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Converting Repeating Decimals to Common Fractions

Try to write each of the following decimals as a common fractions.

(a) 0.432

(b) 0.1 repeating = 0.111111.....

(C)0.01 REPEATING = 0.0111111....

(D) 0.01 REPEATING = 0.010101010....

(E) 0.123 REPEATING = 0.123123123.....

(F) 2.243 REPEATING = 2.243434343...

(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 ...

Solution Summary

Converting repeating decimals to common fractions are analyzed.

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