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# How to Convert Repeaing Decimals to Common Fractions

1) Convert all the following into common fractions:
(a) 0.432
(b) 0.11111...
(c) 0.011111...
(d) 0.0101010...
(e) 0.123123123...
(f) 2.243243243...
(g) 0.939339333933339....

2. Can all decimals (terminating, repeating and the rest) be written as common fractions? Explain how you know, and describe a general procedure for converting, in the cases when it's possible.

#### Solution Preview

I have provided an example problem for each of the ones you submitted. Just plug in your own numbers and you're all set!

(a) 0.432
Example: Convert 0.45 to a fraction.
Step 1: Let x = 0.45.
Step 2: Count how many numbers there are after the decimal point. In this case, there are 2.
Step 3: Multiply both sides by 100, because 100 has 2 zeroes. We get 100x = 45.
Step 4: Solve for x. In this case x = 45/100. Reduce to get x = 9/20.
(b) 0.1 repeating = 0.111111.....
Rule: All single-digit repeating decimal number can be written as a fraction with 9 as the denominator and the repeating single-digit as the numerator.
0.111... = 1/9
0.222... = 2/9
0.333... = 3/9
0.444... = 4/9
0.555... = 5/9
0.666... = 6/9
0.777... = 7/9
0.888... = 8/9
(C) 0.01 REPEATING = 0.0111111....

Example: Convert 0.022222
Since there are 6 digits in 022222, the very last digit is the "1000000th" decimal place.
So we can just say that .022222 is the same as 022222/1000000.
We can reduce this fraction to lowest terms by dividing both the numerator and denominator by 2, since it's the GCD. This gives us 11111/500000.

(D) 0.01 REPEATING = 0.010101010....

Example: Convert 0.131313
Slide the decimal point in this number to the right 2 place(s) (the same number of digits in ...

#### Solution Summary

All types of examples given and general rules provided.

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