In school, we go over converting fractions to decimals, percentages to fractions, and even percentages to decimals, but we are never told how to make complex decimals become fractions. How do you even start? Well, in this post, I’m going to go over exactly how this process is done!

We’re not going to focus on converting common decimals, such as 0.5 or 0.25, into fractions. Instead, we’ll focus on long and even repeating decimals. To begin, let’s look at how this idea is applied to basic decimals:

Essentially, all decimals are fractions with a denominator of 1. This would also explain the idea of adding because our denominators would be the same (or 1). For example, 2+3 is really just 2/1 + 3/1, which would be 5/1 or just 5. Now, going back to our examples, to make these proper fractions, we need to put these values in lowest terms such that the numerator and denominator are both whole numbers:

Thus, that is how we can convert those decimal values into fractions. However, the question still remains, how do we convert repeating decimals into fractions? To answer this, let’s start with some problems:

Now, how do we get these values to become fractions. Well, since the values are repeating, let’s figure out a way where the repeating portion gets cancelled out. Thus, let’s use algebra to get an easier representation of our values:

Wow! We were able to convert repeating decimals into proper fractions. More importantly though, we can actually find a pattern from this information. In particular, in the denominator, the number of 9’s is equal to the number of repeating digits. Moreover, the number of zeroes is equal to the number of non-repeating values (after the decimal point). Finally, to get the numerator, subtract the non repeating value(s) from the whole value after the decimal point. For emphasis, in example 2, we get the numerator as 8246 because 8254 – 8 is equal to 8246. Lastly, let’s apply this concept to some other problems:

For problem 1, the numerator would be: 5813-58 = 5755. For the denominator, we know that there are 2 repeating digits and 2 non-repeating values. Hence, we get 9900. Overall, our fraction would be 5755/9900. Now, let’s check. 5755/9900 = 0.5813131313. Moving on to problem 2, the numerator would be: 5218046-521 = 5217525. Since there are 4 repeating values and 3 non-repeating values, our denominator would be 9999000. Alas, our fraction would be 5217525/9999000. Just to check, we get: 0.52180468046! In conclusion, not only did we learn how to convert repeating decimals into fractions, but we found a pattern and used that to complete the task much more efficiently!