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

Question

99
100
999
1,000

Answer 1

If a repeating decimal has a repeating cycle of three digits, it can be converted to a rational number with a denominator of \(999\).

For example, if you have a repeating decimal like \(0.abcabcabc...\), where "abc" is the repeating part, you can express it as a fraction as follows:

Let \(x = 0.abcabcabc...\)

Then, multiplying both sides by \(1000\) (since the repeating part has three digits) gives:

\[1000x = abc.abcabc...\]

Subtracting the original \(x\) from this equation, you get:

\[1000x - x = abc\]

Which simplifies to:

\[999x = abc\]

Thus:

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

So, the correct answer is \(999\).