Question

If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
A. 999
B. 100
C. 99
D. 1,000

Answer 1

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

To understand why, consider a repeating decimal like \( 0.\overline{abc} \) where \( abc \) represents the three digits that repeat.

When you express this repeating decimal as a fraction, it can be done by setting:

\[
x = 0.\overline{abc}
\]

Multiplying both sides by \( 1000 \) (since there are three repeating digits), you have:

\[
1000x = abc.\overline{abc}
\]

Now, subtract the first equation from the second:

\[
1000x - x = abc.\overline{abc} - 0.\overline{abc}
\]

This simplifies to:

\[
999x = abc
\]

Thus, we have:

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

So, the denominator of the rational number representation of a repeating decimal with a repeating cycle of three digits is \( 999 \).

Therefore, the correct answer is **A. 999**.