If you're repeating decimal has a repeating cycle of three digits it will convert to a rational number with what denominator
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
To convert a repeating decimal with a repeating cycle of three digits to a rational number, you need to determine the number of digits in the repeating cycle and then find the denominator based on that information.
In this case, since the repeating cycle has three digits, the denominator will be 999.
To see why the denominator is 999, let's take an example:
Let's say our repeating decimal is 0.abcabcabc...
To convert it into a fraction, we'll multiply both sides by a suitable power of 10 to remove the decimal point:
10 * 0.abcabcabc... = abc.abcabcabc...
Now, subtracting the original equation from the modified equation gives:
(10 * 0.abcabcabc...) - (0.abcabcabc...) = (abc.abcabcabc...) - (0.abcabcabc...)
Simplifying the equation:
10 * 0.abcabcabc... - 0.abcabcabc... = abc.abcabcabc... - 0.abcabcabc...
9 * 0.abcabcabc... = abc
Dividing both sides by 9:
0.abcabcabc... = abc/9
Therefore, our repeating decimal is equivalent to the fraction abc/9. Since the repeating cycle has three digits, the numerator will be represented by abc, and the denominator will be 9.