If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1 point)

Question

999
99
100
1,000

Answer 1

If a repeating decimal has a repeating cycle of three digits, it can be expressed as a rational number with a denominator of 999.

For example, the repeating decimal \( 0.abcabcabc... \) can be written as:

\[ x = 0.abcabcabc... \] Multiplying both sides by 1000 (since there are three repeating digits) gives:

\[ 1000x = abc.abcabcabc... \] Now, subtracting the original equation from this will result in:

\[ 1000x - x = abc \] \[ 999x = abc \] Thus, we can solve for \( x \):

\[ x = \frac{abc}{999} \]

Therefore, the answer is 999.