How can you write repeating decimals as a fraction?

Answer 1

Here is an example, with the fraction 0.4123123123... Note that the length of the period is 3 (three decimals repeat over and over). Call your fraction "x":x = 0.4123123123... (equation 1)

Multiply x by 1000:

1000x = 412.3123123... (equation 2)

Note: 1000 is obtained as 10 to the power (length of period), in this case, 10 to the power 3. Just write a 1, followed by "period" zeros.

Subtract equation 1 from equation 2. This will give you an equation with whole numbers. If you solve that for "x", you get "x" as a fraction.

Answer 2

Until you become expert at this I suggest you do this in two stages (using c and d separately). Suppose there are c digits after the decimal place where the digits are non-repeating, and then you get a repeating pattern of strings of d digits. Then the numerator is the old original string 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 so d = 3. So the numerator is 12326159 – 123216 = 12313833 and the denominator is 99900 Therefore the fraction is 12313833/99900.