Question

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

99
99

999
999

1,000
1,000

100

Answer 1

If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of **999**.

This is because when you have a three-digit repeating decimal, like \(0.abcabcabc...\), you can express it as:

\[
x = 0.abcabcabc...
\]

Multiplying both sides by \(1000\) (which shifts the decimal point three places to the right):

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

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

\[
1000x - x = abc \implies 999x = abc
\]

Thus:

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

So, the answer is **999**.