How to turn repeating decimals into fractions?

Answer 1

Notice the pattern. One digit repeated is always that number over 9. Two digits repeated is always those two digits over 99 and so on.

0.7777 repeated = 7/9

0.5454 repeated = 54/99

0.125 repeated = 125/999

Answer 2

Here is an example. Convert 0.848484... into a fraction.Call this number "x". Write two equations:

100x = 84.848484...

x = 0.848484...

(Note: the factor 100 is used because 2 digits are repeated. For 1 digit, use the factor 10; for 2 digits use the number 100; for 3 digits use the number 1000; etc.)

Subtract the second equation from the first.

99x = 84

If you solve for "x", you get your original number (which we called "x") as a fraction. In this case, x = 84/99. Simplify the final result, if possible.

Answer 3

Until you become expert at this I suggest you do this in two stages. Suppose there are c digits after the decimal place where the digits are non-repeating, and then you get a string of d digits which repeat. Then the numerator is the original string, without the decimal point, and including one lot of the repeated digits minus the original string with none of the repeating digits. The denominator is 10^c*(10^d - 1), which is a string of d 9s followed by c 0s.
For example 123.26159159… There are 2 digits, "26", after the decimal point before the repeats kick in so c = 2, and the repeating string "159" is 3 digits long do d = 3. So the numerator is 12326159 – 123216 = 12313833 and the denominator is three 9s followed by two 0s = 99900 Therefore the fraction is 12313833/99900. You should then check to see if the fraction can be simplified.