Question

if a repeating decimal has a repeating cycle of three digits it will convert to a rational number with what denominator would that answer be 99 1000 or 999 or 100

Answer 1

A repeating decimal with a repeating cycle of three digits can be converted into a rational number with a denominator of 999.

For example, the repeating decimal \(0.\overline{abc}\) (where \(abc\) represents a three-digit repeating sequence) can be expressed as:

\[
x = 0.abcabcabc\ldots
\]

To remove the repeating part, multiply \(x\) by 1000 (since the cycle is three digits):

\[
1000x = abc.abcabcabc\ldots
\]

Now, subtract the original \(x\) from this equation:

\[
1000x - x = abc
\]

This simplifies to:

\[
999x = abc
\]

So,

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

Therefore, the answer is **999**.