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Here are some tips for Repeating Decimals, which aligns with New York state standards:

Repeating Decimals


A repeating decimal is a number where the decimal part repeats the same digit sequence over and over again.
Here are examples of repeating decimals.
0.7777777... = 0.7
1.85858585858585... = 1.85
253.207207207207207... = 253.207
3.610888888888... = 3.6108

The line drawn over the number(s) shows the repeating part of the decimal.

To convert repeating decimals to fractions, we want to multiply the decimal by a power of 10 such that subtracting the original number from it will cancel the repeating decimal part.
See the examples below to see how to convert repeating decimals to fractions.


Example 1: With only repeating numbers in the decimal

Convert the following decimals into simplified mixed fractions. If the whole number value of the fraction is 0, just leave it blank.
3.74

To make the problem easier, take out the whole number and just work on the decimal. In this problem, we will work on converting 0.74 into a fraction.

Let's assign x = 0.74747474...
The repeating part is 2 digits long so we multiply x by 102 or 100.
So now we have 100x = 74.74747474...

100x = 74.74747474...
-   x = 0.74747474...

99x = 74 
x = 74
99
 
For the answer, we bring back the whole number.
3.74 =


Example 2: With non-repeating numbers in the decimal

Convert the following decimals into simplified mixed fractions. If the whole number value of the fraction is 0, just leave it blank.
2.0146

To make the problem easier, take out the whole number and just work on the decimal. In this problem, we will work on converting 0.0146 into a fraction.

Let's assign x = 0.0146464646...
The repeating part (46) is 2 digits long and the non-repeating part (01) is 2 digits long so we multiply x by 10(2+2) or 10000.
So now we have 10000x = 146.46464646...

Since there is a non-repeating part in the decimal that is 2 digits long, we multiply x by 102 or 100.
Now we have 100x = 1.4646464646....

10000x = 146.46464646...
-   100x = 1.46464646...

9900x = 145 
x = 145
9900
 
  = 29  
1980
 
For the answer, we bring back the whole number.
2.0146 =

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