it does get a whole lot easier. I have added a few more examples below.

I have uploaded a worksheet that has some questions on this, as well as general working with decimals questions. This can be found in the "Worksheets" tab. Work through this and have as much as you can completed by Monday's lesson. 

If you have any questions on this content, feel free to comment on this post, or email or chat to me in person.