Changing Repeating Decimals to Fractions (II)
When Only Part of the Decimal Repeats

Here is an example of a repeating decimal where only part of the decimal repeats:
0.12444...
In this example, the '12' does not repeat. The '4' is the repeating part.

There is one digit that repeats, and two digits that don't.

Here's one more example:
0.2545454...
Notice with this one that the '54' repeats. The digit '2' does not repeat.

There are two digits that repeat, and there is one digit that doesn't.


Now let's look at how to change these partially repeating decimals into fractions.

The method requires the repeating decimal be in the form with the dot over or bar over the repeating digit(s).

Here's another partially repeating decimal:


Notice that it contains one repeating digit (2), and two non-repeating digits (34).

Here's how to make the fraction, using the decimal with the dot over the repeating part:

The Top:
Use the entire decimal, but subtract the non-repeating part.
The Bottom:
Use 9's for the repeating digits, and 0's for the non-repeating digits.
In this example, we need one 9 and two zeros.

This gives a final answer of 308 over 900, which can be reduced:



Let's look at another one.

In this example, there are two repeating digits, and one non-repeating digit.

The Top:
Take the entire decimal and subtract the non-repeating part ...
The Bottom:
Use two 9's (two repeating digits) and one 0 (one non-repeating digit) ...


              Simplify and reduce the result.




Here's another one:

Notice that it contains one repeating digit, and two non-repeating digits.

The Top:
Take the entire decimal and subtract the non-repeating part ...
The Bottom:
Use one 9 (one repeating digit) and two 0's (two non-repeating digits) ...


              Simplify, and reduce the result, if necessary.




Here's a final example, just to make sure you've got it.


This one contains two repeating digits, and two non-repeating digits.

The Top:
Take the entire decimal and subtract the non-repeating part ...
The Bottom:
Use two 9's (two repeating digits) and two 0's (two non-repeating digits) ...


              Simplify, and reduce the result, if necessary.
 






Now look at this additional example; see if you can identify the steps that were carried out.



Incidentally, you can always check your calculation by dividing your final fraction on a calculator...



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