Fractions and Decimals basically represent the same idea. Both are used to represent a part of a whole. That being said, it seems reasonable to convert a fraction to a decimal and back. Decimals are easier to work with, however finding a decimal to represent a fraction can be much more difficult than just writing the fraction itself. The reason for this is that decimals deal strictly with denominators that are divisible by 10. Any fraction that has a denominator that is a multiple of 10 can easily be converted into a fraction:
1/10 = .1
39/100 = .39
598/1000 = .598
This concept becomes more difficult when we are looking for decimal representations of fractions that do not have a denominator that is a multiple of 10. Lets look at the fraction 3/8. In order to do this we need to be able to convert the denominator to a number that is a multiple of 10. This can very difficult. In this case we can multiply the fraction by 125/125 as shown below:
While this seems simple enough, coming up with the 125 is a game of guessing and can be very difficult. Instead we can find this same answer another way without guessing.
In a fraction, the line that divides the numerator and denominator is actually the same as the ÷ sign. 3/8 is the same as saying 3 divided by 8. When we first learn division, we learn that the answer is 0 with a remainder of 3, but that 3 actually represents the 3 in the fraction 3/8. What we can do is rewrite our 3 in our divisor by adding extra 0's after the decimal point and do long division on the number.
We do this by treating the 3.0 as 30 and figure that 8 goes into 30 3 times.
We make sure that our decimal point in our answer lines up with the decimal point in dividend and write the 3 to the right of it in the same spot as if we were doing normal long division. We proceed the same way as a normal division problem and keep adding 0's to the dividend until we get a result that doesn't have a remainder.
As you can see we get an answer of .375 as we found above by guessing.
Repeating Decimals
Sometime when using this method we may find we get to a point where there is an endless loop. No matter how many 0's we add to the dividend we keep getting the same results. To see an example lets look at converting the fraction 1/6 to a decimal. We set up the problem as above and for the first digit in our result we get a 1. After that we see that we get repeating result of 6. Each time we perform a step we repeat what we did the previous step. In this case, there is no way to add enough 0's to get an answer without a remainder.
In this case 1/6 cannot be represented by a decimal that we can write down as there are an infinite number of 6's. Once we recognize this we can write our answer as below:
Here we write the 6 with a line over it indicating that the 6 repeats for ever.