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

Question

100
1,000
99
999

Answer 1

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

For example, if you have a repeating decimal like \(0.abcabcabc...\), it can be represented as:

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

where \(abc\) is the three-digit number formed by the repeating digits. This is because the decimal repeats every three digits, leading to a factor of \(999\) in the denominator. Therefore, the correct answer is 999.