Question

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