Converting Exact Decimals to Fractions
Exact or Non-Repeating Decimals
An exact decimal number is a decimal number that terminates, meaning it has a finite or exact amount of numbers to the right of the decimal point.
All of these are examples of exact decimals. Even though the last example did repeat once, it didn't repeat infinitely many times.
An exact decimal does not have any infinitely repeating patterns, such as
where the line over the numbers represents an infinitely repeating pattern of those numbers.
When we want to convert an exact decimal into a fraction , we must first form a fraction out of that decimal number.
We do this by making the decimal number itself the numerator and placing in the denominator.
Let's choose as our decimal number. Then
We get rid of the decimal point in the numerator by moving it to the right the number of places to where it is not needed anymore.
For , we would move the decimal point to the right two spaces, to the right of the
Next, we add the same exact number of to the right of the in the denominator as the number of places we moved the decimal point in the numerator.
So we add two to the right of the
This process forms an equivalent fraction to our original fraction that does not have the decimal point in the numerator
After we form the correct fraction to represent the initial decimal, we need to simplify it if possible
It helps to remember this rule:
We add the number of to the right of the in the denominator that equals the number of places we moved the decimal point to the right to get rid of it.
Convert into a fraction.
First, we divide by to form a fraction
Then move the decimal point right three places and add three to the right of the
Here we showed all of the intermediate steps that show the processes of moving the decimal point to the right and adding the same amount of simultaneously.
Convert to a fraction
Converting a Repeating Decimal to Fraction
What if we have a decimal that repeats? How do we find the answer to a problem like that?
Say we have . You may know that this repeating decimal's fractional value is just , but how do we actually get to this answer?
There is a Mathematical trick that we can employ where we just divide the numbers to the right of the decimal by the same amount of placed next to each other, like , , ...
We have one repeating to the right of the decimal, so we divided it by one .
Convert to a decimal
We have two numbers in a repeating pattern, so we divided by two to make
If there are one or more numbers to the left of the decimal, we still divide by the same amount of as there are to the right of the decimal, but we will need to subtract the number that appears to the left of the decimal from the number that is formed by combining the numbers to the right and to the left of the decimal point.
Convert to a decimal
The numerator was because was the number formed by combining all of the numbers and was the number to the left of the decimal point, so we had to subtract it from , which results in . Then we divided by one because one number was repeating.
Here is the rule for a decimal that repeats after an initial amount of non-repeating decimals
We subtract from because that is the total number of digits that don't repeat.
We divide by one because of the repeating pattern of and add a to the right for the one non-repeating number to the right of the decimal point .
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