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

Question

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.